Symmetry:

The Quarterly of theInternational Society for theInterdisciplinary Study of Symmetry(ISIS-Symmetry)

Editors:Gy6rgy Darvas and D~nes Nagy

Volume 4, Number 1, 1993

DLA fractal clusterof I C@ particles

INTERNATIONAL SOCIETY FOR THEINTERDISCIPLINARY STUDY OF SYMMETRY

(ISIS-SYMMETRY)

PrestdentD~nes Nagy, lnsmute of Apphed Physics. Umversity ofT~ukuba, Tsukuba Science Cny 305, Japan(on leave from Eotvos Lor~nd University, Budapest. Hungary)[Geometry and Crystallography, H~st0ry of Science andTechnology, L~ngulsttcs|

Honomry PresidentsKo~tantin V. Frolov (Moscow)

andYuval Ne’eman (TeI-Aviv)

Vice- Presiden tArthur L. Lo~b, Carpenter Center for the Visual Arts,Harvard Umvers~ty. Cambridge. MA 02138,U S.A. [Crystallography, Chemical Physics, Visual Ar~s,Choreography, Music]

andSergei V. Petukhov, lnsmut mashmovedeniya RAN(Mechanical Engineenng Research lnsmute, Russian,Academy of Sciences 101830 Moskva, ut. Grib<m_.dova 4, Russia(also Head of the Rusman Branch Office of the Society)|B~omechamcs, Blomc~, [nformatton Mechanics]

Gyorgy Darvas, Symmetrion - The Institutefor k,.d~anced Symmetry StudxesBudapest, PO. Box 4, H-1361 Hungary[Theoret~cal Physics, Phdosophy of Science]

John Hosack, Department of Matherrmtics and ComputingScience. Umvers~ty of the South Pacific, PO Box 1168, Suva, Fiji[Mathematical Analysis, Phdosophy]

Regtonal Chatrpersons I Representattve$:

AFRICAMozambtque: Paulus Gerdes, Inst~tutoSuperior Pedag6gieo, Ca~xa Postal 3276, Maputo,Mozambique[Geometry, Ethomathemat~cs, History of Science]

AMERICASBrazil" Ubiratan D’Ambrosio, Rua Peixoto Gomide 1772, ap. 83.BR-01409 S,~o Paulo, Brazil[Ethnomalhemat~cs]

Canada Roger V. Jean, D6partemetat de math~matiqueset mformauque, Untversit~ du Quebec ~ Pamouski,300 all6e des Ursulines, RimouskJ, Qu6bec, Canada G5L 3AI[Btomathematics]

US A " William S. Huff, Depar~nent of Architecture,State University of New York at Buffalo, Buffalo,NY 14214. U.S A.[Arch~tecture, Design]

Nicholas Toth, Department of Anthropology,Indtana University, Rawles Hall 108, Bloomington,IN 47405, U S.A.[Prehistoric Archaeology, Anthropology]

ASIAChina. PR." Da-Fu Ding, Shangha~ Insmute of B~ochenustry,Academia Simca, 320 Yue-Yang Road,Shangha~ 200031, PR. ChinaITheoreocal Biology]

Le-Xiao Yu, Department of Fine Arts, NanjmgNormal University. Nanjmg 210024, P R China[Fine Art. Folk An, Calligraphy]

ladta. Ktrti Trivedi, Industrial Design Centre. IndianInsmute of Technology, Powa~, Bombay 400076, Intha[Des~gn. lnd~an Artl

Israel. Haman Bruen, School of Education,Umversity of Hafts, Mount Car~el, Hafts 31999, Israel[EducaUon]

Jo~ Rosen, School of Physics and Astronomy,Tel-Aviv Umversfly. Ramat-Avw, Tel-Aviv 69978, Israel[Theo~ehcal Physics]

Japan: Yasushi Kaji]~awa, Synergettcs Institute,206 Nakammur’ahara, Oda’*’ara 256, Japan[Design. Geometry]

Koichiro Matsuno, Department of BtoEngmeering,Nagaoka University of Technology, Nagaoka 940-21, Japan[Theoretical Physics, Biophysics]

AUSTRAUA AND OCEANIA

Austraha. Donald Herbison-Evans, BasserDepartment of Computer Sc~ence, Umversily of Sydney,Madsen Budding F09, Sydney, N.SW. 2006, Australia[Computing, Dance]

~]i" Jan Tent, Department of Literature and Language,Universtty of the South Pacific,PO. Box 1168, Suva, Fui [Lmgms~acs]

New Zealand" Michael C. Corb~Ilis, Department of Psychology,Umversity of Auckland, Private Bag, Auckland 1, New Zealand[Psychology]

Tonga: ’llaisa Futa-i-Ha’angana Helu, Dtrector,’Atemsl (Athens) lnsutute and University,P.O Box 90, Nuku’alofa, Kingdom of Tonga[Phtlosophy, Polynesian Culture]

EUROPEBenelu.~: Pieter Huybers, Faculte~t der Civiele Techniek.Technische Univemne~t Delft(Ctvil Engineering Faculty, Delft University of Technology),Stevinweg 1, NL-2628 CN Delft, The Netherlands[Geometry of Structures, Bu)ld]ng Technology]

Bulgaria: Ruslart 1. Kos~v, Geologicheski Instttut BAN(Geological Insutute. Bulgarian Academy of Sciences),ul Akzd. G. Bonchev 24, BG-1113 Sofia, Bulgaria[Geology, Mineralogy]

Czech Republic: ~-bjt&:h I~pskJ~, Fyzik~lni tlstav ~?AV(lnsmute of Physics, Czech Academy of Sciences), CS-Ig0 40Praha 8 (Prague), Na Slovance 2 (POB 24),Czech Repubhc [Sohd State Phystcsl

France: Pierre Sz~kely, 3bis, ~mpasse Vilhers de l’lsle Adam,F-75020 Par~s. France [Sculpture]

/ICULTURE & SCIENCE I/The Quarterly of the International Society for the

Interdisciplinary Study of Symmetry(ISIS-Symmetry)

Editor~:GyOrgy Darvas and D~nes Nagy

Volume4, Number l, 1995, 1-112

SPECIAL ISSUE:SYMMETRY AND TOPOLOGY IN EVOLUTION, 1

Guest Editors:SzanisT.16 B~rc-zi and B~la Luk~cs

CONTENTSEDITORIAL, B. LuI~cs, S. Bdrczi, L Moln,~r, and G. Pa~l

SYMMETRY." SCIENCE & CULTURE¯ On the mathematics of symmetry breakings, B61a Lul~cs 5¯ Topology of the Universe, GyOrgy Padl 13¯ The evolution of cosmic symmetries, l~la Lul~cs 19¯ Chirality in the elementary interactions, Ferenc Glack 37¯ Symmetry changes by cellular automata in transformations of

dosed double-threads and cellular tubes with M~ibius-band,torus, tube-knot, and Klein-bottle topologies, Szaniszl6 B~rczi 49

¯ Pentamond: A new crystalline modification of carbon, Gdbor G~ay 69¯ Chirality of organic molecules, Ferenc Glack 83¯ On geometric symmetries and topologies of forms, B~la Lukdcs 93

SFS: SYMMETRIC FORUM OF THE SOCIETY 105

Sr~M~T~r: CVI~aNDSCt~VCE is edited by the Board of the International Societyfor the Interdisciplinary Study of Symmetry (ISIS-Symmetry) and published quar-terly by the International Symmetry Foundation. The views expressed are those ofindividual authors, and not necessarily shared by the Society or the Editors.

Any correspondence should be addressed to the Editors:

GyOrgy DarvasSymmetrion - The Institute for Advanced Symmetry StudiesP.O. Box 4, Budapest, H-1361 HungaryPhone: 36-1-131-8326 Fax: 36-1-131-3161E-mail: h492dar@ella.hu

D~nes NagyInstitute of Applied PhysicsUniversity of TsukubaTsukuba Science City, Ibaraki-ken 305JapanPhone: 81-298-53-6786 Fax: 81-298-53-5205E-mail: nagy@bk.tsukuba.ac.jpThe section SFS: Symmetric Forum of the Society has an E-Journal Supplement.Annual membershipfee of the Society: Benefactors, US$780.00;Ordinary Members, US$78.00 (including the subscription to the quarterly),US$30.00 (without subscription);Student Members, US$63.00 or US$15.00, respectively;,Institutional Members, please contact the Executive Secretary.Annual subscription rate for non-members: US$96.00.Make checks payable to ISIS-Symmetry and mail to Gy0rgy Darvas, ExecutiveSecretary, or transfer to the following account number: ISIS-Symmeuy,International Symmetry Foundation, 401-0004-827-99 (US$) or407-0004-827-99 (DM), Hungarian Foreign Trade Bank, Budapest, Szt. Istv~tn t6r11, H-1821 Hungary (Telex: Hungary 22-6941 extr-h; Swift MKKB HU HB).

@ ISIS- .Sym.. metry. No of this be without writtenpart publicationmay reproducedpermmsion from the Society.

ISSN 0865-4824Cover layout: Gunter Schmitz

Image on the front cover. Paul MeakinDiffusion Limited Aggregation (DLA) Fractal Cluster of I 06 particles

Images on the back cover. TaraM Vicr, ekDet~ Model for DLA; Fractal Snowflake

Ambigram on the back cover. Douglas R. HofstadterLogo on the title page: Kirti Trivedi and Manisha Lele

Fot6k6sz anyagr61 a nyomdai kivitelez6st v6gezte:AKAPRINT Kft. E v.: Dr. H6czey I_Aszl6n6

9421657

3~mme~y: Culture and ScienceVo~ 4, No. 1, 1995, 5

EDITORIAL

These two volumesI (special issues Symmetry and topology in e~olution, 1-2) containa selection from the lectures of the symposium held on 29-30th May 1991 under theaegis of the Natural Evolution (Gconomy) Scientific Committee of the HungarianAcademy of Sciences, by the Evolution of Matter Subcommittee of the said Com-mittee. The committee is an interdisciplinary body for the general overview of thenatural sciences to develop new long-range connections among different disci-plines; the Subcommittee tries to follow (or trace back) the steps of self-organisa-tion of matter from the Beginning to (at least) the present.

Unfortunately no constructive method is known to view Natural Science as a unit;the global picture is deduced from the separate disciplines. Still, an overview maygive answers to old questions. For example, on all levels of organisation the actualpath of evolution may be regarded either as accidental or as predetermined. Suchquestions have not only professional connotations but philosophical, religions, etc.,as well. We do not want to leave the pounds of strict natural science and try toshow the most coherent available picture as a starting point of any later deduction.By looking at the neighbouring levels sometimes the causes or consequences can berecognised.As an example: we are chiral (can tell our right hand from the leR one in the vastmajority of cases); the neutrino can always do so, and the electron sometimes. Isthere any connection among these facts? The premature answer can be avoided byrecognising that the chimpanzee is not chiral.The lecturers of the symposium are experts of their own fields; the OrganisingCommittee took the responsibility of selecting the actual persons, but the texts arethe lecturers’ sovereign works. We hope that the written material will promotesome further interdisciplinary work.

The Organising CommitteeB. Luk~cs (president), S. B~rczi, I. Moln~Ir, G. Pa~ll

Evolution of Matter Subcommitteeof the Natural Evolution Scientific Committee

of the Hungarian Academy of Sciences

1 This work was partly supported by OTKA 14104 project.

3~anme~: Culno~ and ScienceVoi. 4, No. 1, 199.3, 5-11

ON THE MATHEMATICS OFSYMMETRY BREAKINGS

Physicist, (b. Budapest, Hungary, 1947).Address. Central Re~.arch Institute for Physics, Budapest 114,P.O.B. 49, H-1525 Hungary, E-mail: lukacs@rmk530.rmki.kiki.hu.I~ieki~ of Interest" Cosmology; General Relativity;, Heavy IonPhysics; SI~zh Acoustics; Economics.

Abstract: Deviation~ from symmetries are discusse~ Wedistinguish two cases: (i) symmetric laws + asymmetricexternal disturbances lead to approximately symmetricactual states; (ii) symmetric laws without otherinfluences lead to states asymmetric in an extentdetermined by the laws themselves via spontaneoussymmetry breaking.

INTRODUCTION

This paper is devoted to the mathematics of the different kinds of symmetrybreaking, considering the aims of the present issue. Our goals are limited: we donot want to go into fine details of definitions, but try to demonstrate that a symme-try may be broken for different reasons and to show how the extent in which it isbroken can be quantified.

1. ON SYMMETRIESInstead of a complete definition let us use the following one:

A SYMMETRY IS A TRANSFORMATION LEAVING SOMETHINGINVARIANT.

Example: conaider a sphere. By rotating it remains the same.

For a precise definition of geometrical symmetries see another paper in this issue(Lukaics, 1993).

2. ON BREAKINGIf there is no symmetry at all, nobody speaks about broken symmetry. We speakabout a broken symmetry if

1) the situation is almost symmetric;2) the situation ought to be symmetric still it is not.

The symmetry can be broken1) by external asymmetric influences;2) spontaneously.

3. ON THE EXTENT OF AN APPROXIMATE SYMMETRYThe measure is slightly subjective. However, if there is approximate symmetry, thenthe ideal symmetric situation can be visualized, and the actual one can be comparedwith it.

Example: the ellipsis. Its equation isr = r(~) = R/(1 + e cos ~) (1)

where e is the excentricity. For ellipses O<_e<_l; for e< <1 the ellipsis is almost a circle.So the ~ctent of the deviation from the symmetric circle (rotational symmetry) is mea-sured by the dimensionless ~If the symmetry breaking is caused by an external influence then generally it is pro-portional to the influence (and then one can compare the acting forces). If thebreaking is spontaneous, this is not necessary at all.

4. AN EXAMPLE FOR NON-SPONTANEOUS SYMMETRYBREAKING: OBLATENESS

A rotating fluid body is often an oblate spheroid (although other solutions of thehydrostatic equations exist as well (Chandrasekhar, 1969)). For small angularmomentum fl one can get simple approximate formulae as follow:The rotation results in an inertial ’force’. Including its ’potential’ one gets

V(r, ~o) = -GM/r + ½l12r2sin2 ~p (2)

Then the equipotential surfaces (of which one is the surface of the fluid (Thorne,1971)) are given by roots of a cubic equation. Again for small 1~ they can beapproximated in a transparent way. Consider a value Vo<0. Then ro = ro(~,):

ro = (-GM/Vo){1 + ½[(GMgI)2/(Vo3)] sin2 ~a} (3)

MATHEMATICS OF SYMMETRY BRF.AKINGS

Then the extent of asymmetry or symmetry breaking is the ratio of the two terms inthe bracket {} for, say, ~ = ~r/’2. Substituting terrestrial surface data, the ratio is0.0017, rather moderate.

$. ON SPONTANEOUS SYMMETRY BREAKING

Again instead of a precise mathematical definition here we adopt the followingtransparent one:

THE SYMMETRY BREAKING IS SPONTANEOUS IF THE LAWS ARESYMMETRIC STILL THE MOST NATURAL CONFIGURATION (SAY

GROUND STATE) FOR THEM IS ASYMMETRIC.The simplest example is mechanical equilibrium in a potential V(x) = V(-x). Thepotential is even, i.e., mirror-symmetric: this will be the symmetry. The most naturalconfiguration is now the equilibrium one; if that is not unique, then the ground stateamong thent Here come the details.The condition for equilibrium is

x = Xo; (dV/d~)o = 0

For simplicity we restrict ourselves to polynomials, such that

v(0) = 0v(_+®) = +®v(-x) = V(x)

We are going up with the degree of the polynomial.

The only possibility compatible with (5) is

Vo(X) = 0

(4)

(5)

(6)

Then (4) holds everywhere; the ground state is degenerate. This case is trivial.

P1Conditions (5) cannot be satisfied: this class is empty.

From (5) we get

V2(x) = ax2, a >0 (7)

B. LUK4C$

~t I I I I I I I ]-0.0 1.0

Figure I: Quadratic potential. Stable equilibrium at the center.

This potential, for a= 1, is shown as Figure 1. The only solution of (4) isx=O (heavydot).

SYMMETRIC EQUILIBRIUM STATE FOR SYMMETRIC LAWS.

Conditions (5) cannot be satisfied: this class is empty.

From (5) we get

V4(x) ffi ax~ + bx4, b>0 (8)This potential, for some values of a and for b--l, is shown on Figure 2. The equilib-rium condition (4) 1cads to

(9)X(a ~ " 2bx2) =0

Now, xoo=O is always a root, but for a<O two other ones exist as well:

Xo± = _+(-a/~)v, (10)Then, as shown on Figure 2, for a >_0 Xoo=0 is the only equilibrium point, and is anenergy minimum (heavy dot). However, for a<0 the situation is qualit~tive~,different.

MATHEMATICS OF $YMMETRY BREAI(dNGS

~ 2: Quartic potentials; a ffi + 1 (dot), 0 (short dash), --0.5 (long dash), -- 1 (solid). Heavy dots atstable minima, circles at unstable maxgma.

As seen, Xoo is an equilibrium point, but energy maximum (circle). A local maxi-mum is an unstable equilibrium: for small perturbations the state goes away andreturns not. On the other hand, the neWXo_’S are stable minima (heav~ dots). Aftersome transient evolution the mass point settles down at one OfXo±.

ASYMMETRIC GROUND STATE(S) FOR SYMMETRIC LAWS.

Now, notice that the actual final state will lack the mirror-symmetry. In the sametime the set of ground states is still symmetric: there is one on the right, one on theleft. Similarly, both are equally probable (50-50 per cent) as final states. Thi~ sym-metry is a consequence of the symmetry of the potential. But the symmetric poten-tial V4 leads to asymmetric final states ff a < O.

This phenomenon is called spontaneou~ symmetry breaking, since in a dynamicalevolution from symmetric laws and initial conditions the system goes into anasymmetric actual state without any external asymmetric influence.

The very simple example shows that no very specific or complicated mechanismsare needed for this spontaneous symmetry breaking. Still, one generally expectssymmetry from symmetric laws and has the tendency to stop at the symmetric solu-tions. So, remember the possibility of spontaneous symmetry breaking.

The necessary condition for such a breaking is the appearance of a ’double-bot-tomed potential’. This is known from various parts of natural sciences. Some caseswill be seen in the following articles. In addition, we mention here three examplesfrom physics.

10 B. LUI~CS

1) Higgs mechanisms in particle physics (Langacker, 1981). Some hypothetical scalarbosons may have quartic self-potentials V(~), where ¯ stands for the wave func-tion, so roughly I ~ I 2 is proportional to Higgs particle number or probability. (Ifthe potential is not polynomial, or higher than quartic, then the theory is notrenormalisable, the infinities from divergences cannot be removed, so the theoryhas no prediction at all, i.e., is not a theory.) So Higgses spontaneously appear in a’number’ determined by the parameters of V(~), except if the average energy ishigher than V(O)-Vmin. (I.e., above the central peak of Fig. 2.) If the Higgses arecoupled to other particles, the interaction mimics other particle masses - I ~ I 2The mechanism was invented for renormalisable quantum field theories of vectorbosom, where particle masses are detected but would prevent renormalisation. Insuch systems phase transition is expected when the specific energy passes through avalue in the order of the central peak: at high energy (temperature) the actual stateis symmetric, and going to low temperature this symmetry spontaneously breaksdown ("the state rolls down from the hill").2) Ferromagnetism (Kittel, 1961). Sometimes the interaction between electronspins or orbital momenta is attractive and strong enough to prefer identical direc-tions for neighbouring atoms. Then the ground state is asymmetric: magneticmomenta pointing into a definite direction in average, although the laws areisotropic in the 3D-space. This direction is, however, random, because the isotropiclaws lead to isotropy in probability. If the environment is not isotropic (e.g., theweak geomagnetic field), then sometimes this anisotropy singles out the direction(but the strength of a ferromagnet is prescribed by its physical parameters and isdefinitely not proportional to geomagnetism).3) Piezoelectricity of BaTiO3 (Kittel, 1961; von Hippel, 1950). BaTiO3 is a cubiccrystal. The expected location of the Ti ion is at the center, the actual one is excen-tric by a fixed shift. In a very simplified way one may say that the Ti ion is fairly bigcompared to O’s. The O’s are located at the centers of the lateral faces of the cube,so restricting the room for Ti just in the center. Then it is ’more convenient’ (say,energetically) for the Ti ion to be excentric, which means a potential of qualitativeform of Figure 2. For a more correct and less transparent explanation see (Kittel,1961); in an approximation a quartic potential is obtained. The configuration has ahysteresis loop, but above a threshold value the external potential makes the Tijump into an oppositely excentric position, with reversed electric dipole moment.This can be done with external pressure as well, and this is the reason for piezo-electricity.

6. CONCLUSIONThe most important message is:REMEMBER: A SYMMETRIC POTENTIAL MAY HAVE MINIMUM IN THE

CENTER BUT MAY HAVE MAXIMUM AS WELL SYMMETRIC LAWSOFTEN LEAD TO ASYMMETRIC GROUND STATES WITHOUT ANY

FURTHER REASON.

Life is not always simple.

MATHEMATICS OF SYMMETRY BREAKING S 11

REFERENCES

Chandrnsekhar, S. (1969) Ellipsoidal Figures of Equilibrium, New Haven: Yale University Press.yon Hipp¢l, A. (1950) Ferroelectricity, domain structure and phase transitions of barium titanate,

Review of Modem Physics, 22, 221.Kittel, Ch. (1961) Introduction to Solid State Physics, New York: J. Wiley and Sons.Langacker, P. (1981) Grand unified theories and the proton decay, Physics Report$, 72C, 185Luk~c*, B. (1993) On geometric symmetries and topologies of forms, Symmetry: Culture and Science, 4,

1, 93-103.Thorne, K. S. (1971) Relativistic stars, black holes and gravitational wave~, In: Proceedings of the 47~h

International School "Enrico Fermi", p. 237.

~mmeoy: Culotte and SciencePot 4, No. 1,199~, 1~-18

TOPOLOGY OF THE UNIVERSEGy6rgy Pafil

Astrophysicist, (b. Budapest, Hungary, 1934, ~’1992).Afft/iation: Konkoly Observatory o[ the Hungarian Academy of Sci-ence, Bndape~t 114, P.O. Box 67, H-1525 Hungary.Fielda og interest: Coamolo~, gala~ cluster, qua~u~, apparent

Publications:. Red ~ and quasara, Science ,~oumal, 1970, June,101; The global structut~ o[ the universe and the distribution o[quasi-stellar object, Acta Physica Hungatica, 1971 (30), 51;Cosmogenesis, (with LukAcs, B.) In: LukAca, B. ~ a/., ed~.,Evo/ution: From ~ to Biosenesis, p. 7, 1990, KFKI-1990-50; Inflation and compactification from galaxy redshifts?(with Horvath, I. and I.,ul~ics, B.), Astrophysics and Space Science,1992 (191), 107.

Abstract~ The unexpected possibility of a multiplyconnected character of the space of the Universe isdemonstrated for the non~pecialists by using simpleeramples. Indications for the realization of this possi-bility in the actual Universe are briefly discussed andmentione~

its cosmological implications

FROM TOPOLOGY TO CRYSTAL STRUCTUREWhen speaking about the ’topology of the Universe’ one may be inclined to thinkof the connectivity properties of material structures within the Universe, which -according to our present knowledge - may be ’filamentary’, ’bubbly’, ’frothy’,’sponge like’ etc. (Melott, 1990). There is, however, a possibility for a deeper kind ofa ’topology of the Universe’, when the space itself (even a possible emptinessalone) does have its own nontrivial connectivity properties. E.g., going alwaysstraight ahead in the space one may find himself at his starting point after havingcovered some characteristic distance depending on the direction of the journey (cf.Ellis, 1971; Wolf, 1967). Recent astronomical observations - and also quantum-cosmological considerations - have made this possibility worth of special attention(Pa~l, 1971; Pa~l et ai., 1991; Fang and Sato, 1984; Fang and Mo, 1987).

In order to make the somewhat mystic or incredible spatial recurrences under-standable for the specialists of other fields of sciences it is customary to considerfirst the case of two-dimensional spaces (surfaces) inhabited by flat beings

(’observers’ or ’travellers’) and start with the simple example of an infinite cylinder.Here there is clearly only one single direction in which the fiat traveller may returnto his original place, if the route chosen by him is the straightest possible one whilecrawling on the surface (without superfluous turns aside) - but such a return isimpossible in all other directions. In the usual Euclidean plane there is no possibil-ity for a linear return at all. The transition from one of these cases to the otherseems fairly easy. One may always roll up a fiat (uncurved) plane into a cylinder somaking a ’multiply connected’ surface out of a ’simply connected’ one. - Obviouslyin the simply connected plane there is only one straight line connecting any twogiven points, while in the multiply connected cylinder there may be infinitely manyof them.This simple example is relatively easy to reformulate in such a mathematical formas to permit generalization to any dimension and space curvature. Imagine first apicture just painted by fresh paint on the surface of a cylinder, which is then sentrolling quickly on a plane sheet of paper before the lapse of time needed for thepaint to dry. Clearly repeated images of the picture will appear painted on thepaper in excactly periodical distances. One may say in a somewhat more abstractlanguage that an infinite set of equidistant pictures on a plane precisely ’represents’a single picture on a cylinder, or that by identifying all the points corresponding toeach other according to a ’group of parallel translations’ in a plane (n times a givenlength d) an equivalent representation of a cylinder is obtained from a two-dimen-sional Euclidean space (plane). The plane is called the simply connected ’coveringspace’ and the translations generate a ’compactified’ multiply connected space, thecylinder. The short range ’local’ properties of these spaces coincide, but the longrange ’global’ ones (e.g., long range returns) differ. The plane and the cylinder aresaid to represent topologically different ’space forms’ of the fiat (Euclidean) space.In a more general language one may say that any ’space form’ of constant curvaturecan be derived from a simply connected spherical, flat or hyperbolic universal cov-ering space (of positive, zero or negative curvature) by introducing into the latter adiscontinuous group of fixed point free isometric transformations and identifyingits points which correspond to each other under the transformations of the group.This identification leads to the multiply connected ’quotient space’. A set ofinfinitely many points of the original space (which are connected by thetransformation group) plays the role of a ’single point’ in the new space.

According to this scheme there are five topologically different space forms of thetwo-dimensional Euclidean space. The first is the trivialplane with no compactifi-cation (recurrences, identifications), the second is the (infinitely long) cylinderobtained by a single sequence of translations, i.e., identifications by equidistantsteps along a single direction, the third is the (finite) torus obtained by two alge-braically independent sequences of translations, i.e., identifications by steps alongtwo different directions, the forth is the (infintely long) M6bius band obtained byone sequence of translations combined with a reflection, while the fifth is the K/einbottle obtained by two sequences of translations, one of them combined with areflection. It is easy to see that if one tried to use two translations combined withtwo reflections, then a fixed point would emerge whose local properties would dif-fer from those of other points of the space. So a two-dimensional Euclidean spacemay have 3 space forms with infinite extent and 2 space forms without infiniteextent! - Note that this finiteness is essentially different from the much better

TOPOLOGY OF THE UNIVERSE 15

known finiteness of the surface of the simple sphere (i.e., two-dimensional space ofpositive curvature). The latter is isotropic, i.e., the distance of returns is the same inall directions, while the former is necessarily anisotropic.In the same spirit but somewhat more generally one can construct a three-dimen-sional ’cylinder’ by introducing one sequence of translations in the three-dimen-sional Euclidean space. In this case not infintely long two-dimensional strips, butinfintely long and wide three-dimensional layers would ’repeate themselves’ (wouldbe identified) in the original three-dimensional covering space. Without enumer-ating all the 18 possibilities for the three-dimensional Euclidean~space forms weonly mention here the three-dimensional torus (designated by T~) generated bythree independent sequences of translations (without reflections or rotations)which is the simplest case when the three-dimensional Euclidean space can be fully’compact’ (finite, closed) and can have finite total volume.For visualization a compactified space may also be thought of as a system ofconguent cells in a crystal lattice with some definite rule of alignment of the ceils.In general the cells may have different forms and the alignment may happenaccording to different combined translations, rotations and reflections as well. Onlyspecific transformations lead to meaningful compactifications. This also applies tospaces of nonzero curvature, although the meaning of ’translation’ or ’rotation’ issomewhat more obscure in these cases. For details see Ellis (1971), Wolf (1967).Obviously the observable properties of a multiply connected universe are identicalwith those of the corresponding ’universal covering model universe’ populated byequal configurations in strictly congruent cells and all the known formulae ofobservational cosmology hold in this crystal-like covering. These models naturallyshow some kinds of periodicities according to spatial distance. This is how they canbe discovered astronomically. The question is then, whether there are indeed suchdiscoveries.

FROM CRYSTAL STRUCTURE TO WORLD MODELHints to actual periodicities in the space distribution of astronomical objects firstappered 20 years ago in connection with quasars (Padl, 1970; Padl, 1971; Fang andSato, 1984; Fang and Mo, 1987 and references, therein), but have failed to pass theproper statistical significance tests. Really slg~ificant periodicity has first beenfound in the space distribution of galaxie~ by Broadhurst et aL 0990). It turned outthat the typical distances between galaxy pair~ are distributed according to a non-random, periodic pattern .wh°se regularity depends on the. world model, more par-tlcularly on the forces acting on the expansion of the Llmverse (PaAl et al. 1992).

The so called autocorrelation coefficient of galaxies indicates the relative excess ofgalaxy pairs with a given spatial separation, This spatial distance of galaxy pairs is,however, not a directly observable quantity, it can only be calculated from the mea-sured redshifts of spectral lines in the light of the galaxies. This calculation in turn~elds different results depending on whether the universe is filled simply by theusual cosmic matter (’dust’), which - according to Einstein’s General Theory ofRelativity - is gravitationally attracting, or it is filled mostly by the modern quart-

tumtheoretical ’vacuum’ (a background sea of energy without particles ofidentifiable form), which is gravitationally repulsive (Patti and Luk:ies, 1990). Ineither extreme case of dust or vacuum dominance the calculated periodicity of thespace distribution of galaxies turns out to be relatively poor, see Figure I and Fig-ure 2 (although some 15 partially irregular periods may be distinguished even inthese cases). However for a vacuum/dust ratio equal to 2/1 one finds 17 fairly regu-lar periods (Fig. 3). The superiority of the latter model becomes even more promi-nent, ff one calculates the periods also by an independent method (i.e., by mini-mizing the squared deviations of the places of maxima of the galaxy distributionfunction from an equidistant set of distances) and indicates the multiples of thisperiod on the diagram of the autocorrelation function (equidistant vertical lines onthe Figures). The periods defined in these two different ways coincide only for themodel of Figure 3, while in Figures 1 and 2 the two kinds of periods differ sostrongly that the phase differences accumulate to a full period. Consequently thistuning of periods singles out the best cosmological model, unless one is willing toaccept that prominent periodicities appear just by mere chance without any deeperphysical reason.

Thus if one looks for a peril/city, which may be a signal for the topologically non-trivial character of the space of the Universe, then one finds a particular world modelfilled with twice as much vacuum as dust. This model also has further importantadvantages for cosmology.Astronomical observations seem to show that the attracting dust gives about 1/3part of the critical density needed to make the space of the Universe almost fiat(uncurved, Euclidean). Now we may have found the remaining 2/3 part in the formof vacuum and so the old ’flatness problem’ may have disappeared. In other words akind of ’missing mass’ seems to have been found. - On the other hand In thesemoderately vacuum dominated models the total expansion time measured from theBig Bang to the present epoch is essentially longer then in the dust models and so itis much easier to find time in them for the oldest stellar systems, i.e., a ’missingtime problem’ is also easier to solve in this case.

FROM WORLD MODEL TO "WORLD’S END"

However the new model may have terrifying consequences as well. The obtaineddensity of vacuum 10O00 times surpasses that of the thermal cosmic backgroundradiation. This implies that - according to the Stefan-Boltzmann law - ourvacuum is already 10 times supercooled and so may explode In any moment. Whenin the yen! early Universe a supercooled vacuum was destroyed by a phase transi-tion, then some bosons mediating interactions acquired rest energy and mass (outof the released vacuum energy). If the present supercooled vacuum gave mass tothe remaining massless bosons, then those bosons could be e.g., the photons medi-ating the electromagnetic interaction, implying that the range of electric forcewould fall from infinity to about a millimeter or so (Pa~l, Horv~th, and Luk~ics,1992). If gluons got masses, then the stability of nuclei and the nuclear fusion pro-cesses might change (e.g., in stars).

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Similar statements could, however, been found in much earlier literature as well."The sun will be darkened, and the moon will not give its light, ... and the heavenlypowers (range of strenghts!) be shaken" and later on "there will be a new heavenand new earth" (new phase after universal cosmic phase transition). So one has toadmit that the priority for such predictions sould be given to the following gentle-men: Matthew (24, 29), Mark (13, 24), Luke (21, 26) and John (Rev. 21, 1).Early in this century (and in the last one) the beginning or creation of the worldhad been regarded as absolutely unscientific, before relativistic cosmology made theBig Bang a fully natural and even n~ry notion. May it be that a similar changeof views will happen in our days in connection with the ’End of the World’ too?!

ACKNOWLEDGEMENTSThe author is greatful to St. Matthew, SL Mark, St. Luke and St. John for theirvaluable written information and to B. Lukdcs and L Horvdth for useful comments.

REFERENCES

Broadhurst, T. J., Ellis, R. S., Koo, D. C., and Szalay, A. S. (1990) Large scale distributions o~ the galax-ie~ at the galactic ~ Nmure, 343, 726.

Ellis, O. F. R. (1971) Topology and cosmology, Gen~alRelad~y and Gravitation, 2, 7.Fang, L Z. and Mo, H. J. (1987) Topology of the univerr, e, In: Hewit, A. eta/., eds., Proceeding~ of the

/AUSym~, No. 124, Dordrecht: Reidel, p. 461.Fang, L. Z. and Sato, H. (1984) Is the periodicity in the distribution of quasar redshifts evidence for the

universe being multiply connected? Chinese ~ andA~, 8, 148.Meiott, A. I. (1990) Evidence for the universe being multiply connected? Physic.v Repot, 193, 1.Pall, G. (1970) Red shifts and quasar, ScienccYoumal, June, 101.Pail, G. (1971) The global structure of the universe and the distribution of quasi-stellar objects, Acta

Physica Hungarica, 30, 51.Pail, G. and LukAc~, B. (1990) Cosmogenesis, In: Luk~c~, B. eta/., eds., Evo/ution: From Co~no~

to~, p. 7, KFKI-1990-50.Pail, G., Ho~/ith, I., and Luk~a, B. (1992) Inflation and compactification from galaxy redshifts? A.~ro-

~ and Space $cience, 191, 107.Wolf, J. A. (1967) Spaces ofCoctmug Cu~,ature, New Yo~k: McGraw-HilL

~,~ry: C~mr~ and ScienceVo~ ~, N~ 1,1993,19-36

THE EVOLUTION OF COSMIC SYMMETRIESB61a Lukfics

Phy~ichtt, (b. Budapest, Hungaw, 1947).Addrm~. Central Research Institute for Physics, Budapest 114,P.O.B. 49, H-1525 Hungazy, E-mail: lukaes@nnk530.rmki.klki.hu.¥ield~ ef Intere~: Cmmology; General Relativity;, Heavy IonPhysics; Speech Aco~tics; Economics.

Abstract: We discuss the changes of symmetries (bothspatial and internal) of the Universe from the beginningto the present age. For this one must clarify what is a’beginning’; what do initial conditions mean for the totaland unique Universe and if any then what may be thecorrect initial conditions. While one cannot settle downthis question with a finality, we discuss the problem indetail~ and try to give a probable answer. The mostprobable scenario contains many subsequentspontaneous symmetry breakings, sometimes with apartial compensation between deteriorating internalsymmetries and improving spatial ones.

1. WHAT IS COSMOLOGY?Cosmology is a pseudo-Greek term of the same structure as geology, meteorologyetc., so roughly it is the science of the Cosmos or Universe. Originally it wasdistinguished from cosmogony, a discipline describing the genesis of the Universe.However, in General Relativity the matter governs the geometry of the space-time,and, except for very special cases, the matter of the whole Universe cannot remainin equilibrium. Therefore any valid theoretical description of the Universe willautomatically contain its evolution as well, and then there is no need for a separatecosmogony.Now the question is if there is an independent science of cosmology at all. Theanswer is no ff

1) there is no such definite entity as Universe; or2) there are no specific laws of the Universe.

And for the present lecture: in the first case we cannot speak about the symmetriesof the Universe; in the second we can but it will have a limited physical relevance.

20 B. LUI~CS

To explain this statement let us see what may be the Universe. If the Universe issimply an incoherent sum of entities then it has no definite global structureincluding symmetries, and the local structure randomly changes from point topoint.. If the Universe has a global structure but all its laws are consequences ofknown local laws, then the actual symmetries and their evolution belong to thepurely descriptive part of science, and therefore one cannot learn too much from it,although the stout may still be interesting.

These questions are rather fundamental and we cannot be expected to be able tosolve them here. However a very brief recapitulation of earlier ideas is edifying. Notclaiming completeness,

1) the Universe of the ancient Greek science is a single finite unit, with itsown laws, and with spherical symmetry.

2) Newton’s Universe is more or less the infinite absolute space + thematter filling it; the empty absolute space would have an E(3) symmetry but thematter may or may not share this. This Universe, which is only a nomen collectivum,has no law of its own.

3) in General Relativity the Universe may or may not exist as a definiteobject. General Relativity can describe the Universe as an object; it may possesssome definite symmetry, for example. However no specific laws of the Universeappear in General Relativity.

4) the so called standard cosmological model contains some cosmologicalprinciples which may be consequences of some unknown cosmological laws.

Methodical and consequent thinking in a model may point to some problematicalpoints where the model is incomplete, self-contradictory or impossible. In thepresent case the result is as follows.

1) Aristotle’s cosmology was free of paradoxes (and this was the reason tolive for two millenia), although finally it has been proven incorrect.

2) Newton’s cosmology had paradoxes, some of which was known from thebeginning. E.g., gravity causes instability in an infinite Universe (Newton, 1756)known later as Seeliger’s paradox.

3) General Relativity removes the old paradoxes. E.g., the Universe may befinite or compact (Paal, 1993); even if it is infinite, gravitational instabilities are notnecessary because of the nonlinearity of gravitational laws; it may be of finite ageetc.

4) The standard model is of maximal spatial symmetry (for simplicity butalso suggested by observations). Applying the known laws the Universe possessessome constants of integration. Now, these constants can be read off fromobservations, and changing them even moderately the present Universe would bequalitatively different from the actual one (Carr and Rees, 1979). So the actualUniverse seems ’improbable’. Of course, this term does not have any meaning here,because there is only a single Universe, so one cannot study a set of them withdifferent initial conditions. However, if one eliminates the probability problem by

THE EVOLUTION OF COSMIC SYMMETRIES

the unicity of the Universe, then he must tacitly accept that something is behind theonly values of the constants, so they are prescribed somehow. Then we may expectunknown laws of the total Universe. We stop here, not discussing the question ifsome of the laws may be given by the unknown unification of general relativity andquantum field theory.

2. PART vs. WHOLEIn any case the present Universe consists of a lot of parts as galaxy clusters,galaxies, stars and particles. It is obvious to classify entities as primary (noconstituents), secondary (with primary constituents), etc. However, thisclassification is insufficient for the present purpose, not telling anything about therelations between the ’higher’ entity and its parts. Since we are interested in theproblem of specific laws, it is worthwhile to list the 3 qualitative different possiblerelations between a higher entity and its parts. Social sciences were long agoconfronted to this question (laws of associations, constitutions of states, etc.) andfrom their results here we take the following cases (names in the canonicalGerman) (Marx, 1964). The discussion and classification will be useful also for alater lecture in this Volume (Luk~ics, 1993b).

1) Einheit. Constitutional analogy is the centralised state. The Whole existsin its own right with its own laws, the Parts have no independent existence or laws.

2) Einigung. Constitutional analogy is the federation. Both the Whole, andthe Parts exist in their own rights with their own laws.

3)Vereinigung. Constitutional analogy is the confederation. Theindependent entities are the Parts with their own laws; the Whole is their sum withinteractions, so can be derived from them.

Obviously any further transitional stages could be identified, but these three willsuffice now for us.The present Universe is clearly not an Einheit. Namely, its parts (e.g., stars) canquite satisfactorily be described or explained without referring to the totalUniverse. There remain the other two possibilities; we could choose between themaccording to the success or failure of explanatory models (see later). However thiswas not necessarily true for the primordial Universe, which probably did not haveseparate parts of permanent identity (cf. later).

3. BUILDING TOGETHER OR TAKING APART?.

Consider the hierarchy of objects in the present Universe. The obscrv~able partcontains more than 10~ galaxy clusters, of which each contains roughly 10~ galaxies.Our galaxy contains - 1011 stars and an average star contains - 1057-nucleons andsimilar number of other particles. Our Sun is 4.6 × 109 ys old (Novotny, 1973), theoldest stellar clusters in our Galaxy have ages (10-20)×109 ys, and extrapolatingback the recession of galaxies the objects of the presently observable volume were

packed with a density higher by orders of magnitude than the present one at~I5×109 ys ago. So the formation of any presently existing structure should beexplained on a scale 1010 y or shorter.Then galaxies cannot build up by random encounters of 1011 independent stars(instead of detailed formulae remember that the solar system cannot have had aclose encounter in the last 4 billion years, therefore such processes are tooinfrequent to collect 10It stars in 15 billion years). Also, it is improbable thatthousands of independent galaxies could have congregated to form a galaxy clusterin 15 billion years. This suggests a fragmentation from protoclusters to stars.To be sure, star formation is fairly explained by gravitational contraction. Howeverthe above probability considerations suggest formation/naMe an existing galaxy;, sowe may try with the idea that first the galaxy is formed, then it fragments intosmaller matter elements, and finally these ’droplets’ contract into stars.If so, then it is obvious to go one step farther by investigating the possibility thatproto-galaxy-clusters dropped out from a primordial unity, indeed, this is thestandard explanation in this years; we shall see why.

4. ON STANDARD COSMOLOGIES

The simplest nontrivial and reasonable cosmological model is a space-time withfull spatial symmetry (i.e., constant curvature k = 0 or _+1 on constant timehypersurfaces). Then (Robertson and Noonan, 1969)

(4.1)

where

Isinx forkffi+l

S(x) ffi ~ x k = O (4.2)

shx k ffi -1

Then the distance of ’naturally moving’ objects change proportionally to R, sodensities are -R-3. This scale function R is governed by the Einstein equationswhich (without cosmological constant) read as

k2 ffi (8’B-]*3)G~R2 - k (4.3)R ---- -(4’tr/3)G(p÷3p/c2)R (4.4)

where G is the Cavendish constant of gravity, o is the mass density and P is the(dynamical) pressure. With an additional equation of state P = P(p) (or of similarform) these equations can be integrated for a given initial condition (discussedlater).

Assume full thermodynamic equilibrium and blackbody radiation for the equationof state

p = N(~/90)~/(~)3 (4.5)where T is the temperature and N is the number of independent helicRy states,roughly the humor of di~crcnt ~ of panicle, - I~. ~en ~ = 3P/c~, and fork = 0 ~s. (4.34) ~n ~ anal~i~lly ~IvM. ~ ~mpadng the derivative of (4.3)~th (4.4) onc g~u

T = T~o IR (4.6)

where R~ = R(to), To = T(to) and to ~ an arbRra~ ~nvcnlent time momcn~ ~cnsubsututmg to (4.3) one ge~R ~ 02~N~)l/4(~-2/(~)3)’l/4T~o~ (4.7)

where t’ ~ a ~nt of ~te~tion. Howler,T ~ (32~N~)1/4(~-2/(~)3)-1/4/~ (4.8)

Obse~e that T ~ ~ at t ~ £. ~ t~e moment ~ the natural ~ro ~t of thet~e ~unting, and in th~ ~nvention ~ ~ee c~ ~ ~ ~ ~~olu~.

Eq. (4.7) deserves some discussion but this will be postponed till the next Chapter.Now we turn to (4.8). It gives a continuously decreasing temperature. AboveT - 10000 K - I eV the photons would have destroyed all the atoms, so acompletely ionised plasma was present. Photons are substantially coupled to freeelectric charges, so before t-t’ -300000 ys they destroyed any virtualgravitationally bound structure. When T goes below 1 eV -10"12 erg, the gasbecomes fairly transparent for photons and structures of sufficiently large massesbecome bound. At this temperature

¯ ffi 0cz - 10 3erg/cm 3 (4.9)For the smallest bound object

Etherm -- PR3 - Egrav - GMZ/R; M - pR 3 (4.10)Hence the first appearing bound objects have the mass - 1051 g - 10is solar mass.Since this value is roughly that of the largest galaxy cluster, the standard cosmologycan explain the appearance of galaxy clusters. Therefore it seems that

1) before 300000 ys from t" there were no identifiable structures aboveelementary panicles in the Universe; and

2) among the objects there was a fragmentation chain Universeprotoclusters -- protogalaxies -* matter for protostellar contraction.

However at 3O0O~ ys the matter probably contained the known elementaryparticles and in the standard model it is so from the beginning. Thus we still have todiscuss the initial conditions, since elementary particles can stand up in differentconfiguratious.

24

5. ON INITIAL CONDITIONSNow we return to cq. (4.7). It is valid only for k = 0, but for early times (hightemperatures) it is a good approximation also for k = _+1. Eq. (4.1) shows that aconstant factor is undefined for k = 0, but it is not so for the other two cases. Noclear evidence for k is seen in the observable part - 1028 cm, therefore now

Rpr -- q x 102s cm, q > 1 (5.1)

From the blackbody radiation

Tpr=3KTherefore in eqs. (4.6-7)

ToRo .~ q x 10~2 ergcm - q x 1028 cmgrad

(5.2)

(5.3)

From mass counting the ]aarticle number of the observable part is - 1078. For thesize R it multiplies with q~.With this data in mind go back to eqs. (4.3-4). They can be rewritten as

YzR2 -- GM/R = -V~k

dE + PdV = 0; E = (4~r/3)R3~; M = E[c2(5.4)(5.5)

F_.q. (5.4) can be read as ’energy conservation’ with ’kinetic’ and ’potential’ energies;the next equation shows that the changes are adiabatic (no exterior) and Mchanges. However going backwards to the past both terms on the left hand side of(5.4) are growing in absolute value as 1/t. We cannot exactly use (4.7-8) nowbecause they would give k -= 0, but it can be seen that for early times the ’initialconditions’ Ro and T_ must be more and more fine tuned to get the constantdifference. Eqs. (4.3-43 of course preserve the difference, but now we try to thinkaccording to causality.At present all the 3 terms in (5.4) are in the same order of magnitude. Going backthe needed tuning is -(t/tpr). If anything prescribed R at a to it must haveprescribed with this accuracy. If the initial conditions were determined in an ageT - 1 GeV, e.g., then the tuning must have been cca 10-3°.

IFTHE INITIAL CONDITIONS WERE RANDOM FOR THE UNIVERSETHEN THE PRESENT UNIVERSE IS ABSURDLY IMPROBABLE.

Now, it is difficult to interpret this statement. First, when should the descriptionstart with an initial condition? In a ballistic problem the proper moment for theinitial conditions is when the stone is leaving the hand. Until that moment theequations of motion are not those of the ballistic problem, and at that momentvarious initial velocities can and used to be prepared (in a reasonable range). Thenthe proper moment for setting the initial conditions for the Universe would be themoment when the evolution was switching from unknown previous ones to the

THE EVOLUTION OF COSMIC SYMMETRIES

Einstein equations (4.3-4) (’creation of the Universe’?). But we do not knowanything about previous, qualitatively different, eras of evolution (with theexception of a slight guess, see the next Chapter). In addition, there is only a singleUniverse. We do not know if it even has any meaning to imagine a set of differentlyprepared Universes. It is quite possible that one could not apply the usualdichotomy of (arbitrary) initial conditions + (fixed) equations of motion on the(total) Universe. But practically all branches of physics use this dichotomicdescription to evolutionary situations. So it is possible that the Universe wouldneed a new type of physics or at least additional cosmological laws prescribing, e.g.,the unique ’initial conditions’. This was the reason to permit the possibility that thepresent Universe might be an Einigung, having its own laws as well as its parts havetheirs. It ispossible that the Universe cannot be totally understood from its parts.

However we do not want to be involved in theology or similar disciplines.Therefore no philosophical questions of creation of the Universe, the will of anyCreator to prepare special initial conditions etc. will be discussed. Here weformulate two questions in the usual physical language.

1) Was any special time moment to in the past which was a ’beginning’ toimpose initial conditions?

2) If there is any room for initial conditions, then what are the quantitiesfor which initial values are to be imposed?

For Question 1) we of course cannot give a final answer. However we list,without the claim for completeness, the possibilities for the kinds of such to’S. Aspecial time moment might have been the starting point of the Einstein evolution.Another possibility would have been the minimum of the radius R(t). The thirdpossibility is the maximum of the temperature or energy density or any relatedquantity. Obviously the above blackbody model does not contain any of these threespecial moments: in it the Einstein equation always governs the evolution, theminimal radius is 0, which is a singular point unsuitable to prescribe data, and themaximal temperature is ~, again in a singular state. We return to Question 1) inthe next Chapter.For Question 2), in contrast, the answer is simple. In the above simple model ofblackbody radiation there are two independent quantities, R and T. So an initialcondition is the prescription of Ro and To. In more complicated models somefurther quantities may appear, e.g., for the total numbers or densities of differentcharges or particle.

6. ON THE ’BEGINNING’The fact that we cannot go beyond General Relativity is not a proof of its ultimatevalidity. The Einstein equation is the simplest equation governing the curvaturewhich is conform to present observations as e.g., planetary motions. However it isquite possible that the equation possesses extra terms unseen in the presentobservations. To be definite, the standard form of the Einstein equation is

Rik -- VzgikRrr - hgik = - (8zrG/c4 )Tik (6.1)

Here gik is the metric tensor, Rik is the Ricci tensor which contains gik together withits first and second derivatives in a covariant manner, linearly in the second ones, ),is the so called cosmological constant, while Tik is the energy-momentum tensor ofthe matter (Robertson and Noonan, 1969). Above ), was neglected, but the presentobservations give upper bounds for [ ), [, and its effects can be felt at lowdensities, so ), is irrelevant for early times. Now, the simplest possibility for adifferent evolution is the presence of higher neglected terms (nonlinear or higherin derivatives).Of course the neglected term may be of any form and may appear just in the nextmeasurement or 20 orders of magnitude above. However there is a natural way togenerate some higher terms.The Einstein equation (6.1) can be obtained from a variation principle whoseLagrangian is

L = Lmatter + Lgrav (6.2)where

Lgrav = f(R) = (c4/8~rG)(R +2X); R =- R rr (6.3)

Now, assume that f(R) is nonlinear. Indeed, L,4nczos (1972) pointed out that apurely quadratic Lagrangian would lead to a dimensionless action integral, a purenumber, which, therefore, seems to be something ’fundamental’, good forasymptotic behaviour. Writing

f = ~rR2 + (c4/8~rG)(R+2),) (6.4)

the Einstein equation, up tofirst order in tr, changes to

Rik - V2gikRrr -- ~gik + °(8~rG/c4)(R;ik - R;rr gik ) = --(8~rG]c4 ) "Tik (6.5)The new coefficient tr has the dimension gcmS/s4. Do we have any guess for thevalue of such a quantity?

Yes, certainly. Contemporary physics knows 3 quite fundamental and generalphenomena, each with its own single and unique characteristic constant. They are(i) gravity with the Cavendish constant G = 6.67 x 10-8 cm3/gs2, (//) relativity withthe velocity of light c = 3.00x10t° cm/s, and (iii) quantization with the Planckconstant h = 1.05 x 10-27 gcmZ/s. Partially unified theories do exist but the finaltriadic unification (’Relativistic Quantum Gravity’) is not at hand. However, itsfundamental scales must be set by G, c and h. Now, the obvious form for tr of theabove dimension is

o = (fundamental number) xhc3 (6.6)

So extra terms of (6.5) type may be expected from any quantum extension ofGeneral Relativity. As an approximation, let us start from (4.1-8), and calculate theextra terms. They become comparable with the old ones at

T- Tpt = hV~cS/G = 1.22 x 10~9 GeV- 10~5 erg (6.7)

THE EVOLUTION OF COSMIC SYMMETRIES 27

which is out of any possibility for observation. But this means that in the unifiedtheory of all present theories this may be the point where Geometry emerges fromthe sea of foaming quantum fluctuations, or the corresponding temperature may bea maximal temperature.

The simplest model anticipating such phenomena is General Relativity + aHawking radiation from the change of the geometry. Then the past history of theUniverse is geodetically incomplete. It appears with infinite temperature but withfinite energy density ¯ - ¯~,l = Tpt4/(hc)~, and with a finite radius R - Rpl,

Rp! = hV/’~c3 - 10-33 cm (6.8)

(Di6si et aL, 1986). According to the uncertainty principle a quantum fluctuationfrom ~ = 0 to ¯ = ¢Pt in a volume Rpt3 survives till tpt = Rpl/c. However, if energy isproduced (which is so for P < 0, and also if Hawking radiation of this type is present(Di6si et aL, 1986)) then during that time the original energy of the fluctuation canbe reproduced and then the Universe can remain for later use.This is a v.ery primitive ’model’. However it contains a ’beginning’ (of the evolutiongoverned by the Einstein equation). There the initial conditions are

Ro - RpI, ¯o - 41"1 (6.9)

i.e., completely prescribed by number constants (number of helicity states, ~-, etc.)and by the 3 fundamental constants. In this scenario no freedom appears in theinitial conditions of our single and unique Universe, which is hopeful.Now let us extrapolate back the standard Universe (4.7-8). There is a state withT - Tit and ¯ - ~Pl. However there R - 0.1 ×q cm - 1032 ×q ×Rpt. This is arather ’unnatural’ initial condition which cannot be expected from any ’RelativisticQuantum Gravity Theory’. This high factor is reflected in the mentioned ’finetuning problem’.AGAIN, THE PRESENT UNIVERSE SEEMS IMPROBABLE IN THE LIGHT

OF FUNDAMENTAL CONSTANTS.Let us note that the present elementary particles seem improbable as well. Namely,the massive ones are below the only natural combination for rest energy - T~,z by 20orders of magnitude, and some possess sizes 20 orders of magnitude above Rpl.While these masses and radii are in natural relations with each other, the massesthemselves remain unexplained from (still unknown) fundamental theories. Thefundamental elementary objects in any ’Relativistic Quantum Gravity’ would haver - Rpl, rn - Tpl/c2 (and maybe a lifetime -tpl), so protons, electrons etc. cannotbe the elementary objects of ’the fundamental’ theory.

WE HAVE REASONS FOR DOUBTS IF THE KNOWN ELEMENTARYPARTICLES HAD BEEN PRESENT AT ’THE BEGINNING’.

We do know that some ’elementary particles’ are composite objects and werecreated at a definite stage of the evolution of the Universe. However, e.g., theelectron seems point-like, so really elementary. Still, its mass does not belong to

28 B. LUK~CS

’fundamental unified physics’. (Its electric charge does, since e2 -hC, thenumerical factor being as moderate as 1/137.)For charges or particle numbers the natural initial conditions are simple enough. Aparticle number is dimensionless, and charges may be defined likewise. Then thefundamental constants G, c, h yield nothing for them; the natural initial conditionis either N,~ - 1 or N,, = 0. The same is true for the total entropy S. In contrast,extrapolating back from the present Universe, Nbaryon - q3 × 1078, and S - 1087(Guth, 1981).

7. ON PHASE TRANSITIONSThe above ’unnatural’ numbers seem to point into a common direction, discoveredby Guth (1981):THERE SEEM TO HAVE BEEN SUBSTANTIAL ENERGY AND ENTROPY

PRODUCING PROCESSES IN THE EARLY PAST OF THE UNIVERSE.

As shown by eq. (5.5), this is possible if P < 0. Negative (dynamical) pressures areunfamiliar, but may appear e.g., in scenarios when a phase transition cannot startbecause of any barrier inducing supercooling of the high temperature phase. Thenthe energy is higher than the equilibrium value, therefore the pressure is lower.Since now P > 0, it was >0 in the past in any not unstable or metastable state, butfor transient periods, followed by reheating, it may have been even negative.To illustrate this we show the simplest nontrivial example; for details see (K,qmpfer,Luk~cs, and Pa~l, 1990). Consider a system without particle numbers. Omitting thedetails we have an equation of statep(T), and

s :P,r (7.1)~ = Ts -p = Tp~. -p (7.2)

Consider a model system of coupled scalar and vector bosons. The scalar Higgsbosom possess a quartic self-potential V(dp), mentioned in a parallel paper(Luk~ics, 1993a). (I) is a multicomponent quantity for a set of scalar bosons. If aparticular Higgs has a nonzero expectation value <~> (in a side minimum of thequartic potential), then some coupled vector bosons get masses.At a given temperature T all the vector bosom with m < < T simulate a blackbodyradiation, while those with m > > T have negligible contribution to the pressure. Soin the roughest approximation

- (7.3)where N* is the number of helicity states for the particles m<T, and V is to betaken at the actual equilibrium state, ~ = ~o or ¯ = ~+_. Then, according to eq.(7.2),

~ = (N%r2/30)T4/(hc)3+ + V(dp)/(hc)3 (7.4)

THE EVOLUTION" OF COSMIC SYMMETRIES

Recapitulate the Figures of (Luk~ics, 1993a). The quartic potential possesses acentral peak. Until the average energy (roughly the temperature) is above thispeak, <~> = 0. For lower energies the state sits down to one of ~___. There is adifference AV between the two cases. In addition, in the spontaneous symmetrybreaking the vector bosons coupled to ~b get masses proportional to <~>. Let usassume that at least some of these masses are >T. Then we have two differentequations of state, one for the high temperature phase in which the state of qb ismirror-symmetric (< ~ > = 0):

po(Z) = (No~2i90)~4/(hc)3 - aV/(hc)3 (7.5)

and one for the low temperature phase where ¯ sits in an asymmetric minimum

p+ = ( (No-AN)~r2/90)T4/(hc)3

The two phases are in equilibrium at Teq

Po(Teq) = P +(Te~)whence

for which

(7.6)

(7.7)

Teq = (90AV/rr2AN)If T is decreasing, there is a symmetry breaking at Teq.

(7.8)

Figure 1: Just above Teq.

Now consider a situation when T is just passing Tea very rapidly. The scenario issketched on Figures 1-3. Just above Teq the symmdtric state is stable. Just afterpassing the state starts to roll towards a side minimum, but if the cooling is very fastthen there are states similar to Figure 2. That snapshot is a state in which T hassubstantially decreased but still <~> is moderate and AV is still almost theoriginal. This is the supercooled symmetric state. Assume that it supercools to

Tea/3. Then the thermal part ofp has decreased by two orders of magnitude, so ispractically negligible compared to AV. In this case, from (7.1-2)

¯ ~ -p --- AV (7.9)Then for k = 0 ¢qs. (4.3-4) give an exponential expansion

R = Rlet/’~ ; 3,z = (3/8,r)((~c) 3c2/GAIO (7.10)called inflation. During this inflation ~ is roughly constant, while the volumeincreases, so energy is produced, while the total entropy is roughly constantbecause of the adiabatic change.

Figure 2: Just below Teq. The state starts to roll down.

Figure 3: Well below Teq. The state starts to settle down. Reheating will follow.

/

THE EVOLUTION OF COSMIC SYMMETRIES

The supercooling ends after some At needed to reach the side minimum. Then theasymmetric phase is established. Comparing (7.4) and (7.6), the energy of theground state becomes lower, so some energy must go again into thermal degrees offreedom, which is the reheating. This is a nonequilibrium process, producing anentropy AS -AE/T~,, where AE is the energy produced in the inflation. IfAt/~, - o ( 10 t ~. 10rr), then the inflation can produce almost any increase ine 87nergy or entropy, e.g., the factor AS/S - 10 needed to eliminate the fine tuningproblem in (Guth, 1981).This is only the simplest possible scenario, but for the present goal it is enough. Letus stop at this moment and summarize the effects of a spontaneous symmetrybreaking preceded by a supercooling.

1)The symmetry of the actual state of some scalar bosom breaks down.2)Some vector bosons get masses.3)R jumps up, with substantial energy and entropy increase.4) The inflation smoothens the existing spatial inhomogeneities, so spatial

symmetries (uniformity) are restored.Therefore in such phase transitions internal and spatial symmetries changeoppositely: the spatial symmetry is restored on the account of the internal one.

8. ON GRAND UNIFICATIONOur particle physical measurements do not extend beyond 1003 GeV energy,except a few isolated reconstructed cosmic radiation events up to 1011 GeV. A ver~bold extrapolation, however, suggests a spontaneous symmetry breaking at -1015GeV. Namely, the low energy particle physics seems to have 3 independentinteractions

1) Electromagnetism, with symmetry group U(1).2) Weak interaction, with symmetry group SU(2).3) Quantum chromodynamics (whose peripheral effect is the strong

interaction), with symmetry group SU(3), for colours. (The earlier literaturementioned another SU(3) symmetry for flavours, i.e., among different kinds ofhadrons. This symmetry is approximate, and has no intimate relationship to thefundamental symmetries. It is a consequence of the fact that the hadrons arecomposed from quarks.)Now, with increasing energy the 3 coupling constants seem to converge, and it ispossible that all the 3 interactions belong to a common SU(5) group, (the smallestone with all the 3 as subgroups) (Langacker, 1981). The resulting theory is calledGrand Unification, in the simplest extrapolation the parameters of the theory canbe calculated from low energy data, and the theory may be valid just above 1015GeV, although up to now no predicted consequence has been observed.

In an SU(5) symmetric theory there is no qualitative difference between quarks andleptons. They can continuously be transformed into each other by exchangingmassless vector bosons of appropriate charges. Therefore in Grand Unificationonly two charges are conserved, namely

32 B. LUI~CS

electric chargebaryon number - lepton number.

If anything new happens between 1015 GeV and Planck energy, we cannot guess itfrom here.

9. AGAIN ON INITIAL CONDITIONS

Now we can rediscuss the problem of initial conditions. We start at Plancktemperature (or energy or energy density) which we take t = 0 (indistinguishablefrom t = tpt. Anarve extrapolation of the present Universe resulted in (4.13), whichled to the unnatural initial condition R = 0.1 cm at T = Tpl. However, as we haveseen, any part of the present entropy or energy may have been produced in aninflation. So there is no evidence against

Ro - Rpt

HenceSo - 1.

For the charges, all observations suggest electric neutrality, so

The other conserved quantity of Grand Unification is the (baryon-lepton) number.Let us count the particles in our neighbourhood. The overwhelming majorityconsists of a few kinds of particles as protons, neutrons, electrons, neutrinos (allwith antiparticles) and photons. Photons do not carry any kind of charge. Theconserved quantity A can be calculated presently as

z~ = (Np + Nn - Ne - Nv ) - (antipart.) (9.1)

Now, N_ = Ne (neutrality; for both particles the antiparticles are negligible).Neutrons "are stable only in nuclei, all the matter ~s practically 90% H and 10% He.Hence Nn ~ No15. Neutrino numbers cannot be measured because low energyneutrinos pradtically do not interact with the measuring apparatuses. Butcosmological models suggest Nv - Ne, and then a slight excess of ant/neutrinoscan compensate Nn. Therefore there is no evidence against

In addition, for the nonconserved numbers one may assume a fully symmetricinitial state.

This starting Universe cannot be distinguished from nothing at all for a period titdue to the uncertainty principle. One may hope that these initial conditions end ~nour Universe; we cannot check this, because (i) the number factors in TO and Ro arestill unknown, (//) we cannot calculate near Tpt, (iii) the fine tuning problem doesnot exist anymore due to inflations, and (iv) we do not know how many phase

THE EVOLUTION OF COSMIC SYMMETRIES

transitions happened until now. Anyway, this is a nice initial Universe and the lestarbitrary one.

But the starting Universe was of the mass and size of a single quantum fluctuation.

THIS INITIAL ’PLANCK’ UNIVERSE CANNOT HAVE CONTAINED ANYPARTICLES, OR EVEN ANY IDENTIFIABLE PARTS.

The less arbitrary initial Universe cannot have beeB anything else than an Einheit,being an undifferentiated unity.

10. A POSSIBLE SCENARIOWe do not know which kind of Grand Unification is true, or if any of them is trueat all. We do not know anything even in this extent at higher energies. However,hypothetical scenarios connecting the discussed initial Universe to the present onecan be drawn in a more or less qualitative manner. Steps of such scenarios werediscussed in (K/lmpfer, Luk~tcs, and Pa~l, 1990); here we concentrate only on thechanges of symmetries and those in the individuality of parts.

1) Beginning. The Universe is one, indivisible, elementary unit, with Planckdata. Maximal symmetry: no observable spatial structure; fields in symmetric states,conserved charges at 0 values. No parts. The whole Universe is one ’particle’.

2) Just after beginning. Some energy production must have happened,otherwise the Universe would have fluctuated back to its absence after tpl. Maybe adelayed phase transition happened, maybe Hawking radiation preserved theUniverse; we do not know.

3) Somewhere not far (?) below Tpr Possibility for individual particles to bemore or less separated and to drop out. (By uncorrelated fluctuations in thegrowing volume?)

4) Somewhere between Tpt and 1015 GeV. Supersymmetry breaks down.Boson and fermion members of superpairs become distinguishable.

5) Just above 1015 GeV. Quantum fluctuations may start to create spatialinhomogeneities.

6) In a range downwards from 1015 GeV. Supercooling, inflation. Again asubstantial part of the energy is created. This energy is new and do not necessarilyfollow the original pattern of inhomogeneity. Spatial symmetry is then restored (insome extent). Finally SU(5) symmetry breaks down to SU(3)xSU(2)xU(1).Afterwards baryons and leptons are practically separately conserved. The actualstate is still symmetric for particle-antiparticle reflection.

7) Just after the SU(5) breaking. An effective (spontaneous?) CP breakingof the broken Grand Unification leads to faster decay of antiquarks andantileptons? (This point is rather obscure, for the details see (Barrow, 1983).)

34 B. LUI~CS

8) Down to -1000 GeV. Probably no phase transition. Fluctuations maygenerate inhomogeneities but no macroscopic permanent structures exist. As far aswe guess the presentpoint-like particles already exist. The antiparticle/particle ratiocontinuously goes down.

9) Somewhere in the range 1000 GeV. The mixed SU(2)xU(1) Weinberg-Salam interaction breaks apart to the familiar electromagnetism and weakinteraction. Again spontaneous symmetry breaking happens for some scalarbosons, and the weak coupling bosons W and Z (observed) get masses. Thetransition may be of first order, but already the time scale of the expansion andcooling is cca. 10-1° s, longer than the characteristic time of the electromagneticinteractions. So no substantial supercooling and inflation is expected.

10) Between 1000 GeV and 200 MeV. Standard expansion and cooling.SU(3) symmetric state forflavour abundances.

11) At cca 200 MeV. Hadronisation of quarks in a probably first ordertransition (between 8 and 15 #.s from beginning). Strong fluctuations, correlated involumes containing ~1 solar mass. At the end quarks and gluons are confined,protons, neutrons, some hyperons, and mesons are present. The state is no moreSU(3) symmetric for flavours (hyperons are less abundant), but there is still anSU(2) symmetry of flavours (equal numbers of protons and neutrons).

12) Between 200 and 1 MeV. Standard expansion and cooling. Still particle-antiparticle symmetry for leptons .

13) At I MeV. (- 1 s.) Neutrons start to decay but the lifetime is - 1000 s.14) At 0.5 MeV. Annihilation of e-e+ pairs. Only the slight (10-8) e- surplus

survives. On the account of the actual CP symmetry the spatial homogeneitysomewhat restores.

15) Somewhere at 0.1 MeV. (- 1000 s.) The free neutrons vanish by decay;bound ones in d, t, "~ and a survive and build up the primordial helium. Since it isdetected, we have observational evidence about separate autonomous parts of theUniverse from t = 1000 s.

16) Afterwards, for a while. Some nucleosynthesis. The matter is an e+pplasma, opaque for photons.

17) At - 1 eV. (300000 ys after beginning). Neutral atoms build up. Thematter becomes transparent.

18) Afterwards. Macroscopic structures become possible. First proto-galaxyclusters are stable. Henceforth macroscopic homogeneity breaks down to random(homogeneous isotropic) distributions of spheres. This epoch ends with the start offusion in protostars, and no great change happens until present in symmetry.

19) Presen~ No symmetry on human scales, spherical symmetry onterrestrial scales, more or less homogeneous isotropic distribution on large scales.

20) Future..9??. (See also (PaOli, 1993) for apossible future.)

THE EVOLUTION OF COSMIC SYMMETRIES

11. CONCLUSIONSThe present paper had a double goal. First, we have emphasized that ourknowledge about the early Universe is very limited. This statement of an expert isprobably accepted by anybody else. However, hence other logical statements follow.For example, familiar models may not be applicable for the early Universe; ournotions may be alien from those situations; very serious problems sleep almostundisturbed under many subsequent layers of familiar and quantitative problems.These statements are rather negative, and one can always question the validity of amodel under strange circumstances, but now we wanted only to demonstrate thatcosmology still needs continuous contemplation about its fundamental notions.However, in any definite time there is a best description, and our second goal was toshow up the outlines of such a description, specially from the viewpoint ofsymmetries. We have shown that (i) the present Universe can be obtained from amaximally symmetric primordial one; (//) the breakings of symmetries werespontaneous, therefore natural, and (iii) at some symmetry breakings internal andspatial symmetries changed oppositely (the breakdown of internal ones droveprocesses partially restoring the spatial ones).In a very symmetric Universe complicated organisms (as ourselves) could not exist.In a very asymmetric one minds would have great difficulties in understanding. Thepresent Universe is convenient for rational beings (according to observations). Wedo not know if it remains so forever. Present General Relativity does not predictfurther deterioration of the remainder of the spatial symmetries, and presentparticle physical theories do not predict the breakdown of the present effectiveSU(3) ×SU(2) × U(1) symmetry. However, they have not been constructed to predictit in the lack of any indication from measurements. One cannot exclude theexistence of further scalar bosons with quartic potentials (Linde, 1984); if theyexist, further symmetry breakings may happen. But we do not know to which vectorboson they are coupled and with what strength.

REFERENCES

Barro~v, J. D. (1983) Cosmology and elementary particles, Fundaments of Cosmic Physics, 8, 83.Cart, B. J., and Rees M. J. (1979) The anthropic principle and the structure of the physical world,

Nature, 278, 605.Di6si, L. eta/. (1986) On the thermodynamics of the vacuum,Astrophysics and Space Science, 122, 371.Guth, A. (1981) Inflationary universe: A possible solution to horizon and flatness problems, Physical

Review, D23, 347.K~mpfer, B., LukAcs, B. and Patti, G. (1990) Fenomenologiya fazovyh l~rehodov v rannei vselennoi,

[Phenomenology of phase transitions in the early universe, in Russian], Zhurna! ElementarnyhChastits i Yader, 22, 63.

Lanczos, C. (1972) Vector potential and quadratic action, Foundation of Physics, 2, 271.Langacker, P. (1981) Grand unified theories and the proton decay, Physics Report% 72C, 185.Linde, A. D. (1984) The new inflationary universe scenario, Reports on Progress of Physics, 47, 925.

B. LUK.4C$

CHIRALITY IN THE ELEMENTARY INTERACTIONSFerenc Gliick

Theoretical physicist, (b. Bhnd, Hungary, 1963).Addre~: Central Research Institute for Physics, Budapest 114,P.O. Box 49, H-1525 Hungary.Fields ofintere~. Particle physics, weak interaction, radiative correc-tions.Pub//cations." Radiative correction to electron neutrino correlationin iambda fl-decay, (with T6th, K.), Phydca/Re~’w, D40, 1989, 119,Order-a radiative corrections for semileptonic decays of unpolar-ized bmyons, (with T6th, K.), Physica/Review, D41, 1990, 2160,Order-a radiative corrections for semileptonic decays o~ polarizedbaryons, (with T6th, K.), Physica/Review, D46, 1992, 2090, Mea-surable distributions o~ unpolarized neutron decay, Physica/Review,D47, 1993, 2840.

Abstract: General survey of the chirality (parityviolation) of the weak interaction is given, with some mathematical detaiis andexperimental consequences.

1. ELEMENTARY PARTICLES AND THEIR INTERACTIONSThe elementary particles are the ’smallest’, most fundamental, structureless anduniversal building blocks of our world. According to the generally accepted theoryof the elementary particles (the so called Standard Model), we classify them into 4groups: leptoms, quarks, intermediate bosoms and the Higgs scalar. Both the leptonsand the quarks (see Tables I and 2) are classed into 3 families. Each family contains2 types of elementary particles. In the lepton families one of the particles has zerocharge and approximately zero mass. They are called neutrinos. The second particlespecies have an electric charge of - 1 (in the unit of the electron’s charge), and theirmasses are shown in Table 1, (1 GeV is approximately the mass of the Hydrogenatom; 1 GeV ffi 1000 MeV). The electron is stable, but the muon and the tau decayinto other particles after a mean lifetime of 2×10-6 s (2 microseconds), and3 × 10-13 s, respectively. All these leptoms carry precisely the same amount of spin(intrinsic angular momentum): 1/2 . For each lepton there is a correspondingantilepton. The antiparticles have the same mass and spin as their respectiveparticles but carry opposite values for other properties, such as electric charge. Theantiparticle of the electron is called positron.

38 E GLOCK

The lepton families are distinguished mathematically by lepton numbers; for exam-ple, the electron and the electron neutrino are assigned electron number 1, muonnumber 0 and tau number 0. Antileptons are assigned lepton numbers of the oppo-site sign. Although some of the leptons decay into other leptons, the total leptonnumber of the decay products is equal to that of the original particle. For example,the muon decays into an electron, an electron antineutrino and a muon neutrino:/z- -~ e- ~’e v ~t"T°tal lepton number is unaltered in the transformation.Electric charge must be conserved in all interactions, and the electron is thelightest charged particle. Therefore it is absolutely stable.The quarks are also classified into 3 families (see Table 2). Their fractional charges(1/3 and 2/3 of the electron’s charge) are never observed, because they form combi-nations in which the sum of their charges is integer. Barions consist of 3 quarks, themesons consist of a quark-antiquark pair. For example, the most well-known bari-ons, the proton and the neutron contain the light u and d quarks: p -uud;n -- udd.

The top quark has not been observed when writing this article in the high energyexperiments. If it exists, it’s mass should be in the 100 GeV - 180 GeV interval,derived from theoretical and experimental investigations.The six leptons and six quarks (with their antiparticles) are now thought to be thefundamental constituents of matter. Four forces (interactions) govern their rela-tions: electromagnetism, gravity, strong and weak interactions. These interactionsof the leptons and quarks are mediated by the intermediate bosons (see Table 3).The strong interaction between two quarks is mediated by the gluons, the electro-magnetic force between two electrically charged particles is mediated by the pho-ton. The heavy W+, 14~ and Z bosons are responsible for the weak interaction. Theexistence of the graviton is uncertain.

Name Letter Mass Charge

electron neutrino ue ~ 0 0

electron - 0.5 MeV -1

muon neutrino v~ ~ 0 0

muon /~- 106 MeV -1

tau neutrino ~,~ ~ 0 0

tau r- 1.78 GeV -I

Name Letter Mass [ Charge

up u ~ 4 MeV 2/3

down d ~ 8 MeV -1/3

charm c 1.5 GeV 2/3

strangei s ~ 150 MeV -1/3

top t ? 2/3

bottom b 5 GeV -I/3

Table 1: Leptons. Table 2: Quarks.

There is another hypothetical particle that has not been observed experimentallyyet: the Higgs particle. It is neutral (0 electric charge), and its spin is also 0.According to the generally accepted theory of the elementary particles (the Stan-dard Model), the interaction of the Higgs with the leptons, quarks, W"x-- , Z bosons isresponsible for the masses of these particles. They get their masses through the so

CHIRALITY IN THE ELEMENTARY INTERACTIONS 39

called spontaneous symmetry breaking (for further details see e.g., (Halzen andMartin, 1984; Quigg, 1985; Luk~cs, 1993a)).

Interaction

Strong

Electromagnetic

Week

Gravity

Intermediate boson

gluons

photon

graviton ( ? )

0

0

80 GeV, 91 GeV

0

Spin

1

1

1

2

Table 3: Fundamental interactions and their intermediate borons

2. SYMMETRY PRINCIPLES IN PHYSICSThe notion of symmetry is central to the theories of the elementary particles. Atransformation which does not alter the laws of nature is called symmetry of thenature (or symmetry transformation). The phenomena (events) of nature takeplace exactly in the same manner in the transformed world as in the original world.For example, the gravity or the Coulomb force between two particles has transla-tion symmetry: the F = x1 - x2 force is unaltered after the x -~ x + a translation(here x1 and x2 are the coordinate vectors of the particles).It is an experimental fact that certain physical quantities are not observable(unmeasurable). From these facts we can infer some symmetry principles, and fromthese principles we can deduce mathematically the conservation laws of nature(Lee, 1974). The table below contains some examples for these relations:

Not observable Symmetry transformation ] Conservation law

absolute space space trauslation : momentum

coordinate z -, ~ + a

absolute time time trausl~tion : energy

absolute direction rotation: angular momentum

in space

absolute space reflection : parityright (left)

absolute phase gauge transfornmtion : de:tHe charge

of wave function

Table 4: Unobservable quantities, symmetries and conservation laws

40 F. GLITJCK

The symmetry principles in particle physics led to the discoveries of new laws ofnature. For example, each of the four fundamental forces is now thought to arisefrom the invariance of a law of nature, such as the conservation of charge or energy,under a local symmetry operation, in which a certain parameter is altered indepen-dently at every point in space. The resulting theories are called gauge theories(Halzen and Martin, 1984; ’t Hooft, 1980). The gauge group (local symmetrygroup) of the Standard Model is the SU(3)c x SU(2)L X U(1) group.

3. SPACE REFLECTION SYMMETRY IN QUANTUMMECHANICSLet ~k be an atomic wave function, H the Hamilton operator of the system, and Pthe space reflection operator:

--: ¢’(x) =If H(x) = H(- x) (space re^flection is symmetry of the system), and ~b is eigenfunc-tion of the Hamilton~ ( H~b..= E~ ), then P~ is also eigenfunction with the sameeigenvalue (E), and HP = PH.

The eigenvalues of P are P = _+ 1 (because from P~k = P~/, " ~1~ =/#~ = if).P is called parity of the system. If the physical system has symmetry under spacereflection, then its parity is conserved.

According to the empirical Laporte-rule, the atomic wave functions change theirparity while the atom emits a photon. In 1927 Wigner showed that the Laporte-ruleis a consequence of the space reflection symmetry of the electromagnetic interac-tion.

Various investigations of atomic and nuclear transitions demonstrated unam-bigously that both the electromagnetic and the strong interactions have exact spacereflection symmetry (they are parity conserving).

4. THE PARITY VIOLATION (SPACE REFLECTIONASYMMETRY) OF THE WEAK INTERACTIONNot only the atomic wave functions, but also the elementary particles have parity(intrinsic parity), similarly to their intrinsic angular momentum (spin). The parityof the leptons, barions, mesons, and photon can be measured with elementary pro-cesses proceeding through the electromagnetic and strong interactions. For exam-ple, the photon and the ,r±, ~r° mesons have negative intrinsic parity.

The weak interaction was supposed to be also parity conserving until 1956. ThenLee and Yang pointed out that the conservation of parity as a universal principlewas very inadequately supported by experimental evidence. They were first led tothis finding by consideration of the various decays of the K-meson. Both the K -~"rr~" and K-~ ~rrrr decay modes had been observed, while the ,r,r final state hadP = + 1 parity, the ,r,r,r state had P = - 1.

CHIRAL ITY IN THE ELEMENTARY INTERACTIONS

Space reflection symmetry of the world would require the ’mirror reflected’ world(with the x -*-x transformation) to be indistinguishable from the original world.Therefore, to check the mirror symmetry, we have to carry out two experimentsthat are mirror images of each other. If mirror symmetry holds, they should givethe same results. On Figure 1 below we show the layout of the famous Co6°experiment (accomplished in 1956).

Figure 1: Co60 experiment.

The Co6° nucleus decays into Ni6° nucleus, electron (e-) and electron antineutrino(~e)" Mirror symmetry can be checked by measurement of the correlation betweenthe outgoing electron’s direction and the spin of the Co6°.

At the left hand side of Figure 1, the spin of the Co6° nucleus points upwards, atthe right hand side downwards. This arrangement corresponds to mirror reflection.The different number of electrons going upwards in the two cases shows the viola-tion of space reflection symmetry for the weak interaction.Various experiments after 1956 showed that the spin of the antineutrino is alwayspointed towards its direction of motion (it is always right-handed), and the neu-trino’s spin is always opposite to the direction of motion (it is left-handed). This isanother example of space reflection asymmetry (parity violation). After spacereflection the left-handed neutrino would become right-handed, and right-handedneutrino does not exist in nature. World and mirror-world are distinguishable byphysical experiments!

Let us introduce the notion of charge reflection (denoted by C). Thistransformation changes all particles to their antiparticles (e- * e+, ~, * ~, ...).The charge reflection is symmetry of the electromagnetic and strong interactions,and (similarly to space reflection) it is not symmetry of the weak interaction. Theexperiments show. however, that the combined CP transformation (charge + spacereflections together) is very good symmetry of the weak interaction (for example,the right-handed 7goes under this transformation into the existing left-handed v).

42 F. GLOCK

In 1964 a small CP violation in K-meson decays was discovered. This CP violationcould explain the matter-antimatter asymmetry of our world (according to theSU(5) grand unified theory, theX- bosons with masses of = 1015 GeV could havedecayed asymmetrically into quarks-antiquarks and leptons-antileptons, aboutt --- 10-35 s after the big bang) (Wilczek, 1980; Luk~ics, 1993b).

5. MATHEMATICS OF THE WEAK INTERACTION AND OFPARITY VIOLATIONIn the following we shall present some mathematical details of the weak interactionand its parity violation. First we introduce the notion of the Feynman-graph. Thesediagrams can illustrate the processes of the elementary particles, and also, one canuse them to read the building blocks necessary for the calculations of the measur-able quantities (according to some definite mathematical rules). One can deducefrom each Feynman-graph a complex number: the amplitude of the correspondingelementary process that is illustrated by the graph. This complex amplitude can beused to calculate the measurable quantities of the process.Figure 2 below shows one of the simplest Feynman-graphs of the e e --, e ecollision process. The interaction of the electrons is mediated by the photon (3’),which is unobservable here (virtual).

Figur~ 2: Simple Feynman-graph of the e’e" --* e-e" process

We need 3 types of mathematical expressions for the calculation of the complexamplitudes :

- wave functions- propagators- vertices

5.1 Wave functions

They correspond to the outer lines of the graphs (on Figure 2 they are the e-lines). The ~b(x,t) wave function of the electron has 4 components:

CHIRALITY" IN THE ELEMENTARY INTERACTIONS 43

t)/ 3(x, t)/ 4(x,

Here x is the space vector, t is for time. The space-time dependence of ~b for freeelectron is given by the Dirac-equation ."

.oO .

me is the elect ron ma.,s. "~ ". "~ ~ ¯ 7 are the Dirac-matrices :

.to 1 0.): 0 1

= 0 -1 ; = -1 00 0 1 1 0 0

(for further details see e.g., (Halzen and Martin, 1984)).

5.2 Propagators

They correspond to the inner lines (virtual particles, which are not observed at thegiven process). For particles with mass M and p = (E, p) four-momentum (E is theenergy, p the 3-momentum of the particle) the propagator is proportional to

1/(t,2 -M~)The wave functions and the propagators characterize the particles, but not theirinteractions!

5.3 Vertices

The vertices are referred to the points where the outer and inner lines meet eachother. They are complex matrices, and they determine the interactions of the parti-cles in the process.The vertex of the electromagnetic interaction is given by the

ie7 ~ (/z = 0, 1, 2, 3)4 x 4 complex matrices (e2 ~ 4w/137). This vertex corresponds to those points ofthe Feynman-graphs where 2 electron lines and a photon line meet each other:

Figur~ ~a

44 F. GLOCK

It can be shown that this kind of interaction, given with the 7 ~’ (# = 0, 1, 2, 3)matrices, is parity conserving.

Let us now consider the ~’e e - -~ v~ e- collision process. One of the Feynman-graphs can be seen below:

¥1gur~ 3: Feynman-graph for the vee" -’~ P¢~- process.

The collision here is mediated by the W boson. The complex amplitude corre-sponding to this graph contains the

1 / (p2 _ Mw2)

factor, coming from IV propagator. The IV mass is very large: Mw ~ 80 GeV =80x 10 9 eV, therefore the above propagator factor is very small (/# is negativehere). This explains the fact that the weak interaction, mediated by the heavy Wboson, is very weak at small energies.

The v e e - IV vertex (corresponding to the points of the above graph where thelines of these particles meet each other) has the following form:

iCwT t~(1 -7 5 )where ."

The presence of the 7 5 matrix is responsible for the parity violation of the weakprocesses!The weak interaction vertices of the quarks have the above form, but with differentcwnumbers :

CHIRALIT"Y IN THE ELEMENTARY INTERACTIONS 45

Figure 3a

The neutron decay (n -~ pe-~ e ), for example, can be reduced to d -~ ue ~ ~" equark decay (with some complications coming from the strong interaction of thequarks and gluons).

The e e -, e e ~.ollision process can be mediated not only by photon, but alsoby the Z boson. The corresponding two Feynman-graphs :

T

gigure 4: Photon and Z exchange graphs of the e’e- --~ e’e" process

For the calculation of the observable quantities of the e-e- -~ e-e- process we haveto add the two complex numbers coming from the two graphs :

The eeZ vertex has the following form :

The e, cw, cv, ca coupling constants are real numbers, and have the same order ofmagnitude. The uuZ and ddZ vertices have similar forms, with different cV and cAconstants. The propagator factors in the two amplitudes are the following :

photon (~) -~ 1/p 2 ; Z boson -, 1/(p ~ - Mz2)

46 F. GL~rCK

The mass of the Z boson is very large (Mz ~ 91 GeV = 91 x 109 el/), therefore atsmall energies (where I p2 I < < Mz2) the Mz amplitude is very small comparedto Me:

IMzl << IM~IIn the atoms, the atomic electrons interact with the quarks in the nucleus via bothphoton and Z exchange:

Z

Figure $: Photon and Z exchange graphs for the electron-nucleus interaction

The N nucleus contains neutrons (with udd quarks) and protons (with uudquarks). The eeZ, uuZ and ddZ interactions are parity violating (due to the 3’ 5matrix in the vertices). Therefore, there are parity violating effects in atomic pro-cesses (Zeldovich, 1959). These effects are very small (due to the large mass of theZ boson), but can be observed. We mention that the cw, Cv, cA constants in theabove vertex formulas are completely predicted by the SU(2)L x U(1) unified the-ory of electromagnetic and weak interactions (Weinberg-Salam model) (Halzenand Martin, 1984).

6. EXPERIMENTAL DEMONSTRATION OF THE ATOMICPARITY VIOLATIONThe effect of the small parity violating contribution of the Z boson exchange inatomic processes can be shown by measurement of the optical activity of atoms (forother methods, see (Bouchiat and Pottier, 1984)).Light is transverse wave motion - the electric field vector vibrates perpendicularlyto its direction of propagation. The vibration can be arranged to take place in onlyone direction. This is the linearly polarized light. The direction of propagation andthe electric field line determine the polarization plane. When the electric field vec-tor rotates along a circle, the polarization of the light beam is called circular. Thecircularly polarized light contains photons with definite angular momentum (right-

CHIRALITY IN THE ELEMENTARY INTERA CTIONS 47

or left-handed photons). The linearly polarized beam is superposition of right- andleft-handed circularly polarized beams with equal electric field amplitudes.

If a medium interacts differently with the right-handed and the left-handed pho-tons, we call it optically active. When a linearly polarized light beam passes throughsuch a medium, the polarization plane of the beam is rotated through some angle,and the beam emerges from the medium linearly polarized in a different direction.This rotation of the polarization plane is the consequence of the phase delay be-tween the right-handed and left-handed circularly polarized beams.

Many crystals and molecular compounds exhibit rather large optical activity. This isdue to the asymmetric arrangement of their atoms. The mirror images of these crys-tais and compounds rotate the polarization plane in the opposite direction andwith the same angle. Optical activity here has nothing to do with parity violation.The mirror image of a gas of atoms, however, is identical with the original gas.With mirror symmetric elementary interactions the atoms look the same in amirror as they do in reality. Therefore, any optical rotation observed in an atomicgas is not caused by handedness (left-right asymmetry) in the geometry of theatoms, as it is in molecular gas, but by the handedness embodied in the laws ofnature that govern the weak force.

The angle of optical rotation predicted by the electroweak theory (Weinberg-Salammodel) is extremely small, about 10-5 degree under the most favourable experimen-tal circumstances. The rotation degree enhances roughly with the cube of theatomic number (Bouchiat and Pottier, 1984). Therefore, the heavy atoms arepreferred from the experimental point of view. Unfortunately, the theoretical cal-culations are rather difficult for the heavy atoms.

The first experiments performed in 1976-78 in Washington and Oxford with atomicBismuth showed serious discrepancy between the predictions of the Weinberg-Salam model and the experimental results (Baird et al., 1976; Sandars, 1977). Theresults of subsequent experiments carried out in 1980-82 were, however, in goodagreement with this model (Bouchiart and Pottier, 1984; Barkov, 1981), which isnowadays the generally accepted unified theory of the electromagnetic and weakinteractions (for historical details see Picketing, 1984, p. 294).

REFERENCES

Baird,P. E. G., a a/. (1976) Search for parity non-conserving optical rotation in atomic Bismuth, Nature,264, 528.

Barkov, L M., (1981) Parit/ts~rt6 folyamatok atomokban [Parity Violation in Atoms, in Hungarian],Fizikai Szem/ 31, 4.

Bouchiat, M. and Pottier, L. (1984) An atomic preference between left and right, Sciendfic American,250, 6, 76.

Halzen, F. and Martin, A. D. (1984) Quarks andLeptons, New York: John Wiley and Sons.’t Hooft G. (1980) Gauge theorie~ of the force~ between elementary particle~, SciendficAmerican, 242,

6, 90.

~. 6L~)CK

Lee, T. D., (1974) Szimmetriaelvek a fgik~ban [Symmetry Principles in Physics, in Hungarian], FizikaiSzem/e, 1.

Luk~lcs, B. (1993a) On the mathematics of symmetry breakings, Symmetry: Culture and Science, 4, 1,5-11.

Luk~lcs, B., (1993b) The evolution of cosmic symmetries, Symmetry: Culture and Science, 4, 1, 19-36.Picketing, A. (1984) Conaructing Quarks, Edinburgh: Edinburgh University Press.Quigg, C. (1985) Elementary particles and force~, SciendficAmaican, 252, 4, 64.Sandar~, P. (1977) Can atoms tell left [rom right?,New Scientist, 73, 764.Wilcz~k, F., (1980) The cosmic asymmetD’ between matter and antimatter, Scientific American, 243, 6,

60.

SYMMETRY CHANGES BY CELL~ AUTOMATA INTRANSFORMATIONS OF CLOSED DOUBLE-

THREADS AND CELLULAR TUBES WITH M~)BIUS-BAND, TORUS, TUBE-KNOT, AND KLEIN-BOTI’LE

TOPOLOGIESSzaniszl6 B6rczi

Abstract: The definition and classification of double-frieze structures in ornamentalart from archaeology (lk~czi, 1985, 198(~ 1989), and the introduction of symmetryoperations as local type, cellular automatic operations (B~rczi, 1985, 1987, 1989)opened the possibility of using these concepts in the crystallography of different surface-mosaic structures.

The first application of these concepts was in the transformations of the double-threadcellular M6bius-band to torus. This transformation preserves the half of the doublethread in the form of cellular bana~ but rearranges its neighbourhood in the directionperpendicular to the direction of the cellular band- This transformation - fromM6bius-band to torus, to and back - rearranges the pattern of the double-thread sys-tem and it is invariant to the knot-structure of tube-knots. Surface structure of Klein-bottle built from M6bius-bands is also discussed.

50 S. BI~RCZI

INTRODUCTION

To develop the topics of this paper it was necessary to unify the achievements inthree different directions of investigations. One direction was: symmetry as a local(cellular automatic) operation (B~rczi, 1976, 1985, 1986, 1989). The second onewas: the recognition of the role of double-threads in ornamental constructions(mainly in archaeological finds; B~rczi, 1986, 1989). The third one was the intuitiverediscovery of the M0bius-band to torus transformation (B~rczi, 1990). Construc-tion of a cellular automatic model to these developments formed a framework tobuild together and summarise them in this paper.

SYMMETRY BY LOCAL OPERATIONThe rich set of Avar-Onogurian ornamental structures (Fig. 3) in which there werefrequently double friezes, suggested to the author, that for the classification ofthese double friezes a new meaning of the classical symmetry concept should beneeded. The classical symmetry concept used global-local connections: symmetrywas the order of the ordered whole on its repeating, congruent elements(represented by symmetry operations). The symmetry concept connected withAvar-Onogurian structures modified the role of operations. Double friezesrequired a local and one-step generator type operation concept. (We may call ittechnological symmetry concept or cellular automatic operation concept because ofits step-by-step effect in structure building.) This symmetry concept was a local one,which recognlsed the global order (the cells were conscious of the global order),but considered operations as generators of the ’state’ of the neighbourhood.

OPERATIONS

translation with unit distance - t

,@lide reflectlon w. unit dist. - gmirror reflection w. unit dist.- m

half turn with unit distance - 2

Figure 1: Combinatorical construction of the four congruencies which work as operations to generatethe neighbour pattern in order to build a line in the plane. A combination of the three last of themresults in a fifth one: mg. These five line-patterns form a s~t of basic frieze patterns(t,g, m, 2, mg).

M~B IUS-BAND TRANSFORMATIONS B Y CELL ULAR A 13"TOMA TA 51

o

z

o

THE FIVE BASIC FRIEZE PATTERNS

~7 ~7 ~7 ~7

~7 ~7 ~7 ~7

Figure 2: The matrix of the 20 double frieze patterns (B~rczi 1986, 1989.) The matrix is organized(woven) from the five basic frieze patterns: t, g, m, 2, and mg. doubled by the four simple frieze generat-ing operations: t,8, m, 2, applied as local operations. Avar-Onogurian representatives are shown in Fig-

ure 3. Celtic representatives are shown in Figure 4.

There are four simple congruency operations which may generate different friezepatterns (with unit width) along the line (embedded into the plane). They are thefollowing ones: translation, mirror reflection, glide reflection and half turn (Fig. 1).The basic frieze patterns are those which were generated by these congruency oper-ations plus one more frieze: that which was their combination: rag. The doubling ofthem needs a local generator of the neighbourhood thread from the given friezepattern. This operation can be carried out by the four line-generator simple con-gruencies (Fig. 1). Local congruency operations determine the neighbourhoodpositions of a repeating element in a net of perpendicular and horizontal rows ofthe patterns of the matrix in Figure 2.

CELL~ AUTOMATIC FRAMEWORKThe cellular automaton model has a characteristic framework of description. It iscomposed from two parts of conditions. The first one gives the structure of the cel-lular background, the second one gives the transitional functions. Both parts of con-ditions form a pair of approach: a local and a global one, as follows:

A. CELLULAR BACKGROUND(Aa) Local characteristics of the cell-mosaic system give the form of cells, theirconnections and neighbourhood relations.(Ab) Global characteristics of the cell-mosaic system give the surface and the enclo-sure of the local relations to form a whole.

B. TRANSITIONAL FUNCTIONS(Ba) Local transitional function for cell mosaic elements which are individualautomata (discrete function in space and time, step by step transforming cell-states).(Bb) Global transitional function for the whole surface populated by the cell.mosaic system (it forms a sequence of stages of the surface taken step by step, as aconsequence of summarised - for all cells - local transitional functions).Although the points a and b are not independent of each other, the advantages ofthe cellular automaton model come from this separability of local and global pic-ture: both for conditions and operations, and from the expressed connectionsbetween the local and global characteristics of the phenomenon.

THE INDIRECT WAY OF CONSTRUCTION OF CELLULARAUTOMATON MODEL: THE INDIRECT VON-NEUMANNPROBLEM

The classical way of the development of a cellular automaton model was the con-struction of Aa and Ab background and the Ba local transitional function, at first.Then followed the deduction of the global transitional function Bb, which hold theprimary goal of the construction. We may call this way of model construction to thedirect way. (The principal aim of yon Neumann’s cellular automaton construction

"~ lWa!SS~l~ ~ u! molqo~d Otll ~oqs o~ luotudo-lo^op lopom uo~mom~ a~InllOa oIojoq ~n8 "0oqlo~ol s~a~d oni~ ~oJ-~I ’suo!lom

ioJ-!-]’mo ~OJ-A) ~I -V-~- A :S~OllO~ s~ tuoq~ loqu~s o1 oanpo~u! OA~so~g~s ooaq~ pu~ sdo~s o~ s~t uo!lounj i~uoD!su~n l~qOlff Otll aod~d ~uoaow otI~ uI

¯SO^lOsmoql SlIOO oql~oj uo!l~unj IgUO!l!sugn lWaOl og oql l~nnsuoa o~ ,~llUUg pug :mol~s ~!gsom-IlOOOtll ~Io suo!lgtu~o~Isu~l ~IO sdo~s oloJas!p ~Io so~ls ~Io oauonbos ~z s~ uo!lounj IgUO!l-!sug~l IgqOlg qfir Olglntua0j o~ qV pug t,p’Jo slu!od jo uo!1gu!tu~olop Otll Xq puno~-~logq otO ~u!qo~o~ls ~oW¢ "(996~ ’uurtunoN uo^) uo!~onnsuoa Jo uo!~ao~!p s,uugtu-non uo^ jo ~oodso~ u! Xgs ~oo~!pu! oq~ sT. ~odrd s!ql u! uo!~onnsuoa jo Xg~ ~nO

(’suo!!-aunj l~UO!l!su~n l~qOl~ jo 1o^oi oql uo oanlanals ~u!anpoadoa-jlOS ~ pl!nq ol

CRYSTALLOGRAPHY OF THE M~)BIUS-BANDIt is well known that M6bius band can be constructed from a finite long, unit wideband (belt) in the following way. Cut this normal band (belt) perpendicular to itsedges and attach the two ends after a half turn rotation of one of its end around themiddle axis of the band. This half turn transforms normal beltband to MObius-band. Let us assume that the belt band was adorned with a frieze pattern. Whatkind of frieze patterns may remain invariant after the transformation shown ear-tier?It is important to notice, that the band is built up by transparent pattern of thecells, what means: that both sides of a cell are shown with the same pattern figure.During the given normal-belt to M0bius-band-belt transformation, half turn acts asif it were a glide reflection for the neighbouring cells at attachment position. As aconsequence of the construction of the M6bius-band from a normal belt-band wecan conclude, that frieze patterns should have glide reflection generator in order tothey should be fitted onto the M0bius-band. This is a local condition for the cells.But there is a global condition, too. It is a number condition. Normal belt-bandshould have a pair number of cells in order to fit its pattern with glide reflection.The transformation procedure (cutting and attachment) shown earlier to constructM0bius-band from a normal-belt-band needs the elimination of one cell at theposition of attachment after half turn, because the operation referred destroys theorder there: after half turn there will be two cells with the same (i.e., only transla-tional) position, so lacking the needed glide reflection for these two neighbours. Tocorrect this failure of the order of the pattern we have to ’cut out’ one of these twocells, during an in sire production of the transformation described. Therefore theglobal condition for the number of cells in a belt with M6bius-band structure is thefollowing: M0bius-band should contain odd number of cells with g (glide reflectiontype) generators. Considering the case of mg structure, too, we may conclude thatnot the number of cells, but the number of units suitable for glide rcfiection (nowpairs of cells) should have the number odd, on a M6bius-band. Of the double-friezeand basic frieze patterns given in Figure 2, those which are suitable to fit onto aM0bius-band are given with black colour in Figure 5. (The transformations of thesenine double frieze patterns in the further parts of our paper will be always given inthe matrix of double frieze patterns, first given in Figure 2.)

THE MOBIUS-BAND TO TORUS TRANSFORMATIONAs we mentioned in advance, the M0bius-band - to - torus transformation con-sists of three steps: V -Fi~l - R. Instead of classic description we give these steps asa global transitional function.(Bb-1) Cut the MObius-band at middle between its edges, along its circle. Colouringof half-band helped to see this operation in Figure 6. (comment: this cutting sepa-rates the two half bands locally, but does not do it globally: the L-long MObius-band becomes a 2L-long one, 4 times half-turn-twisted belt band.)(Bb-2) Move the opposite sections (cells) of the 2L-long, twisted band, so, thatcoloured sides be the outer surface, and the two edges of the opposite sections

M~BIUS-BAND TRANSFORMATIONS BY CELLULAR A UTOMA TA 57

(cells) may be attached. (Comment: this operation may be substituted in a way asfollows: do not remove far the cut half-bands (with unit width) from each other, butinstead of it, slide one of the half bands behind the other, and so form the positionto contact opposite edges of the slidden cells.)

(Bb-3) Attach and glue the contacting edges. (Comment: two gluing lines areresulted in locally; the two opposite edges of any cells, but globally only one gluedline run around the torus. This glued single line globally turns ~- in cross sectioncircumference of the toms when it run 2~- along the great circumference. The fulllength of this single line is 4,n’:2~r - small:great circumferences. This is the charac-teristics to be generalized when used for tube-knots in the inverse transformations:torus (or tube-knot)-to-M0bius-band.)

GLOBAL TRANSITIONAL FUNCTION LOCAL TRANSITIONAL FUNCTION

Bb-l. Ba-l.

cuttingat middleline

Ba-2.

Bb-3. Ba-3.

slidingone ofhalf-bandsbehindthe other

gluing theoppositeedges ofcontactedcells

Figure 6: The global (left column) and the local (right column) transitional function in the cellular auto-matic formulation of the M6bius-band -- to -- torus transformation.

Figure 6 summarizes visually these steps both for the global and the local transi-tional functions. The program of our paper is to formulate the indirect problem: totranscript the global transitional function to the local transitional function. The

steps in the local transitional function can be easily followed according to the movements given for Bb-2 in comment. This is the Ba function:

(Ba-1) Contacts along band direction are preserved, separation in the perpendicu- lar direction is executed between neighbouring cells.

(Ba-2) 'Sliding behind' one half band (cell-ribbon) to the other.

(Ba-3) Contact and glue the free, opposite cell edges. The twolayered - glued - structure should be blown up to form a torus. This blown up tube exhibits the plane-symmetry pattern according to the neighbourhood relations between the cells in it (Fig. 7). (Gluing results in a kind of Born-Karman boundary condition for the cellular bands regarding their patterns.)

Figure 7: Transformations of Mobius-band -to -torus rearrange the pattern of the surface. These rear- rangements are summarized in the double frieze. matrix projected to the corresponding 'woven' plane- symmetry pattern matrix. The process how the transformation rearranges the pattern is shown form-m

double frieze pattern in figure 8. The inverse rearrangements are referred in Figure 9.

M~BIUS-BAND TRANSFORMATIONS BY CELLULAR A UTOMA TA 59

Ba-2.Bb-2.

Ba-3.Bb-3.

Figure g: Unification in a single sequence of sketches of the global and local transitional functions of thethree steps (given earlier in Fig. 6) in the M6bius-band --to --torus transformation for the case ofm-m

double frieze on M6bius-band -- towards thepm plane-symmetry structure on torus (or t-m doublefrieze pattern, if projected). Cells of the local model are represented by their abstract figure of triangles.The critical step of Ba-2 and Bb-2 shows how the sliding of one of the half bands behind the other forms

pm structure after the third step from m-m frieze structure.

SURFACE PATTERNS /IN THEIR DOUBLE FRIEZE REPRESENTATIONS/ OF THETORUS /OR TUBE-KNOTS/ SUITABLE TO THE INVERSE TRANSFORMATION

Figure 9: The initial surface patterns (given with their skinned and smoothed double frieze forms) on thetorns (or on the tube-knots) suitable to the inverse M6bins-band -- to -- torns transformation. All thesetransformations and initial patterns are valid for tubeknots which fulfil the conditions given in theBb-3

point of global transitional function. Two cases for the simple knot are shown in the right column.

THE TORUS- (OR TUBE-KNOT-) -TO-MOBIUS-BANDTRANSFORMATION

In the inverse (or reverse) of the M0bius-band - to - torus transformation (i.e., inthe torus-to-M0bius-band transformation) the surface cell-mosaic patterns givenin the lower matrix of Figure 7 are the initial conditions. These patterns are thosewhich may form a correct double frieze pattern on the M0bius-band after the in-verse M~bius-band- to - torus transformation.

M~BIUS-BAND TRANSFORMATIONS BY CELLULAR A UTOMA TA

Figure t0: Idealized drawing of a tube-knot with C5 rotational global symmetry withp2 type surface pat-tern (or 2-2 corresponding double frieze pattern), on its surface which suitable to the inverse M6bius-

band --to--tot-us transformation. This transformation forms g-2 double frieze structure on the surface ofthe M6bius-band-knot. Compare it to the simple tube knots of Figure 9.

The group structure of a knot can be understood easily, when we consider the knotas a representation of a frieze pattern woven from a strand, wound up around acentral point. Here on Figure 11 the two most simple knots are shown with theircorresponding frieze patterns.A (top) The simple knot is the representation of the frieze pattern of 2 (half turngenerated) wound up around a point: radius vector crosses two strands at once, butglobally one strand runs around two times, according to the suitable alternation ofthe 2 (half turn) frieze pattern. The 12 (twelve) repeating elements (correspondedto sections between strand-crossings) of the corresponded frieze pattern prove thatthey may not be commensurable with odd number of cell units of the double-friezepattern of the tube-knot with this knot-structure, if tube-knot is parcelled to cellsaccording to a double frieze pattern suitable to be transformed by inverse MObius-band-to- (tube-knot) -torus transformation.B (bottom) The knot with C4 rotational global symmetry (carrick band coaster) isthe shortest frieze pattern representation of g (glide reflection) type structurewound up around a point: radius vector crosses three strands at once, but globally

s. BP.RCZt

one strand runs three times, according to the alternation in the woven g frieze pat-tern (coloured with three tones). The 24 repeating elements (corresponded to see-tions between strand crossings) of the corresponded frieze pattern prove, that thefrieze pattern of the knot structure may not be commensurable with the structureof the double frieze pattern of the tube-knot with such structure, if its parcelling tocells happened so that the double frieze pattern was suitable to have been trans-formed by inverse M0bius-band-to-(tube-knot)-torus transformation. Bothexamples shown on Figure 11 intuitively prove that structural hierarchy of tubeknots contains hierarchy levels independent of each other.

A The knot in stretched form

Two color frieze pattern representation of 2 frieze pattern

A

-

I

Three color frieze pattern representation of g frieze pattern

¥1gure 11: Wound up frieze pattern representation of the woven structure of the two most simple knot-structures.A Simple knot with C3 global rotational symmetry and wound up 2 (half turn) frieze patternstructure around a point when considered its structure locally between strand-crossings. B Knot with C4rotational symmetry (global one) and wound up g frieze pattern structure around a point when consid-ered its structure locally, between strand-crossings. Global and local structure is incommensurable in

respect of the inverse M6bins-band --to--(tube-knot)- toms transformation if strand is a tube.

M6BIU$-BAND TRANSFORMATIONS BY CELL~ A UTOMA TA

CONSTRUCTION OF KLEIN-BOTI’LE BY DOUBLING M~BIUS-BANDS

them at their edges one by the other in mirrorsymmetric position result in known topologicalsurfaces: the Klein-bottle (see Fig. 12). Doublingconsists of two parts. First one is the doubling ofM0bius-band cellular background by a mirrorreflection. Second one is the doubling of the suitablepatterns: this may be carried out according to fouroperations shown in Fig. 2 (by t, g, m, and 2 by-generators). In this operation pairs of cells work asgenerators.On the other hand, doubling of the pattern may beaccomplished in two variants. In the case of normaldoubling pattern of the doubled pair is in phase withthe initial one. A modification of this normal case isthat doubling may not be not-in-phase. In the case ofdoubled t, g, m, and 2 lines there is one (for mg three)possibility to slide doubled pattern one cell unit. Thisvariation results in new doubled MObius patterns onthe Klein-bottle, (see Fig. 15).

Figure 12: Construction of Klein-bottle bymirror reflection o[ a M6bitts-band.

SUMMARY AND CONCLUSIONSDifferent ways of pattern generations from a single, ~(~constructed, basic frieze pattern were constructed andanalysed in this paper. After the constructive ~.o,.definition (in combinatorical way) of basic frieze ~patterns, first the double frieze patterns were ~generated by a neighbour-state generator localoperation. Then those double friezes were selectedfrom the double frieze matrix, which are suitable to fit them onto M6bius-band.The most important and interesting results shown in this paper were the cellularautomatic description of the M6bius-band-to-torus transformation and theimplicated pattern rearrangements in the double frieze patterns constructing them.These results may have importance in description and modelling oftransformations and reproductions of molecular double-threads (especially that ofviruses).Different developments of the construction by local doubling operations wereshown. It was shown for tube-knots, that their knot-hierarchy level is independentof the surface cell-mosaic structure of the tube, and the group structure of the knotis incommensurable with the numerosity of the cell-mosaic pattern of the tube,

64 ~. B~RCZI

when it is suitable to the inverse M0bius-band-to-(tube-knot)-torustransformation. Finally, further constructions from M0bius-bands with doublefrieze cell-mosaic patterns were shown to build Klein-bottle from them. There wereused also a local doubling operation to repeat the structure of the initial MObius-band. It was sketched how can be constructed new Klein-bottle surface patternsfrom M0bius-band double friezes by t, g, m, and 2 doubling, and in-phase and not-in-phase cases were also shown to enrich the number of pattern variations.

Figure 13: Doubled double frieze patterns on M6bins-bands form patterns on the Klein-bottle whichmay be produced by gluing a pair of M6bius-bands in a mirror symmetric position. The two examplesshow the initial and the final situation for such operations, accomplished according to the deformation

sequence sketched in Figure 12.

66 $. BI~,RCZI

All the constructions shown in our paper are independent of structural hierarchylevel in the evolution of matter. Therefore phenomena which can be modelledusing up such constructional or transformation processes may be found ondifferent levels of structural hierarchy. But mainly molecular level of structuralformation is the hierarchy level in focus, when these transformations are discussed.On this level it is an important conclusion from our paper, that M0bius-bandstructures are very rarely observed, because of the strong criterion of g symmetryoperation which is necessary condition for the existence of M0bius-band doublethread structure. But g operation contains reflection, which is not a motion, somolecular enantiomorphy should be used up in such constructions. Alternatingselection of enantiomorphous molecules seems almost impossible process innature. Only small number of elements long M0bius-band structures may beexpected or may be hoped to be found in natural circumstances. But the possibilityof constructing them is open. Construction of molecular double thread structureswith M0bius-band global structure is a challenge to chemists and may open a newfield in chemistry. On the other hand it may be interesting field for theoreticalcalculations (with small number of elements) for quantum chemistry, too. All these’predictions’ are valid for structures with both M0bius-band structure and with aknot superstructure, too, but it seems a far future program.

IN-PHASE NOT -IN-PHASE

Flgur~ 15: In-phase and not-in-phase doubling of the 2-m pattern of the initial M6bius-band (left part ofeach pairs in the figure). In-phaso doubling results in cram pattern (left double-M6bius-band), not-in-phase doubling results inpmg pattern (right double-M6bius-band) according to local neighbourhoodrelations of cells (with the triangle pattern element) in the final pattern of the Klein-bottle. The local

structure can be observed when the patterns were stretched onto the plane; the global structure can beexpressed, when the earlier pattern were given with the so called Born-Karman boundary conditions,

which represent the feedback of the structure on itself, when it were closed.

M6BIUS-BAND TRANSFORMATIONS BY CELLULAR A UTOMA TA 67

REFERENCES

B6rczi, Sz. (1976) N6v~nyi szimmetri~ik, [Plant symmetries, in Hungarian], Fizikai Szemle, 26 (2), 59-62.B~.rczi, Sz. (1979) A szalMlyos ~s f~ligszab~ilyo~ (platoni ~ archimed~szi) testek ~ mozaikok ped6dusos

rendszere, [The periodic system of regular and semi-regular solids and mosaics, in Hungarian],Kfz~piskolai Matanatikai Lapok, 59 (5), 193-199.

B~rczi, Sz. (1980) The periodic system of Platonic and Archimedean solids and tessellations, Acta Geo-logica Academia Sciencia Hungarica, 23 (1-4), 184-200.

B~rczi, Sz. (1985) Symmetries in the plant surface lattice systems: Development of Fibonacci numberedstructure in a cellular automaton model, Lecture at Intuitive Geometry Conference, Balaton-sz6plak, May, 1985.

B6rczi, Sz. (1986) Escherian and Non-E.schedan developments of new frieze types in Hanti and oldHungarian communal art, In: Coxeter, H. S. M. et al., eds., M. C Escher:Art and Science, Amster-dam: North-Holland, pp. 349-358.

B&czi, Sz. (1986) New double-frieze types in the Avarian communal art, lecture at the Symmetry Sym-posium and Exhibition, Darmstadt Technical University, Darmstadt, June 13-17, 1986.

B~.rczi, Sz. (1987) Symmetry constraints in development and evolution of Fibonacci-plants, paper pre-sented at the Symposium on Organizational Constraints on the Dynamics of Evolution, Budapest,June 29 - July 3, 1987.

B~rczi, Sz. (1989) Symmetry and technology in ornamental art of old Hungarians and Avar-Onoguriansfrom the archaeological finds of the Carpathian basin, seventh to tenth century A.D., In: Hargit-tai, I., ed., Symmetry 2, Computers and Mathematics with Applications, Oxford: Pergamon Press, 17(4-6), 715-730.

B6rczi, Sz. (1990) Symmetry constraints in development and evolution o1~ Fibonacci plants, In: Maynard-Smith, J., Vida, G., eds., Proceedings in Non Linear Science (1V), Organizational Constraints on theDynamics of Evolution, Manchester. University Pre~s.

B6rczi, Sz. (1990) Szimmetria ~s Stndaura(pit~s, [Symmetry and Structure Construction, in Hungarian],Lecture Note Series, J3-1441, Budapest: TankSnyvkiad6, 260 p.

B~.rezi Sz.(1990) Local and Global Model o[ Fibonacci Plant Symmetries, In: Gruber, B. and Yopp, J.H.,eds., Symmetries in Science 1~, Biological and Biophysical Symmetries, New York: Plenum Pre~s.

B&czi Sz. (1990) Szimmetria ~ topol6gia: R~cs~trendez6d~ek a M6bius-szalag -- t6rusz transzform~-ci6 sor~in, Term(~zet Vil~ga, 10, 464-466.

B~rczi Sz., LukAcs B., and Moln~Ir I. (1991) On Symmetry and topology of the organisms in macroevolu-tion, In: Lulratcs, B., B6rczi, Sz., Moln~ir, I., and Pmtl, Gy. eds., Symmetry and Topolog~ in Evolu-tion, Budapest: MTA-KFKI-1991-32/C, pp. 73-79.

B6rczi, Sz. (1991) Symmetry and topology in a ’cell-mosaic-automata’ model of the Fibonacci-plantstructures, In: LukAcs, B., B6rczi, Sz., Moln~tr, I., and Pafil, Gy. eds., Symmetry and Topolog~ inEvolution, Budapest: MTA-KFKI-1991-32/C, pp. 80-90.

B6rczi, Sz. (1991) Platonic-Archimedean spherical cellular automata in the solution of the indirect Von-Neumann problem on sphere for transformations o1~ regular tessellations, In: Luk,~cs, B., B6rczi,Sz., Moln~ir, I., and Pa~il, Gy. eds., Symmetry and Topolog~ in Evolution, Budapest: MTA-KFKI-1991-32/C, pp. 111-116.

B~.rczi, Sz. (1991) Symmetry by cellular automata, Lecture on the Intuitive Geometry Conference,Szeged, September 1991.

Bilinski, S. (1952) Homogene Netze geschlo~sener orientierbarer Flachen, RAD Bulletin International,277, Zagreb: Academie Yugoslave des Sciences et des Beaux-Arts, 129-164.

Codd, E.F. (1968) Cellular automata, New Yorlc Academic Press.Coxeter, H.S.M. (1969) Introduction to Geometry, 2nd ed., New York: John Wiley and Sons.

s. S~RCZ~

Orunbaum, B. (1987) T’dingx and Patterns, New York: W. H. Freeman and Co.Hargittai, I. ed., (1989) Symmetry 2, Oxford: Pergamon Pre~.Koch, S. A. (1990) The evolution of viruses, In: Lak~cs, B. et al., eds., Evoitaion.. Cmmogene~ to Bio-

gen~ds, Budapeat: MTA-KFKI-1990-50\C, pp. 69-76.LAbos, E. (1991) Attractotx and Imots, In: Luk~cs, B. et al., eds., Symmetry and Totxdog~ in Evolution,

Budapest: MTA-KFKI-1991-32\C.Lak:lc~, B. (1993) On geometric symmetries and topologies of forms, Symmetry: Culture and Science, 4, 1,

93-103.yon Neumann, J. (1966) Theory of Self-Reproducing Automata, (ed. and completed by A. W. Burks),

Urbana, Illinok.Schattschneider, D., and Walker, W. (1977) M. C F~cher ga/eido~/es, New York: Ballantine Bool~.Senechal, M. (1986) F.~her designs on surfaces, In: Coxeter, H. S. M., eta/, ed~., M. C E~ch~: An a~d

Science, Amsterdam: North Holland, pp. 97-109.Vollmar, R. (1978) Al~,ithmen in Ze//u~’maomaten, Stuttgart: Verlag B. G. Teubner.Weyl, H. (1952) Symmaq, Princeton: Princeton University Press.

Symmavy: Culture and ScienceVot 4, No. 1,1993, 69-81

PENTAMOND: A NEW CRYSTALLINEMODIFICATION OF CARBON

Gfibor G~vay

Chemist, cry~tallographer, (b. Kapmvtr, Hungary, 1952).Addm.~. Bolyai Institute, Unive~ity o~ Szeged, Aradi v~-tan0k terc1, Szegcd, H-6720 Hungmy.Fields of intere~. Ct3~tallography, convex geometry, pt-edictivegeometric modeling in chemistff, c~tallography and biology.Pub//cations:. Growth and characterization of Bi4Ge3012 singlecrystals, Pro~e~ in Oy,Jm/Grott~h and Characteriza6on, 15 (1987),145-186; A structural model of the sporopollenin based on dodeca-hedrane units, (with M. Kedves), Acta Biologica, Szeged 35 (1989),53-57; Non-metallic quasicrystais: Hypothesis or reality? Pha~eTra~’itions, 40 (1993), 47-50;, Ieosahedral morphology, In: Hargit-tai, I., ed., Fivefold S~nmu~, Singapore: Wortd Scientific, 1992;Kepler hypersolidrb ~ o~ethe Mathenmfica/Society J. Bo/yai,63, Intuitive Geometry, Amsterdam: North Holland, 1993.

Abstract Slight modifications of a crystal structure atdifferent org~ levels result in new structures related to each other. This possi-bility seems not to be exhausted entirely by Natur~ In this contribution a hypotheticalcarbon polymorph is described which consist$ of five-membered (C5) ring~ in contrastto the known existing polymorph$. Some simple calculations make the model moreplausible.

1. INTRODUCTIONAccording to the well-known point of view, a crystal is a (theoretically infinite)triply periodic material pattern in which the repetitive motifs are atoms (or sets ofatoms) (Feynman, Leighton and Sands, 1964; Kittei, 1961; Buerger, 1965). Thenature of the transition from the atomic (or molecular) organizational level to thesolid state level is an intricate problem and is far from being well understood; recallfor example the high temperature superconductivity.

Prior to going into the physical details, one may scrutinize the purely structuralaspects of this spatial order. This is, admittedly, the typical standpoint of the’classical crystallographer’ (Buerger, 1965). In this approach, the emphasis is on theconcept of symmetry, in fact, that of the global symmetry (the mathematical outputof this is the group theoretical formulation of geometrical crystallography). How-ever, one may not and must not ignore that there is an underlying topology that

precedes the global symmetry. Indeed, the traditional classification of the polyhe-dral-frame structures of minerals (especially those of the silicates) is an implicitapplication of this topological aspect (Bragg, Claringball and Taylor, 1965; Zoltaiand Stout, 1984).By way of illustration, take some representative of covalent crystals. In an abstractsense, the crystal structure of diamond can be interpreted as an (infinite) regulargraph of degree four (the C-C covalent bonds being the graph edges and the Catoms being the vertices). Of course, this particular topology of the network of car-bon atoms is made possible b~ the nature of the vertices. Here the valence of thevertices is four, due to the sp~ hybrid state of the carbon. Although this valenceadmits an infinite variety of linkages, the number of possible cases is strongly lim-ited by the loca/symmetry of the vertices. This local symmetry results from thehighly directional character of the covalent bonds, and in the case of carbon sp3orbitals it is equal to that of the regular tetrahedron. As for the realization, theactually known cases are: diamond with its cubic (global) symmetry and a rarehexagonal polymorph called a lonsdaleite (Zoltai and Stout, 1984). (They exemplifythe fact that the valence and the local symmetry of the vertices together do notdetermine the actual crystal structure uniquely even on the level of topology.)

The relation of these hierarchy levels can be summarized in the following diagram(Fig. 1).

Votence of thevertices

13lobol symmetryof the structure

Fight= t: Hierarchy levei~ in a crystal structure.

PENTAM OND 71

What is the heuristic significance of such considerations? Well, it may draw atten-tion to the inexhaustible possibilities of matter. In the present contribution a hypo-thetical model is given which will perhaps exemplify these ideas.

The recent discovery of the ’third modification of carbon’, (’C60 fullerene’,’footballene’, etc.) (Kroto, 1989) demonstrated, among other things, that the car-bon atom in sp2 hybrid state is able to build up a structure topologically distinctfrom that of the graphite. Both structures form, in an abstract sense, a regulargraph of degree three, but in the C60 structure the original local D3h symmetry ofthe carbon atoms is somewhat violated. At the same time, among the 6-circuits ofthe graph 5-circuits (five-membered rings of atoms) occur as well. All this occurs atthe expense of some bond angle distortions which the system obviously tolerates.Now the question is whether the topology of the diamond can be changed so thatthe valence is preserved but the six-membered rings (6-circuits) are replaced byfive-membered ones.

2. THE PENTAMOND STRUCTURE

Take a tiling composed of pentagonal tiles as shown in Figure 2 (solid line; planegroup: p4g). Let a pentagonal unit correspond to the skeleton of the cyclopentane(CsH10) molecule. It turns out that such a quasi-two-dimensional net can be con-structed with relatively small angle distortions (with respect to either 109° 28’ or108°). So, our tiling is merely a plane projection of a spatial structure in which thesymmetrically repeated units are not the cyclopentane skeletons but those of alarger molecule shown in Figure 3a (plane projection) and in Figure 3b(perspective view). This latter molecule is a bridged-ring system to which thesystematic name tetra-cyclo-[6,2,1,13,6,011,12]-dodecane may be given (for thenomenclature, see e.g., (Barton, 1979)). Figure 3b shows that it resembles a cradle.If the cradles in Figure 2 are thought to be oriented as in Figure 3a, in the samestructure one finds cradles upside-down which are oriented in the other diagonaldirection.

Now reflect this structure in an appropriate horizontal mirror plane and at thesame time shift it in a diagonal direction by a length half of the cradle (glide reflec-tion). Another horizontal net is obtained (dashed line in Fig. 2). Connect these twolayers in an appropriate way (dotted lines in Fig. 2). Infinite repetition of this pro-cess results in a three-dimensional structure (note that repetition of the horizontallayers can be carried out by various other types of compound symmetry transforma-tions).

It is like diamond (an infinite regular graph of degree four) and different from dia-mond (composed of 5-circuits instead of 6-circuits in the horizontal layers). In fact,the structure of diamond can be conceived of as an infinite repetition of cyclohex-ane skeletons of chair conformation, in a similar manner (for comparison, see twoof such layers of diamond projected onto a (111) plane in Figure 4; note that lons-daleite only differs in that the layers completely cover each other when viewed fromabove (Zoltai and Stout, 1984; Merlino, 1990), accordingly, the cyclohexane rings

G, GI~VAY

Figur~ 2: Projection of a layer of pentamond structure o~ the (001) plane (with the unit cell indicated).Dotted fines represent bonds connecting the uppe~" pentagonal net (solid lines) to the lower pentagonal

net (dashed lines); for eask~ visualization, they are only drawn within the marked unit cell.

PENTAMOND 73

providing the interlayer connection are of eclipsed conformation (Balaban, 1989)). Referring to the similarity and dissimilarity, we coined the name pentamond (pentagon + diamond) for this structure.

Figure 3: A repetitive unit of pentamond consisting of 12 C atoms (4 of type a, 8 of type p); (a) plane pmjection; (b) perspective view.

One can distinguish two types of carbon atoms according to their role in the struc- ture: C(a) and C@) (cf. Fig. 3a). The C(a) atoms belong completely to a horizon- tal layer, i.e., they have no interlayer bonds (check this type of vertex in Fig. 2). Accordingly, they are shared by four cyclopentane rings within one and the same layer. The function of the C@) atoms, on the contrary, is twofold: not only are they common vertices of (three) cyclopentane rings in a layer, but they are responsible for the interlayer linkage as well (cf. again Fig. 2).

Figure 4: Projection of a layer of diamond structure on the (111) plane.

3. CRYSTALLOGRAPHIC DESCRIPTIONIt is not difficult to see that we are given a tetragonal structure with a unit cell suchas indicated by a square in Figure 2. The height of the unit cell is twice the inter-layer distance (the latter measured by the distance between two C(a)-s one over theother in adjacent layers).

The atomic positions are as follows:

PENTAMOND 75

4 (~,): O, O, O; ~, ½, O; O, O, ½; ½, ~, ’A;8c0 ): x, y, z; yc, y, z; y, y, g

.F_, y, lh+z; x, y,, lh+z; y,x,z,, y,x,zThus the unit cell contains 12 C atoms.By inspection, one can find the following symmetry elements in the unit cell.1) 7fvertical rotoinversion axes: with fixed point at C(a) atoms;2) 42 vertical screw axes: through the midpoints of horizontal edges;3) 21horizontal screw axes: parallel to edges, at the one fourth and three-fourths ofthe horizontal edges, at elevation 0 and ~;4) 2 horizontal rotation axes: parallel to horizontal face diagonals and betweenthem at elevation 1/4 and 3/4;5) n horizontal glide planes: at elevation 1/4 and 3/4;6) c vertical glide planes: coinciding with vertical cell walls as well as halfwaybetween them;7) m vertical mirror planes: perpendicular to horizontal face diagonals at onefourth and three-fourths of them;8) 21 horizontal screw axes: parallel to face diagonals, over midpoints of horizontaledges, at elevation 1/4 and 3/4;9) n vertical glide planes: through horizontal face diagonals;10) ~" inversion centres: in the centre of 1/8 cells.Thus the space group is:

D164h (Schoenfiies symbol) orP4/ncm (international symbol) orP~n-z ~ ~m---~omplete Mauguin symbol).

The corresponding space group diagram is given in Figure 5. (We note that the listof symmetry elements, the diagram in Figure 5 and the space group pairwise deter-mine each other taking into account the standard interpretation of the symbolsespecially that given in the International Tables (Buerger, 1971). Although theseare well known for crystallographers, here we risked the redundancy in order tomake our contribution self-contained.)Accordingly, the point group of pentamond is D4h - 4/mmm (the crystal class is atetragonal holohedry).

glgu~ 5: Space group diagram of pentamond.

A three-dimensional view of the unR cell of pentamond is shown in Figure 6. It isdrawn in accordance with the numerical values given to the parameters a, c, x; y andz in the next section. It is seen that the six-membered rings are not completely’eliminated’ from the structure. In fact, each C6 ring is linked by its opposite edgesto a next one, forming infinite ribbons of width c/2. The orientation of these rib-bons is (110) and (110) (cf. upper half and lower half of the unit cell, respectively).Two perpendicularly oriented ribbons are joined by a common carbon atom sharedby an upper and a lower C6 ring.

4. TENTATIVE CALCULATIONS

One can easily observe in Figure 2 that there are three distinct types of bondsbetween the two types of carbon atoms: a-/~,/3-/3 and//-B’ (a prime denotes C~)type atom located in an adjacent layer). The number of combinatorially possiblebond angles on an atom of valence four is six, which is reduced by the local symme-try on the atom in question.

PENTAMOND 77

Figure 6: The tctragonal unit cell o[ pcntamond, with coordinates ofa C(fl) atom indicated.

Now observe that on the C(a) atoms the original local symmetry reduces accordingto the scheme: 43m -~ 42m (or Ta.,,~D2d in Schoenflies notation). Hence, we havethe following two types of angles here:

~(fla~, ,n’): it is projected into a straight angle on the (001) plane;

*0~, ~r/2):Fig. 2).it is projected into a right angle on the (001) plane; (cf.

On the C(fl) atoms, at the outset, we can only assume the local symmetry Cs-mwhich is part of the global symmetry. Accordingly, we have the following angles:

,I,(o¢.),where the last two occur twice owing to the presence of the mirror plane m.Our approach is based on the assumption that the well-known carbon-carbonsigma bond length remains unchanged throughout the structure of pentamond. Onthe other hand, the bond angles listed above are allowed to deviate from the idealtetrahedral value. It turns out that the conditions

At) ~onstant bond length

A2) the given local and global symmetry

only allow one fre~ parameter for the structure calculation (global point groupD 4h - 4 /mmm ).The starting point is the angle ~(~a), where the deviation from the ideal arccos(-1/3)ffi109° 28’ value is measured by a ~ parameter. With this starting angle, somecalculations yield the following formulas.

cos4,(~,~r) ffi ½+2~’-4~e

cos~(~,~’/2) ffi -3/4-8’+2~

where ~’= {2/3-~)W.

O)(2)(3)(4)

0e

0.19

0.30

V(~’)

-69

Figure 7: The dependence of angle distortions (left ordinate axis) and unit cell volume (right ordinateaxis) on the ~ parameter.

PENTAMOND 79

Using the formulas (2-4), the functions 6 -~ I A I, where A = q~ - arc cos (- 1/3),are plotted in Figure 7. The intersection points of curves (~Sa) and (fl/3/3’) may beconsidered as that representing an optimum of the structure (within the limits ofour approach). The corresponding value, 6 = 0.1911, was determined by Newton’smethod taking the difference of the two functions.For this ’optimal’ structure, the cell dimensions can be obtained using the followingformulas:

a = 2.2½4.8’ (5)C ---- 2/({3+z1~’-8~’2}½+2{8’-~’2}½), (6)

where I is the C-C sigma bond length. Its actual value, 1 = 1.5445 A, is calculatedfrom the lattice parameter of diamond (a=3.5668 [ (Zoltai and Stout, 1984)),assuming the carbon atoms to be hard non-overlapping spheres with radius r=l/2.Hence, the cell dimensions are:

a -- 3.3014 ~, c = 6.3795 g (7)Expressing the C(fl) coordinates used in the preceding section in these units oneobtains:

x=0.3346, y=0.1654, z=0.1460. (8)(We note that these numerical values have been applied in Fig. 6.)The bond angle values are obtained partly by formulas (1-4). Those for which noexplicit formula is given are calculated from the coordinates (8). The results aresummarized in Table 1.

Results of bond angle calculation in pentamond at 8= 0.1911

Symbol of angle ¯ A

(a~a) 98* 11’ - 11" 17’(flail,w) 105" 51’ - 3* 37’(fl~,~r/2) 111° 19’ 1" 51’(fl/3fl’) 120° 45’ 11° 17’(~fl) 1~0 49’ - 39’(~fl’) 112" 47’ 3" 19’

Equality of the last two numerical values (o43/3 and aflfl’) in this Table would meana C2v-mm2 local symmetry on the C(fl) atoms. Since they differ, the originallyassumed symmetry does occur (viz. Cs-m).

It is seen that the angle strains are relatively small, except for the case of _+11° 17’.Unfortunately these latter cannot be simultaneously reduced, as we saw just now(cf. Fig. 7). Here we have to suppose that the structure will tolerate this amount ofstrain. The unit cell volume has also been calculated as follows:

(9)Its dependence on the 8 parameter is shown in Figure 7. It has a maximum in thevicinity of our optimum point. At this optimum, its value is V -- 69.551 ~3.

We compare the packing efficiency in the structure of pentamond and diamond.This number is usually given as the percentage of the volume occupied by (hardspheres of) atoms in the unit cell. It is well known and is easily checked that in thediamond structure it is equal to (q~’~-/16) x 100% = 34.01%. The ratio of packingefficiencies in the two structures can be obtained by the quotient

(10)

where Vp, VD and Zp, ZD are the unit cell volume and the number of atoms in theunit cell of the two structures, respectively (note that this ratio can be considered asthe reciprocal ratio of molar volumes as well). By appropriate substitution weobtain: q(P/D) = 0.9786. Hence, the packing efficiency in pentamond is 33.28%,that is, a bit ’worse’ in comparison with diamond.

If q(P/D) were greater than one, it would provide the theoretical possibility ofpreparing pentamond by some high-pressure technique. Our model, based on theassumptions A1 and A2, does not feed such hopes. However, as we shall see atonce, the situation is not quite hopeless.

5. CONCLUSIONSIt is to be emphasized that our approximation is merely a first approximation andone of the simplest possible. Indeed, the assumptions A1 and A2 are relativelystrict. They can be relaxed, both separately and jointly. Of course, this can be doneat the expense of an increased number of free parameters. A simultaneous treat-ment of more than one degree of freedom requires finer methods which would bethe object of subsequent contribution(s).

Such ’structure refinement’ will possibly result in a more efficient packing of Catoms (while preserving the topology established in the present paper). A physicalconsequence of this would be the location of the region of existence in a high-pres-sure part of thep-T phase diagram of carbon. At the moment, this is an interestingand quite open problem both geometrically and physically.

Of course, the problem of locating the region of existence is preceded by the prob-lem of the possibility of existence. This refers not only to the natural occurrence ofpentamond but to its technical realizability as well.

Here a disturbing factor may be the amount of angle strain (in particular that origi-nating from ~(o#~a) and ~(j3flfl’)). However, we expect that refined (hence more

PENTAMOND 81

realistic) structure models would exhibit a more tolerable amount of strain (on theother hand, if carbon atoms are inclined to build up such a structure at all, theywould find the true optimum automatically).Since carbon atoms obviously do not overexert themselves to organize into penta-mond structure spontaneously (otherwise pentamond would be a well known min-eral such as diamond and graphite), here we refer to a relatively recent level (atleast in our terrestrial environment) of the evolution of matter: this is the’controlled evolution’ or technological stage. Well, here is a challenge to materialsscientists, solid state chemists and/or other experts on modern synthetic methodssuch as of catalytic vapour deposition, laser fusion and so on.

ACKNOWLEDGEMENT

The author is indebted to Szaniszl6 B~rczi for the discussions and assistance whichled to the present form of this contribution.

REFERENCES

Balaban, A. T. (1989) Carbon and its nets, Journal of Computers and Mathematics with Applications, 17(1-3), 397-416; Reprinted in: Hargittai, I., e~l, Symmetry 2: Unifying Human Understanding, Ox-ford: Pergamon Press.

Barton, D., series ed. (1979) Stoddard, J. E., ed., Comprehensive Organic Chemistry, Vol. 1, Oxford:Pergamon Press.

Bragg, L., Claringball, G. F., and Taylor, W. H. (1965) Crystal Structures of Mincrals, London: G. Belland Sons I./xl.

Buerger, M. J. (1965) X-Ray Crystallography, New York: John Wiley and Sons, Inc.Buerger, M. J. (1971) Introduction to Crystal Geometry, New York: McGraw-Hill Book Company.Feynman, 1L P., Leighton, R. B., and Sands, M. L (1964) The Feynman Lectures on Physics, Voi. 2,

Reading, M_A: Addison-Wesley Publ. Co., Inc.Kittel, Ch. (1961) Introduction to Solid State Physics, 2nd edition, New York: John Wiley and Sons, Inc.Kxoto, H. W. (1989) cB60 buckminsterfullerene, other [ullerenes and the icospiral shell, Journal of

Computers and Mathematics with Applications 17 (1-3), 417-423; Reprinted in: Hargittai, I., ed.,Synunctry 2: Unifying Human Understanding, Oxford: Pergamon Press.

Merlino, S. (1990) OD structures in mineralogy, in: Loreto, L. and Ronchetti, M., eds., Topics on Con-tempora~ Crystallography and Quasicrystais, special issue of Periodica Mineralogica, 59,69-92.

Zoltai, T. and Stout, J. H. (1984) Mineralogy: Concepts and Princip/es, Minneapolis, Minnesota: BurgessPubl. Co.

3)mmetry: Culture and ScienceVo~ 4, No. 1,1993, 83-92

CHIRALITY OF ORGANIC MOLECULESFerenc Glfick

Theoretical physicist, (b. Band, Hungazy, 1963).Address: Central Research Institute for Physics, Budapest 114,P.O. Box 49, H-1525 Hungmy.Fields of intere~. Particle physics, weak interaction, radiativecorrections.Pub//cations: Radiative correction to electron neutrino correlationin lambda//-decay, (with T6th, IL), PhysicalReview, D40, 1989, 119,Order-a radiative corrections for semileptonic decays ofunpolarized baryons, (with T6th, K.), Physica/Review, D41, 1990,2160; Order-a radiative corrections for semileptonic decays ofpolarized bazyons, (with T6th, K.), Physica/Review, D46, I992,2090; Measurable distributions of unpolarized neutron decay,Physica/Review, D47, 1993, 2840.

Abstract: A summary of the optical isomerism oforganic molecules and of the bioorganic optical purity (biomolecular handedness) isgiver~ The origin of the optical purity is explained by means of spontaneous symmetrybreaking and the pari~y violating weak interactior~

1. OPTICAL ISOMERISM IN ORGANIC CHEMISTRY

Generally, isomers are compounds with the same molecular formula which differ inat least one chemical or physical property. For example, two isomers correspondingto the C2H60 empirical formula:

H H H HI I 1 I

~-C----C--O H H-c--o-- C-F41 I I IH H H H

Ethanol Dimethyl ether

84 F. GLOCK

There are several kinds of isomerism (chain, position, geometrical, etc.). Thedetermination of the shape of the molecules and the relative arrangement of itsparts in space is the most important task in organic chemistry.

Optical isomerism

Molecules that exist in two different forms related as object and mirror image arecalled optical isomers (enantiomers). The spatial structure of these molecules isasymmetrical: they have neither planes of symmetry, nor centers of symmetry. Theyare said to be chiral. Chirality refers to the ’handedness’ or the screw sense (left orright) of an object. Those objects that are not identical with their mirror images aresaid to possess chirality, or handedness (familiar examples are hands or screws).Molecules containing four nonidentical atoms or groups of atoms linked to acentral carbon atom provide a typical example for optical isomerism. The carbonatom in these molecules is often called asymmetric. The picture below shows thegeneral structure of the two enantiomers:

Figure 2

Here the A, B, C and E groups sit in the vertices of a regular tetrahedron, the Ccarbon atom is in the centre. The tetrahedral orientation of the carbon valencies isessential for this molecular structure.The planar representation of the above configurations is the following (Bentley,1969):

iFigu~3

I

CHIRALITY OF ORGANIC MOLECULES 85

Optical activity

The optical isomer molecules are capable of rotating the plane of polarization oflinearly polarized light (Gliick, 1993; Section 6). They are said to be opticallyactive. The two enantiomers rotate the plane of polarization in opposite directions,with the same angle. We denote the enantiomers rotating the plane in clockwisedirection by the + symbol, the others by -.

The optical activity of organic compounds provides many important applications. Ifa linearly polarized light beam passes through a solution containing an opticallyactive compound, the rotation angle of the plane of polarization dependsapproximately linearly on the concentration of the compound. One can use thisdependence for concentration measurements. The rotation angle (a) also dependson the wavelength (X) of the incident light. The a(X) function gives the ORD(optical rotation dispersion) spectrum. The right circularly polarized light isdifferently absorbed from the left circularly polarized light by optically activecompounds. The different absorption is also wavelength dependent, and thisdependence gives the CD (circular dichroism) spectrum. The ORD and CD spectraare characteristic properties of the optically active molecules, and theirmeasurements provides an excellent tool for structure analysis of complicatedorganic molecules (Crabb6, 1972; Damjanovich, 1976; Garay, 1970).

L and D enantiomers

The two enantiomers of optically active compounds are denoted by the L and Dletters. They come from the Latin words laevus (left) and dexter (right), byconvention, and have nothing to do with the direction of the plane rotation (boththe L and the D enantiomers can have + and - rotation properties).

The D and L enantiomers of the glyceraldehyde have the following configurations:

H1

---C--oH HO---c--.HcH OH

D(+) L(-)

Figure 4

~. GL~ICK

The absolute configuration (the spatial arrangement of the atoms) of themolecules can be determined by X-ray crystallography. This is, however, a ratherdifficult measurement. Fortunately, many molecular configurations can be tracedback to the configuration of the glyceraldehyde.

The absolute configuration of the L-a-amino acids (the most important buildingblocks of life) is the following:

COOk/i

zN--c -W1

Here R denotes any groups of atoms. For the giycine, R=H, this molecule isoptically inactive. The other ~-amino acids have optical activity.

The L and D enantiomers of the optically active compounds have the same physicaland chemical properties (except of the direction of optical rotation). This is theconsequence of the mirror (space reflection) symme,try of the electromagneticinteraction (Glfick, 1993; Section 3), since this interaction determines the chemicaland the most important physical properties of the molecules. Laboratory synthesisof optically active compounds from inactive sources produces the L and Denantiomeric forms in equal quantity (the result of this synthesis is called racemicmixture of the compound or racemate).

2. BIOORGANIC OPTICAL PURITY (BIOMOLECULAR IImI)NESS)Living beings contain many optically active compounds (amino acids, sugars, etc.).It i~ rather strange and puzzlin~ however, that they are built only from oneenantiomeric form of these compounds. All living organisms use the sameenantiomers! For example, proteins are built only from L-a-amino acids, DNA (thecarrier of the genetic information) contains only the D forms of sugars. We call thisproperty of life opt/ca/purity. It could be one of the most important evidences ofthe common origin of life on the Earth.

Measurements of optical rotation and CD spectra show that proteins of livingbeings (with L-a-amino acids) have right-handed helical structure (~-helix).

CHIRALITY OF ORGANIC MOLECULES 87

Synthetic proteins built from D-~-amino acids have left-handed helix. Proteinsbuilt from amino acid racemate contain both L- and D-amino acids, and have eitherleft- or right-handed helix. The growing rate of these mixed proteins is, however, 20times slower than the growing rate of the pure proteins which contain either L- orD-amino acids only. Also, these mixed (LD) hehces are unstable, and their catalyticactivity is very feeble. On the other hand, the DNA and RNA molecules cannot bebuilt from mixed nucleotides (containing both L and D sugars).It seems from these and many other examples (Garay, 1970): life requires almostperfect optical purity (chiral asymmetry).There are also many examples to show the connection between optical purity andbiological organization. Cancerous tumours contain many D-amino acids. D-aminoacids play an important role as components of bacterial ceil walls. There are otherinferior living beings (e.g.: funguses, insects) that contain also D-amino acids. Itappears, however, that no D-amino acid has been isolated from a properlycharacterized protein, and that D-amino acids have not been isolated frommammals (Bentley, 1969, Garay, 1970).Because of the chirality of its key molecules, human chemistry is highly sensitive toenantiomeric differences. An extreme example came to light in 1963 when horriblebirth defects were induced by thalidomide. The defects were caused by the fact thatwhereas one enantiomer of this chiral compound cured morning sickness, the othercaused birth defects. This example shows the importance of the separation ofenantiomers in pharmaceutical industry. Another example is the limonene: oneenantiomer of this compound smells like lemons, the other like oranges.

3. ORIGIN OF THE OPTICAL PURITYDue to the mirror symmetry of the electromagnetic interaction, the chemicalreactions are mirror symmetric. Therefore, the L and D enantiomers of chiralmolecules arise from achiral molecules in equal amount. How could then thebiomolecules have arisen with complete chiral asymmetry?To answer this question, we distinguish two possibilities: the biomolecularasymmetry could have arisen either before or after the appearance of life.

(a) After the genesis of life

The first cell could have contained, by chance, proteins composed entirely of L-amino acids. This is, however, rather unprobable. Another possibility is that thefirst cell was created, also by accidental fluctuations, with a small excess of L-aminoacids or D-sugars, and so incorporated only a slight chiral asymmetry. Then thetotal asymmetry developped by natural selection: the optical purity could beselective factor during the evolution (we have seen above: the biological processesfavour the chiral asymmetry) (Hegstrom and Kondepudi, 1990).

88 F. GLOCK

(b) Before the genesis of life

(I) Spontaneous symmetry breaking with perfect symmetryLet us assume first that the physical and chemical processes have perfect chiral(mirror) symmetry. Paradoxical though it may seem, mirror-symmetric chemicalreactions can produce, under special circumstances, unequal amounts of L and Denantiomeric forms through a phenomenon called spontaneous symmetrybreaking. In this case, a symmetric state is one with equal number of L and Dmolecules; the asymmetric state is one in which one form dominates. Spontaneoussymmetry breaking is a mechanism by which a system, with symmetric laws ofnature, ’spontaneously’ goes from a symmetric state to an asymmetric one (Luk~cs,1993).Let us consider the following model scheme of reactions (Hegstrom andKondepudi, 1990; Kondepudi and Nelson, 1984); Kondepudi and Nelson, 1985):

A + B ** XL or XD (1)A+ B+)~LD **

@~D(2)

A + B + ~* (3)XL + XD =* P (4)

In this scheme, the chiral species X in the two enantiomeric forms XL and Xo isproduced from the achiral substrate A and B directly through reaction (1) andautocatalytically through reactions (2) and (3). XL and XD may also annihilate eachother by producing a product P. With a suitable supply of theA and B compounds(to maintain their concentrations at a fixed level), the system can be driven far fromthe chiral symmetric state.Let us define the 8 asymmetry and the X critical parameters of the system as:

~ := (% - co)/(cz + co) ; X := ca¯ c,

a~>0(A<K) : A=0 a2 < 0 (A > K) : A =-4-d

Figure 6

CHIRALITY OF ORGANIC MOLECULF~

Here eL, cD, cA and cB are the concentrations of the XL, XD, A and B compounds.Let us assume that the free energy of the system has the following 8 dependence:

The system has perfect chiral symmetry: E(~) = E(-6). The a2 and a4 coefficientsmay have the following X dependences: a 2(X)=K-X, a, (X) = const. The systemreaches its equilibrium when its free energy is minimal. AS it is shown in the figuresbelow, the E(~) curve has only one minimum for a2>0 ()~ < K; ~ mi :---- A -- 0:symmetric state), and there are two minima for a2 < 0 (X > K; A = -+ ~.For X < K the system is in chiral symmetric state. When X is increased past thecritical value K, the system will flop into a state where XL or XD is favoured,although which state is chosen is entirely random. The symmetry of the system fork > K is broken spontaneously (Lukacs, 1993). For large k the completelyasymmetrical states (A = + 1 or A = - 1) will be reached:

-4

The A = 0 state for X > K (dashed curve on Figure 7) is unstable.Since the chiral symmetry of the system in this model is perfect, the final L or Ddominance for large k will be accidental: the system chooses either the A > 0 orthe A < 0 direction at the X = K bifurcation point, but it is completely arbitrarywhich one of the two possibilities will be chosen: the L and the D dominanceoutcomes have equal (50 per cent) probabilities.

(1I) Spontaneous symmetry breaking, with a small asymmetry

Let us add a small asymmetry term to the previous E(8) function:E(~) - a2(h)~ 2+ a 4()~)~ 4_ ~

90 F. GL~1CK

The system is now asymmetrical: E(8) is not equal to E(-8). The figures belowshow the E(~) curves for 3 different cases (~ > 0):

Figure 8

The system has some small preference for 8 > 0 (more L molecules than Dmolecules). The stable and unstable minima (4) have the following X dependencesnow:

Figu~ 9

Let us increase the X critical parameter. Then, if we neglect the statisticalfluctuations, the system always goes into the A = + 1 asymmetrical state (with totalL dominance). The small chiral asymmetry of the system determines the direction ofthe chiral symmetry breaking (Kondepudi and Nelson, 1985); Zeldovich andMihajlov, 1987; Zeldovich, 1988).

CHIRALITY OF ORGANIC MOLECULES

If only mirror symmetrical (parity conserving) interactions were present in nature,then the particular choice of the dominance of the L-a-amino acids and the D-sugars in terrestrial organisms would appear to be a matter of chance. One couldimagine another planet in the universe where the opposite enantiomers (D-aminoacids and L-sugars) would dominate in living beings. There is an interaction,however, which has no mirror symmetry: the weak interaction. In the usualchemical processes its effect is much more weaker than the influence of the mirrorsymmetric electromagnetic interaction. The weak interaction can play, however, therole of the small chiral asymmetry in the spontaneous symmetry breaking of thebiomolecules, thus determining unambiguosly the dominant enantiomers.One possibility is the influence of the parity violating force mediated by the Zboson (Gliick, 1993) (the so called weak neutral current) on the energy values ofthe quantum states of biomolecules. The effect of the Z force has beendemonstrated experimentally for atoms, but not yet for molecules. An interestingtheoretical result, however, has been obtained by S. F. Mason and G. F.. Tranter.Between 1983 and 1986 they performed detailed calculations of the energies ofseveral L- and D-amino acids, taking into account the asymmetric Z force. Theyfound that, in all cases, the biologically dominant L-enantiomers have a lowerground-state energy than the corresponding D-enantiomers. The relative energydifference is of the order of 10-17 (Mason and Tranter, 1984; Mason, 1985). Thiseffect seems to be very small when we take into account the statistical fluctuations.It has been shown by computer simulations, however, that under special conditions(L.and D compounds competing with each other in 108 m3 water, over a period of103 years) the small systematic effect of the weak force overcomes the fluctuations,and determines the outcome of the symmetry breaking: nearly all the amino acidmolecules will have the L enantiomeric forms (Hegstrom and Kondepudi, 1990;Kondepudi and Nelson, 1984); Kondepudi and Nelson, 1985).

Another possibility for the small chiral asymmetry could be the effect of the betaelectrons coming from weak decays of radionuclides. These electrons aredominantly left-handed (their spin is opposite to their direction of motion), andthe two enantiomers of a compound in a racemic mixture are differentlydecomposed by them (Ulbricht, 1959; Garay, 1970). The relative difference in therates of such decomposition of L- and D-enantiomers could be about 10-6. A.Garay had demonstrated experimentally this phenomenon (Garay, 1970; Garay,1968).Finally, we would like to make a remark on the connection between the handednessof the biomolecules and the chiral morphological asymmetries of the livingorganisms. These morphological asymmetries (like the difference between the left-and right-hand sides of the human brain) could have developped during theevolution by spontaneous symmetry breaking, if they gave some advantages to theorganisms (i.e., these asymmetries appeared as selective factors for the biologicalevolution) (Holba and Lukfics, 1993). It might happen that in many cases thehandedness of the biomolecules determined the directions of these morphologicalsymmetry breakings (similarly to the possible relationship between the parityviolating weak interaction and the biomolecular handedness)

CONCLUSIONS1. There are many organic molecules existing in two different forms, related to eachother as object and mirror image (optical isomers, enantiomers).2. Optical purity (almost perfect chiral asymmetry of biomolecules) is acharacteristic property of life: living organisms use only one form of theenantiomers (e.g.: L-amino acids, D-sugars).3. Optical purity is necessary condition of life-functions.4. Optical purity could have developped during the evolution of life by spontaneoussymmetry breaking. The parity violating weak interaction might have unambigouslydetermined the direction of this symmetry breaking.5. The chiral morphological asymmetries of living organisms may be in causalrelation with the chiral asymmetries of the biomolecules.

REFERENCES

Bentley, IL (1969) Mo/ccu/ar Asymme~y in Bio/ogy, New York: Academic Press.Crabb4~, P. (1972) ORD and CD in Chemismy and Biochemistry, New York: Academic Pr~s.Damjanovich, S. (1976) Mai~omolekul~ bioftzik~ja, [Biophysics o[ Giant Molecules, in Hungarian],

Budapest: Akad~miai Kiad6.Garay, A. (1968) Origin and role of optical isomcry in life, Name, 219, 318.Garay, A. (1970) Optikai forgat6k~pesr~g biol6glai jelent6c~gc ~ cruder , [Biological significance and

origin o[ ability fox optical rotation, in Hungarian], F’u~cai Szcm/e, 106.Glilck, F. (1993) Clfirality in the elementary interactions, Symmewy: Culture and Science, 4, 1, 37-48.Hegatrom, 1L A. and Kondepudi, D. K. (1990) The handedn¢~ o[ the univcra¢, SciemificAmerican, 262,

1,98oHolba, A. and Luk~cs, B. (1993) Hominid cerebral latcralisation as spontaneous symmetry breaking,

Symme~: Culture and Science, 4, 1.Kondepudi, D. K. and Nelson, G. W. (1984) Chirai-symmctry-breaking states and their sensitivity in

nonequilibrium chemical ~ystems, Physica, 125A, 465.Kondepudi, D. K. and Nelson G. W. (1985) Weak neutral currents and the origin of biomolecular

chirality, Nature, 314, 438.Lal~k~, B. (1993) On the mathematics o[ symmetry brealdngs, Symnway: Culture and Science, 4, 1, 5-11.Ma~on, S. F. and Tranter, G. E. (1984) The parity-violating energy diffex~nce between enantiomcric

molecules, Molecular Physics, 53, 1091.Mason, S. F. (1985) Biomolecular handedncss, Chemistry in Brim/n, June, 538.Uibficht, T. L. V. (1959) Beta-decay and the biomolecular chirality, Quart~’/y Review of Chemica/

Secie~y, 13, 48.Z~-~ch, Ya. B. (1988) Left-right asymmetry in biology, [Hungarian translation: Bal-jobb aszimmetria

a biol6gi~ban, F’~i/~ai Szem/e, 222].Zeldovich, Ya. B. and Mihajiov, A. C. (1987) Fluktuacionnaya kin’etika reakcij, [Fluctuationai Reaction

Kinetica, in Ru~an], U~hi Fizicheskih Nauk, 153, 469.

Symme~: Culture and ScienceVo£ 4, No. 1, 1993, 93-103

ON GEOMETRIC SYMMETRIESAND TOPOLOGIES OF FORMS

B61a Lukfics

Physicist, (b. Budapest, Hungary, 1947).Address:. Central Reaearch Institute for Physics, Budapest 114,P.O.B. 49, H-1525 Hungazy, E-mail: lukacs@rmk530.rmki.kfki.hu.Fields of Interesl: Cosmology;, General Relativity;, Heavy IonPhysics; Speech Acoustics; Economics.

ABSTRACT: A brief bird’s eye overview is given aboutpossible symmetries and topologic structures of bodiesin the 3 dimensional Euclidean space. Thi~ is done forbasis of understanding some further papers of this issue.

1. INTRODUCTIONThe notion of symmetry is widely used forcharacterizing patterns of biological objects. However, it is often used purelyqualitatively. It is not so in mineralogy and crystallography, but their symmetriesare almost exclusively discrete. Crystallographic analogons are very useful in manycases (e.g., for the radial symmetry of Ctenophora, radial pseudosymmetry ofEchinodermata, left-right symmetry of higher evolved animals, etc.). Still,continuous symmetries are also possible and they need another approach.There are more complicated cases as well. Is anAmmonita symmetric? Its chambersgrow along a spiral line in a regular manner. There is no left-right symmetry,neither radial symmetry, nor rotational symmetry. Still a regularity is dearly seen.Is it a symmetry, and ff it is, in what sense?The question cannot be fully answered. Any well-defined transformation may becalled a symmetry. However, one knows what is a geometrical symmetry. Thepresent paper applies the formalism to situations relevant for biological objects.Section 2 is a brief recapitulation of the Killing vector technique of Riemannspaces. Section 3 applies the equations in the 3 dimensional Euclidean space. Firstall the Killing vectors (including conformal ones) of this space are calculated andthen, now excluding the conformai ones, all possible cases are listed when theactual symmetry of the matter distribution is smaller. Section 4 contains someremarks for discrete symmetries.

94 B. LUK,4C$

Section 5 illustrates the possible subgroups of continuous symmetries on examples,while Section 6 mentions some principles of topologic classifications of objects.Some explanations and commenting remarks, which might also properly befootnotes as well, are included into the text, but for showing the specific r61e inCAPITALS.

2. ON KILLING SYMMETRIESThis Chapter is a brief recapitulation of the theory of continuous symmetrytransformations for pedestrians. For more details see (Eisenhardt, 1933) andcitations therein.

Consider a Riemann (pseudo-Riemann) space of n dimensions. There is acoordinate system {x/} on the manifold. Because of the Riemannian structure anyregular recoordination is permitted (Eisenhardt, 1950), but this possibility will bemainly ignored in this paper.A domain o.f the space is filled with matter. This matter is characterized by a set offields (@a(x~)}. This set contains the relevant data. of the matter, say ~1 may be thedensity, @2 the colour, ~3 classifies the tissue at x~ as liver, muscle, kidney, etc. Thedomain is generally finite, but we restrict ourselves to the interior and neglect theboundaries. For the present we ignore the discontinuous nature of the matter inbiological objects. Then the cellular structure is averaged away. This is satisfactoryif the number of cells is > > 1.Now identify a number of points in the matter; 3 will be sufficient, in aninfinitesimal neighbourhood

Xli = Xi + drii

In a Riemarm space the distances of infinitesimally close points can be written as

drlK = dr.l-dxK (2.1)

(NOTE THE EINSTEIN CONVENTION (Eisenhardt, 1950): THERE IS AUTOMATICSUMMATION FOR INDICES OCCURRING TWICE IF BOTH ABOVE AND BELOW.) In aRiemann space gik is positive definite; in a pseudo-Riemann one this does not hold.Then in a pseudo-Riemann space (as the spacetime around us) 0 distance ispossible between points which are not neighbours. Then in such a space the metricgenerates a so called light-cone structure: the lines of 0 distance build up a kind ofconnection. Here we will not have pseudo-Riemann spaces, but still the problem ofneighbours is to be discussed, since the metric tensor does not show the topology.(E.g., the same metric tensor is valid for a plane, a mantle of a cylinder and that of acone.) This problem is relegated to Section 6, and until that we regard the topologyas known.

So we have our selected infinitesimal t.riangle around x/. Let us try with atransformation. Introduce a vector field/~(x), and shift all the points along thisvector field in the manner

GEOMETRIC SYMMETRIES AND TOPOLOGIES OF FORMS

x/’ = x/ + ~(i(x) (2.2)

with the same ~ for all the points. Then, depending on the vector field, there arethree possibilities:

1) The new triangle is geometrically identical with the original one, i.e., the threeangles have remained unchanged, and so for the lengths of the sides. Then we callthis shift a Killing symmetry.

2) The new triangle is similar: the angles are unchanged and the lengthsproportional. Then the shift is a conformal Killing symmetry.3) Neither is true. Then the shift is not a symmetry. (To BE SURE, WEAKERSYMMETRIES CAN BE DEFINED AND SOMETIMES ARE USED (Katzin et aL, 1969).HOWEVER SOMEWHERE ONE MUST DRAW A LINE, AND WE STOP AT CONFORMALKILLING SYMMETRY.) In addition, for a symmetry we require that at the new pointsthe matter remain the same (symmetry) or at least similar (conformal symmetry),i.e., that ~a show some scaling.

Now we are going to discuss the geometry in details: then some (ransformations aresymmetries from geometric viewpoint, and later the consumer can check if ~ashows scaling or not.We start with infinitesimal shifts, and this will be enough, because one can repeatthe transformation in any times. So now [ ¯ I is infinitesimal, and we can calculateup to first order in it. Note that

dr/’ = dr/ + .Ki,rdr" (2.3)where the comma stands for partial derivative. (THE PARTIAL DERIVATIVEGENERALLY DESTROYS THE DEFINITE TENSORIAL STRUCTORE IN RIEMANN SPACES,SO FORMULAE OF RELEVANCE MUST BE REFORMULABLE IN TERMS OF COVARIANTDERIVATIVES, DENOTED BY SEMICOLON. FOR THE DEFINITION OF THE LATI’ERDERIVATIVE SEE THE STANDARD LITERATURE, E.G., (Eisenhardt, 1950). ALL OURFINAL FORMULAE COULD BE WR.rFFEN INTO SUCH A FORM.) Our previousconditions for a symmetry can be written as

(2.4)

where 12=0 if the symmetry is not conformal.

Now, eqs. (2.2-4), in first order, lead to

g.ul~,~ +g~,i+g.~ = flgi~

By means of covariant derivatives this equation can be rewritten as

(2.5)

(2.5’)

which is the Killing equation (Eisenhardt, 1933); conformal if 12 differs from 0. Interms of Lie derivatives (Eisenhardt, 1933) the equation gets the even morecompact form

S. LU~C$

(2.5")expressing the fact that the change of the metric tensor is proportional to itselfalong the vector field K (0 if the symmetry is not conformal). This is just the reasonfor the similarity between the triangles.If the metric of the space is given (which we assume henceforth), then eq. (2.5) is asystem of linear partial differential equations, which can be solved somehow.Combinations with constant coefficients are again solutions, so the generalsolution is

Id(x) = q~K~(x) (2.6)

wh.ere the q’s are constant; provided that we have got all the independent solutionsK:(x). For conformal symmetries the corresponding fl’s add up in a similarmanner.

For more than one Killing vectors the corresponding transformations form agroup, since a sequence of symmetry transformations is a symmetry transformationby its result. Then one can calculate the commutator of two infinitesimaltransformations by using eq. (2.3); the result reads as

[Ka, K#li -- KarK~,r - I~r Kaiv (2.7)

(WHERE ALL THE COMMAS COULD BE SUBSTITUTED BY SEMICOLONS AS WELLBECAUSE THE DIFFERENCE CANCELS DUE TO THE ANTISYMMETRIC COMBINATION).Since the commutator again would generate a symmetry transformation,

[Kw Kg]i = c~K~ (2.8)

where the c’s are the structure constants of the group.

3. CONTINUOUS SYMMETRY TRANSFORMATIONSIN THE 3 DIMENSIONAL EUCLIDEAN SPACEBiological objects live around us in a 3 dimensional Euclidean space. There

(3.1)

This is a zero curvature space. For constant curvature the number of independentKilling vectors is Vm(n + 1); including conformal ones ½(n + 1)(n +2) (Eisenhardt,1933). So in our case the number of possible independent symmetries is 10, ofwhich 6 keep distances, while 4 are only similarities. The complete list is as follows:

t,= ; ~,=o

GEOMETRIC SYMMETRIES AND TOPOLOGIES OF FORMS 97

with

lIQ -~-X

(3.2)

r2 ~ x2 + y2 + z2 (3.3)

where boldfaces stand for 3-component entities colloquially called vectors of axialvectors. Generally P is called translation, J rotation, D dilatation and sometimes Qinversion, being in some connection with the inversions to the unit circle. The mostgeneral vector field for a symmetry transformation is built up from the above oneswith 10 constant coefficients.

By evaluating the commutators (2.7) one gets

[e~,Pa] = 0[~,~,~a] = ~,.a~, .oe o[Pa, D] = Pa

~a, J#] = "~ r¢j a (3.4)

[~.,Ol = 0

[a., aal = 0~ ~ the confo~al ~oup of 3-space. I~oring the ~o~al s~met~ we getthe s~met~ group of the Euclid~n spa~ in stricter sere, {P,~, which ~ theE(3) group of 3 tra~latio~ and 3 rotations.Now we are ready ~th the ~d~t possible group of ~ntinuo~ s~met~transformatiom for biologi~l object. ~en ~m~ the s~nd ~ndition that in thenew poinm the matter be the same (or similar) ~ in the original on~. So a~ntinuo~ ~nfo~al s~met~ transformation ~11 have the form (2.2), where ~~ ~ in (2.6), ~th the pa~icular ~s ~om (3.2), so that the material characte~ti~

(density, colour, composition, etc.) show a scaling as well; a continuous strictsymmetry will contain only P and J from (3.2) and the material characteristics willbe strictly unchanged. (IT IS POSSIBLE TO BE MORE GENERAL. ONE MAY DEFINETRANSFORMATION FOR THE MATERIAL DATA AS WELL, SAY CHANGES IN COLOUR,AND THEN A COMBINED SHIF~ AND COLOUR CHANGE MAY BE A SYMMETRY. THISCONSTRUCTION IS KNOWN IN SOME BRANCHES OF PHYSICS, AS E.G., THESUPERSYMMETRY OF PARTICLE PHYSICS (GOL’FAND AND LIHTMAN, 1974) BOT THISWILL NOT BE DONE HERE.) In what follows we assume that there is no problem todecide if the new point is materially identical with (or similar to) the original one.

Then by observation one can select the actually existing subgroup. The theoreticalpossibilities are limited because generally the commutator relations (2.7) cease tobe closed ff we omit symmetries from a group. To the end of this Section we ignorethe conformal symmetries.

Now I try to list all the possibilities. (OF COURSE THE POSSIBLE SUBGROUPS OFE(3) ARE LISTED IN THE STAUDARD LrrEI~,ATORE (PETROV, 1966). HOWEVER,THAT IS NOT THE COMPLETE ANSWER FOR THE PRESENT QUESTION, AS WILL BESEEN.) I hope that no case is forgotten, but if the Gentle Reader were to discover amissing one then he would be honoured. To begin with, observe that we have 3independent ’directions’ (say, i = - ¯ -)" Now, select, e.g., We have Px and Jx inthis direction. There are then 5 pos~fities: x"

1)Both Px and Jx are symmetries.

2)Only Px is a symmetry.

3) Only Jx is a symmetry.

4)Neither Px nor Jx is a symmetry, but a special combination ,of theirsnx = a ’x + b xis.

5) No combination ofPx andJx is a symmetry.

Here Hx is a helical symmetry, a combined rotation + translation along therotational axis. Now let us start from the maximal symmetry downwards. Observethat the numbering 1, 2, 3 of the components is arbitrary.

6 symmetries:

(P,J). The matter is homogeneous and isotropic; no preferred point ordirection.

$ symmetries:

~b. This class is empty, according to the Fubini lemma (Petrov, 1966).

4 symmetries:

(P, J1)"rotation.

Homogeneity, with a preferred direction, which is an axis of

GEOMETRIC SYMMETRIES AND TOPOLOGIES OF FORMS

3 symmetries:

a) (Pt, P2, P3)" Homogeneity without isotropy.

b) (P1, P2,./3). Planar symmetry, with a rotational symmetry around an axisorthogonal to the planes.

c) (P1,P2,H3). Planar symmetry, with helically rotated but otherwiseidentical planes.

d) (’/1,./2, J3)" Spherical symmetry (rotations around 3 orthogonal axes).

e) (HI, H2,J3). Rotational symmetry around the z axis with helicalsymmetries along any axis orthogonally intersecting the z one.

2 symmetries:

a) (P1, P2)- Sequence of planes with homogeneity within each plane.

b) (P1,./1)" Cylindrical symmetry.

I symmetry:

a) (P1). Translational symmetry in one direction.

b) (J1). Rotational symmetry around one axis.

c) (HI). Helical symmetry along one axis.

These are 12 different possibilities for non-conformal continuous symmetries ofliving organisms; in the cases when H’s appear, there is a scale constant connectingtranslation and rotation. Including the 4 conformal symmetries the constructionwould go likewise, but this will not be done here.

4. ON THE DISCRETE SYMMETRIESAs mentioned earlier, the discrete symmetries in 3 dimensions are extensively listedin crystallographic literature. (See e.g., (Kittel, 1961) and further citations therein.)So we do not go into details, only classify the possible discrete symmetries into twoclasses. Namely

1) In addition to the E(3) group the Euclidean space possesses 3 independentdiscrete symmetries, which can be chosen as reflexions to 3 orthogonal planes.

2) Any possible continuous symmetry, which is actually absent, can still appear atdiscrete steps. E.g., it is possible that P~ is not a symmetry with arbitrarytransformation parameter ~, but still it is a symmetry in steps NA.

100 B. LUK/[C$

Again, a lot of combinations may exist. E.g. rotations and reflections to planes maybe combined to reflections to sequences of rotated planes. Now, the discreteversion of this is the familiar radial symmetry.

5. EXAMPLESUp to now we restricted ourselves to the Cartesian coordinates x; ~ z. Now, forexplicit examples, it is useful to use other special coordinates, but we do not wantto go into the details of coordinate transformations (for which see e.g., (Eisenhardt,1950)). Only let us note that by performing the coordinate transformationxi’ ffi x"(xk) a vector is transformed as

(5.1)Two special coordinate systems will be mentioned. The first is the spherical polarsystem (r,0,q,):

x = rsin0cos~y ffi rsin0sin~ (5.2)Z ---- rcos0

with the corresponding inverses. The second is the cylindric system (~,~o,[’):xy = psin¢, (5.3)z=~

Now let us see some symmetries. First we return to the list of possible continuousstrict symmetries.

3d, spherical symmetry.

By transforming all the 3 J’s in (3.2) into spherical polar coordinates (not givenhere) one obtains that the transformations act on r = const, surfaces. Any point onsuch a surface can be rotated into any other. Therefore the symmetry exists ff

By other words, any material date can change only with the radial distance from acenter.

3b, planar symmetry with rotation.

It is the most convenient now to use cylindrical coordinates. The rotation axis istaken in the ~o direction. In the planes orthogonal to this axis there act twotranslations and one rotation, which is the E(2) group. Such a plane is then

GEOMETRIC SYMMETRIES AND TOPOLOGIES OF FORMS 101

maximally symmetric. Therefore nothing depends on the coordinates o, ~,.Consequently the symmetry exists ff

2b, cylindrical symmetry.

The symmetry direction is taken as z. Then, by transforming Pz and Jz one gets

Consequently there remains

% =lc, helical symmetry.

As mentioned, Hi = P1 + q J1. The symmetry direction is taken as z. Then incylindric coordinates

The condition that a scalar obey the symmetry is, for the analogy of eq. (2.5")

LK~= = 0which reads as

= 0(Eisenhardt, 1950). Substituting the actual K/ and solving the partial differentialequation, the result is

÷= = ÷=(p, ~ - q~-)Finally, consider a case when conformal Killing vector is also included. Let the onlysymmetry be

We take the symmetry direction z. Then in cylindric coordinates

= [qrjAgain, requiring this as a strict symmetry on the material fields, from the vanishingof the Lie derivative one obtains

B. LUK~iCS

By other words, in one 36ff’ rotation the equivalent p and [" values increase by afactor eq. Consequently the structure is spirallic, with proportionally growingelements. $imi!ar structures, but of course only with discrete symmetries, arc wellknown (e.g., Ammonites). However, if this symmetry exists, chamber sizes, suturelines etc. must obey a very definite growth law with a single scaling constant. Such ascaling can and ought to be checked.

6. TOPOLOGYAs told earlier, the metric gik does not define the neighbours on a manifold. As anexample, consider the mantle of a cylinder of radius # o. Points (~ = 2~r-~,and (~ = ~, 1"o) are very near to each other ff ~<<1. On the other hand,performing the transformation (5.3) the two points get at a distance (2¢r-2~)#o,because the mantle is cut just between the neighbouring points and they are rolledapart. So the coordinates and the metric are not enough; some extra information isneeded about the possible "compactification" of the surface.In the 3 dimensional Euclidean space we do have this information. However,possible configurations still are to be classified according to neighbourhoodrelations (connectivity). The problem has an extended literature, so here only a fewsimple examples will be mentioned; for the details cf. e.g., (Patterson, 1956).Consider a compact 2 dimensional entity (for such case the complete classificationis known), e.g., the unit disc ~ < 1, 0 _< ~ < 2~r. This disc is simply connected. Thisterm means that any closed curve on it can be reduced to any point or can betransformed into any other closed curve by continuous distortion (see the notion ofhomotopy groups (Patterson, 1956)). This is shown on Figure 1.

Figure I Figure 2 Figure 3

Now, punch a small hole around the center. The punctured disc is no more simplyconnected (Fig. 2.). The closed curves now are classified into 2 disjoint classes.Curves around the hole (say A and B) can be distorted into any other such curve

GEOMETRIC SYMMETRIES AND TOPOLOGIES OF FORMS

but cannot be distorted into curves not surrounding the hole (C or D) and viceversa. In addition, a disc with a central hole can be distorted into another disc witheccentric hole. (THE DISTORTION, ON THE OTHER HAND, IS DETECq’ED BY THEMETRIC STRUC’I~RE, BECAUSE DISTANCES BETWEEN POINT PAIRS ARE CHANGING.NEIGHBOURHOOD AND METRIC RELATIONS TOGETHER ARE ENOUGH FORCOMPLETE DESCRIPTION.) SO for topology all the discs with one hole are equivalent.Punch a second hole (Fig. 3). This disc you can deform into any other disc with twoholes. Again, on it closed curves classify into four classes: surrounding the first hole(E), the second hole (F), both holes (G) or neither of them (H). And so on. Theseobjects are inequivalent with each other.Things go similarly for bodies. Consider a sphere. Each internal closed curve can bedeformed into any other. This will not change if one digs a pit into it. However,driving a shaft completely through the interior is no more simply connected: curvessurrounding the tunnel cannot be deformed into curves not surrounding it. At thispoint we deliberately stop.It may obviously be important the possible ways of internal connections for thestructure and operation of an organism.

7. CLOSING REMARKThis paper is by no means complete, and in addition at this point may seem ratherpointless. However, it contains just the necessary mathematical background for therest of the articles in the next issue. Our present purpose was not to enjoy puremathematics but to help the understanding of some following articles.

REFERENCES

Eisenhardt, L. P. (1933) Continuous Groups of Transformation, Princeton: Princeton University Press.Eisenhardt, L. P. (1950) Riemannkm Geomary, Princeton: Princeton University Press.Gol’fand, Yu. A. and Lihtman, E. P. (1974) F_.xtension of the Algebra of the Poincar~ Group Generators

and Violation ofP lnvariance, Pisma ZhurnalEksperknoUal’noi i Teoreticheakoi Fiziid, 13, 323.Katzin, G. H. et al. (1969) Curvature Coilineations, lournal of Mathematical Physics, 10, 617.Kittel, Ch. (1961) Introduction to Solid State Physics, New York: J. Wiley and Sons.Patterson, E. M. (1956) Topology, Edinburgh: Oliver and Boyd.Petrov, A. Z. (1966) Novye metody v obshchei teorii otnositel’nosti, [New Methods in General Relativity,

in Russian], Moscow:. Nauka.

Symmetry: Culture and ScienceVoL 4, No. 1, 1993, 105-109

SFS: SYMMETRIC FORUM OF THE SOCIETY(B ULLETIN BOARD)

All correspondence should be addressed to the editors: Gydrgy Darvas or £Mnes Nag),.

ANNOUNCEMENTS

FIRST CIRCULARCALL FOR PAPERS, WORKSHOP TOPICS, AND EXHIBITION ITEMS

SYMMETRY."NATURAL AND ARTIFICIAL

Third Interdisciplinary Symmetry Congress and Exhibition of the

INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OFSYMMETRY (ISIS-SYMMETRY)

August 14- 20, 1995Old Town Alexandria (near Washington, D. C. ) U.Sdl.

FIELDS OF INTEREST

SYMMETRY." NATURAL AND ARTIFICIAL

The congress and exhibition present a broad interdisciplinary forum where the rep-resentatives of various fields in art, science, and technology may discuss and enrichtheir experiences. The concept symmetry, having roots in both art and science, helpsto provide a ’common language’ for this purpose. The new ’bridges’ between disci-plines could inspire further ideas in the original fields of participants, as well asfacilitate the adaptation of existing ideas and methods from one field to another.The title of the congress emphasizes the presence of symmetry (dissymmetry, bro-ken symmetry) both in nature and in the objects created by artists, scientists, and engi-neers.

106 S FS

Exhibition: Ars Scientifica

There have been several exhibitions representing the specific impact of certainfields of science and technology on art, but ISIS-Symmetry has initiated a regularforum for a broader interface of art and science. The exhibition will consist of twoparts: a professional exhibition and an informal one, based on the objects illustrat-ing the lectures given by the participants. Some workshops will be conducted in theexhibition rooms. SPECIAL INTERESTS ofArs Scientifica are, among others: kalei-doscopes, polyhedra, model designs, new media.

CALL FOR PAPERS

A lecture proposal should include a maximum 4-page extended abstract in a cameraready version. Keeping in mind the interdisciplinary goals of the congress and thecomposition of the participants, please try to help the readers outside of your maindiscipline e.g., by explaining some special concepts, using intuitive approaches, orgiving comprehensive tables and illustrations. The extended abstracts should either(a) describe concrete interdisciplinary ’bridges’ between different fields of art, sci-ence, and technology using the concept of symmetry, or (b) survey the importanceof symmetry in a concrete field with an emphasis on possible ’bridges’ to otherfields. Note, please, that the central topic of the present congress Symmetry:Natural and Artificial opens a wider door towards technological applications.Papers discussing links between any form of symmetry-asymmetry phenomenon orlaw in nature on the one side, and artistic, technical achievements on the other, arepreferred. Please consider that the meetings of ISIS-Symmetry are informal and donot substitute for the disciplinary conferences, only supplement them with abroader perspective.The extended abstracts should be submitted in 2 copies, mailed 1 each to G. Darvasand D. Nagy, on A4 or letter size pages, printed on one side of each sheet, with atleast 2.5 can (1 inch) margins both sides, top and bottom, double spaced, 12-pointcharacters.Sample:

TITLE WITH CAPITAL LETIERS[two line-spaces]

Joe Symmetrist and Josephine AsymmetristDepartment of Dissymmetry, Fibonacci University

San Symmetrino, SY 12358, SymmetrylandE-mail: symmetrist@fibonacci.edu

[two line-spaces]The text should be printed in one column. Figures (black-and-white only) mayinterrupt the text. Please avoid using any other heading (e.g., ’Extended Abstract’,’submitted to ...’). Page numbers should be marked with pencil.References [at the end of the abstract]:Alphabetical order, full bibliographic description.For more details refer to the "Instructions for contributors" on pp. 110-111.

SYMMETRIC FORUM OF THE SOCIETY 107

CALL FOR EXHIBITION ITEMS

Items for the exhibitions should be introduced in the same form as lecture-abstractson A4 or letter size sheets, in black-and-white camera ready, reproducible form.Please mark with pencil at the top of the sheet: (EXHIBITION). A short descrip-tion and/or explanation of the items, as well as the connection to the main theme ofthe congress and exhibition, is preferred. Please give the dimensions of each item.Art works, models, demonstration materials, etc. are welcome, e.g., in the followingsections: Kaleidoscopes, Polyhedral symmetry, The beauty of molecules, Aestheticsof man-made constructions, Mechanical structures inspired by nature: Artificialand natural structures, Design principles, New Media. Proposals for further sec-tions are encouraged.

CALL FOR WORKSHOP TOPICS

Please give the approximate title, short description (how do you plan to organizethe workshop), other expected/proposed contributors, etc. Proposals emphasizinginteractions, mediated by symmetry, between different disciplines; science, art, andtechnology; cultural origins, and relying upon the interest of participants with dif-ferent backgrounds, are preferred.

CALL FOR PROPOSALS FOR EVENING ACTIVITIES OR PERFORMANCES

Music, dance, video, laser, etc. programs are welcome. Please submit your propos-als, similar to the lecture abstracts, with descriptions of the feasibility and the tech-nical requirements. Please mark with pencil at the top of the sheet:(PERFORMANCE), (VIDEO), etc., respectively. (Formal requirements are thesame as above for papers.)

DEADLINF~

for application and short description of contribution and other proposals:December 15, 1994;

for submitting final (camera ready) versions of the extended abstracts:March 31, 1995.

THE FORMAT OF THE CONGRESS AND EXHIBITION

The tradition, initiated by ISIS-Symmetry, to facilitate interdisciplinary dialoguesamong scientists, engineers, and artists will be continued. There will be no parallelsections (which would lead to disciplinary separation of the participants), but eachmorning there will be plenary sessions, while the main ideas will be discussed anddeveloped in afternoon workshops. For the evenings there are scheduled perfor-mances and informal meetings, including recreational, and ars scientifica programs.

The working language of the congress is English.

The Scientific Advisory Committee of the Congress and Exhibition is the Board ofISIS-Symmetry (see inside front and back covers).

108 SFS

CONTACT PERSONS (for the Congress and Exhibition)

America:Martha Pardavi-Horvath, Site CoordinatorGeorge Washington UniversityDepartment of Electrical Engineering and Computer ScienceWashington, D.C. 20052, U.S.A.Phone: 1-202-994-5516; Fax: 1-202-994-5296; E-mail: pardavi@seas.gwu.eduEurope:GyOrgy Darvas, Executive Secretary, ISIS-SymmetrySymmetrion - The Institute for Advanced Symmetry StudiesP.O. Box 4, Budapest, H-1361 HungaryPhone: 36-1-131-8326; Fax: 36-1-131-3161; E-mail: h492dar@ella.hu

Asks:Ddnes Nagy, President, ISIS-SymmetryInstitute of Applied Physics, University of TsukubaTsukuba Science City, Ibaraki-ken 305, JapanPhone: 81-298-53-6786; Fax: 81-298-53-5205; E-mail: nagy@bk.tsukuba.ac.jp

ISIS-SYMMETRY

ISIS-Symmetry organized its first congress and exhibition "Symmetry of Structure"in Budapest, Hungary (August 1989), while the second one "Symmetry of Patterns"was held in Hiroshima, Japan (August 1992). This forthcoming triennial congressand erh~h~ion will be hosted on a third continent.ISIS-Symmetry events demonstrate the emphasis on internationality and interdis-dplinarity. Indeed, the Society has members in 41 countries, on all continents, andits main purpose is to bridge art and science (different disciplines), East and West(different cultures). (Cf. the "Aims and Scope" on p. 112.) Backgrounds of themembers of the Society represent variousfie/ds of science, art, and technology. Theiractivity is linked by the concept of symmetry. Application of symmetry is also a gen-eral tool and method influencing and fertilizing the creative thinking of each other.

APPLICATION FORM

Name: ......................................................................................................................................

Affiliation: ..............................................................................................................................

Mailing Address: ....................................................................................................................

City: ............................................................... State/Country: ...............................................

Fax: ................................. Phone: ................................. E-mail: ............................................

I intend to: C) attend the Congress (3 submit a paper C) exhibit

Tentative title of my contribution: ......................................................................................

SYMMETRIC FORUM OF THE SOCIETY 109

OBITUARY

GYORGY PA~L

1934 - 1992

Our colleague, one of the organisers of the 1991 symposium, from whose materialthis issue has been selected, GyOrgy Pa~l, astronomer, long since member of theGconomical Scientific Society of the Hungarian Academy of Sciences, died inMarch, 1992 after 5 years of ailment.His scientific career started at the end of the fifties, with observations for galaxyclusters and quasars. In this study he formulated a hypothesis - from quasar aggre-gatious at some red-shifts - about the non simply connected topology of the Uni-verse, in 1970. Such a topology, quite possible according to the General Relativity,would result in regular, although non-trivial, recurrences of multiple pictures, and,for very simple expansion laws of Universe, in a rather crystal-like symmetric visualappearance of the large-scale picture of the Universe. Then, for a while, he turnedto "internal" symmetries: regularities about fundamenta.l constants, patterns in thedata of celestial objects and observed characteristic quantities of the Universe. Inthe eighties he concentrated on the cosmological consequences of the particlephysical theory of Grand Unification. This theory claims that the 3 fundamentalinteractions (electromagnetism, weak interaction, and strong interaction), whosecombined symmetry seems to be U(1)×SU(2)×SU(3), are rather three projectionsof a higher scheme of a higher symmetry at least SU(5), broken spontaneously atsome energies.In 1987 his serious illness started, and physicians were very pessimistic about hisfuture. He chose to fight, not only for his life but rather for the possibility to con-tinue his scientific work. He doubled his professional activity. In 1991 the newestcosmological observations about the existence of repeated "Great Walls" separatedby hundred millions of light years suggested him to return to non-trivial topologies;several papers were published about this possibility, and the present issue containsone brief report on this. He died during this work, after several years of slow butcontinuous worsening of his health.His co-authors are continuing this work. At this point it is impossible to decidewhich is the exact symmetry of the Universe from among the 3 possible ones, andwhich is its topology from amongst the 10 possible ones for the symmetry E(3) orfrom amongst the infinite or uncounted possibilities of the cases SO(4) or S0(3, 1).For any case his strong ability for a global viewpoint is seriously lacked.But his works remain, and we remember his thinking pattern and working style asexamples for our later work.

B~la Luk~lcs

11o INSTRUCTIONS FOR CONTRIBUTORS

Contributions to Sr~oaz~. CvLrv~ ~ SCIENCE are welcomed from the b~d~t inte~ationalcircl~ and from ~p~n~tiv~ of all ~bolar~ and agistic fiel~ whe~ s~met~ ~nside~tions play anim~nt rol~ ~e ~ should have an interdi~iplina~ cha~cter, d~ling ~th ~met~ in a~nc~te (not only me~pho~l[) ~n~, ~ di~u~ in ’~ms and ~’ on p. 3~. ~e qua~erly has as~ial intcr~t in h~ dis~nt fields of a~, ~ien~, and t~hnolo~ may influen~ ~ch other in thef~m~ork of ~met~ (s~met~lo~). ~e pa~ should ~ addr~ to a broad non~ialist publicin a [o~ which would en~u~ge the dialogue ~n di~iplin~.Man~p~ may ~ submitt~ di~t~ to the ~ito~, or though mem~m of the Board of ISIS-S~met~.

Contributors should note the following:¯ All papers and notes are published in English and they should be submitted in that language. Thequarterly reviews and annotates, however, non-English publications as well.¯ In the case of complicated scientific concepts or theories, the intuitive approach is recommended,thereby minimizing the technical details. Hew associations and speculative remarks can be included, buttheir tentative nature should be emphasized. The use of well-known quotations and illustrations should belimited, while rarely mentioned sources, new connections, and hidden dimensions are welcomed.¯ The papers should be submitted either by electronic mail to both editors, or on computer diskettes(5 ¼" or 3.5") to Gy6rgy Darvas as text files (IBM PC compatible or Apple Macintosh); that is,conventional characters should be used (ASCII) without italies or other formatting commands. Of coursetypewritten texts will not be rejected, but the preparation of these items takes longer. For any method ofsubmission (e-mail, diskette, or ~ript), four hard-copies of the text are also required, where all thenecessary editing is marked in red (inserting non-ASCII characters, underlining words to be italicized,etc.). Three hard-copies, including the master copy and the original illustrations, should be forwarded toGy6rgy Darvas, while the fourth copy should Ix: sent to D~nes Nag3’. No manuscripts, diskettes, or figureswill be returned, unless by special arrangement.

¯ The papers are accepted for publication on the understanding that the copyright is assigned to ISIS-Symmetry. The Society, however, aiming to encourage the cooperation, will allow all reasonable requeststo photocopy articles or to reuse published materials. Each author will receive a complimentary copy ofthe issue where his/her article appeared.

¯ Papers should begin with the title, the proposed running head (abbreviated form of the title of less than35 characters), the proposed section of the quarterly where the article should appear (see the list in thenote ’Aims and Scope’), the name of the author(s), the mailing address (office or home), the electronicmail address (if any), and an abstract of between 10 and 15 lines. A recent black-and-white photo, thebiographic data, and the list of symmetry-related publmcations of (each) author should be enclosed; seethe sample at the end.

¯ Only black-and-white, camera-ready illustrations (photos or drawings) can be used. The required(approximate) location of the figures and tables should be indicated in the main text by typing theirnumbers and captions (Figure 1: [text], Figure 2: [text], Table 1: [text], etc.), as new paragraphs. Thefigures, which will be slightly reduc~I in printing, should be enclosed on separate sheets. The tables maybe given inside the text or enclosed separately.

¯ it is the author’s responsibility to obtain written permission to reproduce copyright materials.¯ Either the British or the American spelling may be used, but the same convention should be followedthroughout the paper. The Chicago Manual of S~yle is recommended in case of any stylistic problem.¯ Subtitles (numbered as 1, 2, 3, etc.) and subsidiary subtitles (1.1, 1.1.1, 1.1.2, 1.2, etc.) can be used,without over-organizing the text. Footnotes should be avoided; parenthetic inserts within the text arepreferred.¯ The ~ of references is recommended. The citations in the text should give the name, year, and, ifnecessary, page, chapter, or other number(s) in one of the following forms: ... Weyl (1952, pp. 10-12) hasshown...; or ... as shown by some authors (Coxeter et al., 1986, p. 9; Shubnikov and Koptsik 1974, chap. 2;Smith, 1981a, chaps. 3-4; Smith, 1981b, so:. 2.12; Smith, forthcoming). The full bibliographic descriptionof the references should be collected at the end of the paper in alphabetical order by authors’ names; seethe sample. This section should be entitled References.

111

Sample of heading (Apologies for the strange names and addresses)SYMMETRY IN AFRICAN ORNAMENTAL ARTBLACK-AND-WHITE PA’YI’ERNS IN CENTRAL AFRICARunning head: Symmetry in African ArtSection: Symmetry: Culture & Science

Susanne Z. Dissymmetrist and Warren M. Symmetrist8 Phyilotaxis Street Department of Disaymmetry, University of SymmetrySunflower City, CA 11 Z35, U.S.A. 69 Harmony Street, San Symmetrino, CA 69869, U.S.A.

E-mail: symmet rist @symmetry.eduAbstract

The ornamental art of Africa is famous ...Sample of references

In the following, note punctuation, capitalization, the use of square brackets (and the remarks inparentheses). There is always a period at the very end of a bibliographic entry (but never at other places,except in abbreviations). Brackets are used to enclose supplementary data. Those parts which should beitalicized -- titles of books, names of journals, etc. -- should be underlined in red on the hard-copies. Inthe case of non-English publications both the original and the translated titles should be given (cf.,Dissymmetrist, 1990).Asymmetrist, A. Z. (or corporate author) (1981) Book Title: Subtitle, Series Title, No. 27, 2nd ed., City

(only the first one): Publisher, vii ~ 619 pp.; ~further data can be added, e.g.) 3rd ed., 2 vols., ibid.,1985, viii + 444 + 484 pp. with 2 computer diskettes Reprint, ibid., 1988; German trans., GermanTitle, 2 vols., City:. Pub tsfier, 1990, 98~ pp.; Hungarian trans.

Asymmetrist, A. 7.., Dissymmetrist, S. Z., and Symmetrist, W. M. (1980-81) Article or e-mail article title:Subtitle, Parts l-2,.loumal Name Without Abbreviation, [E-Journal or Discussion Group address:journal@node (if applicable)], B22 (v.olume number), No. 6 (issue number if each one restartspagination), 110-119 (page numbers); B23, No. I, 1 I7-132 and i48 (for e-journals any appropriatedata).

Dissymmetrist, S.Z. (1989a) Chapter, article, symposium paper, or abstract title, [Abstract (ifapplicable)], In: Editorologist, A.B. and Editoro]ogist~ C.D., eds., Book, Special Is.me; ProceedinB~or Abstract Volume Title, [Special Issue (or) Symposium organized by the Dissymmetry Society,University of Symmetry, San S~nnmetrino, Calif., December 11-22, 1971 (those data which are notavailable from the title, if applicable)], Vol. 2, City:. Publisher, 19-20 (for special issues the data ofthe journal).

Dissymmetrist, S. Z. (1989b) Dissertation Title, [Ph.D. Dissertation], City:. Institution, 248 pp. (ExhibitionCatalogs, Manuscripts, Master’s The~es, Mimeographs, Patents, Preprints, Working Papers, etc. ina similar way;, Audiocassettes, Audiotapes Compact Disks Computer Diskettes, ComputerSoftware, F ms, M crofiches, Microfilms, Slides, Sound D sks, V deocasettes, etc. with necessarymodifications, adding the appropriate technical data).

Dissymmetrist, S.Z., ed. (1990) Dissirrunetriya v nauke (title in original, or transliterated, form),[Dissymmetry in science, in Russian with German summary], Trans. from English byAntisymmetrist, B. W., etc.

Phyllotaxist, F. B. (1899/1972) Title of the 1972 Edin’on, ’[Reprint, or Translation, of the 1899 ed.], etc.[Symmetrist, W. M.] (1989) Review of Title of the Reviewed Work, by S. Z. Dissymmetrist, etc. (if the

review has an additional title, then it should appear first; if the authorship of a work is not revealedin the publication, but known from other sources, the name should be enclosed in brackets).

In the case of lists of publications, or bibliographies submitted to Syraraetro-graphy, the same conventionshould be used. The items may be annotated, beginning in a new paragraph. The annotation, a maximumof five lines, should emphasize those symmetry-related aspects and conclusions of the work which are notobvious from the title. For books, the list of (important) reviews, can also be added.

Sample of biographic entryName: Warren M. Symmetrist, Educator, mathematician, (b. Boston, Mass., U.S.A., 1938).Address: Department of Dissymmetry, University of Symmetry, 69 Harmony Street, San Symmetrino,Calif. 69869, U.S.A. E.mail: symmetrist@symmetry.eduFields of interest: Geometry, mathematical crystallography (also ornamental arts, anthropology -- non-profe~ional interests in parentheses).Awards: Symmetry Award, 1987; Dissymmetry Medal, 1989.Publications and/or Exhibitions: List all the symmetry-related publications/exhibitions in chronologicalorder, following the conventions of the references and annotations. Please mark the most importantpublications, not more than five items, by asterisks. This shorter list will be published together with thearticle, while the full list will be included in the computerized data bank of ISIS-Symmetry.

112 AIMS AND SCOPE

There are many disciplinary periodicais and symposia in various fields of art, science, and technology, but broadinterdisciplinary forums for the connections between distant fields are ve~, rare. Cons~uently, the interdisci-plin.ary papers.are d!spersed in very different journals and proceedings. This fact makes the cooperation of thea.uthors difficult, and even affects the ability to locate their papers.In our ’split culture’, there is an obvious need for interdisciplinary journals that have the basic goal of buildingbridges (~symmetries’) between vanons fields of the arts and sciences. Becans~ of the variety of topic~ available,the concrete, but general, concept of symmetry was selected as the focus of the journal, since it has roots in bothscience and art.

SY~O~.T~Y:._ C~L~URg .~aVD SCIENCE is the quarterly of the l~rrF.~o~ SOClETY ~OS T~ INTERDI$¢~PLIHARYSTUDY OF ~’Y~O, fETRY (abbreviation: ISIS-Sy6une~y, shorter name: Symmeoy Sociay). ISIS-S]~,mmetry was foundedduring the symposium Symmetry of Sm~cture (First Interdi~cipFma~ Symmetry Symposu~n and Exhibition),Budapest, August 13-19, 1989. The focus of ISIS:Symmetry is not only on the concept of symmetry, but also itsas.socrates (asymmetry~ dissymmetry, antisymmetry, etc.) and related concepts (proportion, rhythm, invanance,el,’) in aninterdisciphnary and intercultural context. We may refer to this broad approach to the concept assymm~_.o_lo~. The suffix -/ogy can be associated not only with knowledge of concrete fields (cf., biology, geol-ogy, philology~ l~ychology, sociology, etc.) and discourse or treatise (cf., methodology, chronology, etc.), butalso ~ith the Greek termtnology of proportion (cf., logos, an4alo~ia, and their Latin translations rado,propo~do).The basic goals of the Society are

(1) to bring t.ogether artists and scientists, educators and students devoted to, or interested in, the researchand understanding of the concept and application of symmetry (asymmetry, dissymmetry);

(’2) to provide regular information to the general public about events in symmetrology;(3) to ensure a regular forum (including the organization of symposia, and the publication of a periodical) for

all thes~ interestedin symmetrology.The Society organiz~ the triennial lnt~dlsciplinary Symmet~ Symp_~_si~_ and Exhibition (starting with the sym-posium of 1989) and other workshops, mcetmgs, and exhibitions.~he forums of the Society are informal ones,which do not substitute for the disciplinary conferences, only supplement them with a broader perspective.The (~uarterly - a non-commercial scholarly journal, as well as the forum of ISIS-Symmetry - publishes originalpapers on symmetry and related questions which present n~v results or new connections between known results.The papen; are addre~ed to a broad non-speciahst public without becoming too general and have an interdis-ciplinary character in one of the following senses:

(l) they describe concrete interdisciplinary ’bridges’ between different fields of art, science, and technologyusing the concept of symmetry;fi_(,2)-_theYc~u.~, survey the importance of symmetry in a concrete field with an emphasis on possible °bridges’ to other

The Ouarterly also has a special interest in historic and educational questions, as well as in symmetry-relatedrecreations, games, and computer programs.The regular sections of the Quarter/y:¯ Symmetry: Culture & Science (papers classified as humanities, but also connected with scientific questions)¯ Sy~nmetry: Science & Culture (papers classified as science, but also connected with the humanities)¯ Symmetry in Education (articles on the theory and practice of education, reports on interdisciplinary

projects)¯ Mosaic of Symmetry (short papers within a discipline, but appealing to broader interest)¯ SFS: Symmetric Forum of the Society (calendar of events, announcements o~ ISIS-Symmetry, news from

members, announcements of projects and publications)¯ S .y~unet~-gruphy (biblio/disco/software/I-udo/historio-graphies, reviews of books and papers, notes on

anmversanes)¯ Rellections: Letters to the Editors (comments on papers, letters of general interest)Additional non-regular sections:¯ S3~metrospcw.t|v¢: #, Historic View (survey articles, recollections, reprints or English translations of basic

papers)¯ Symmetry~ A Sl~¢ial Foc~ oa ... (round table discussions or survey articles with comments on topic~ of

special interest)¯ ~,vrametr~ An Interview with _ (discussions with scholars and artists, also introducing the Honorary

members ot ISIS-Symmetry)¯ Symmetry: The Interfac~ otArt & Selene® (works of both artistic and scientific inter~t)¯ R~v.at|onal Symm©try (problems, puzzles, games, computer programs, descriptions o[scientific toys;

for example, tiling~, polyhedra, and origami) -Both the lack of seasonal references and the centrosymmetric spine design emphasiz~ the international charac-ter of the Society;, to accept one or another convention would be a ’symmetry violation’. In the first part of theabbreviation ISIS-Sym:~_nc._ay all the letters are capitalized, while the centrosymmetric image iSIS! on the spine isflanked by ’Symmetry’ from both directions~ T[iis Convention emphasizes that ISIS-Symmetry and its quarterlyhave ao direct connection with other organizations or journals which also use the word l~is or ISIS. There aremore than twenty ~den~ical acronyms and more than ten such periodicals, many of which have already ceased toexist representing various fields, including the history of science, mythology, natural philosophy, and orientalstudio. ISIS-Symmetry has, however, some interest in the symmetry-related quest ons of many of these fields.

Germany. FR Andre.as Dress, Fakutt~t fOr Mathemallk,Unlvers~tal B~elefeld.D-336L5 B~elefeld 1, Postfach 8640, F R. GermanyIGeometry. Mathemat~zation of Soence]

Theo Hahn, Insorut fOr KJ-istallograph~e.Rhemtsch-Westf’absche Techniscbe Hochschule,D-W-5110 Aachen. F R. GermanyI M meralogy. Crystallography]

ttungary" Mlh~lly Szoboszlai, I~plUSszm~’rnokl Kar,Budapest= M6szakl Eg)’etem(Facuhy of Arcfotecture, Technical Umverslty of Budapest),Budapest. PO Box 91. H-1521 Hungary[Archnecture, Geometry, Computer Aided Architectural Design]

Italy Giuseppe Caglioti, Ist,tulo dl Ingegnena Nucleate -CESNEF. Pohtecmco d] Milan, Vm Portz~o 3413,1-20133 Mdano. Italy[Nuclear Physics. V~ual Psychology]

Poland Janusz Rebielak, Wydzial Architektury,Pobtcchntka Wro¢ tav¢~ k~(Department of Architecture, Techmca[ Umversity of Wroctaw),ul. B Prusa 53/55. PL 50-317 Wroclaw, Poland[Architecture, Morphology of Space St~ctures]

Portugal: Jos~ Lima-de-F-aria, Centro de Cristalografiae Mineralog~a. lnsotuto de Invest~ga¢~Io Ctentifica Tropical,AJameda D Afonso Henr~ques 41. 4.*Esq., P-J000 Lisb~a,Portugal[Crystallography. Mineralogy. History of Scmnce]

Rornanta Solomon Marcus, Faculmtea de Matematica,Universflztca d~n Bucurc.~li(Faculty of Mathemalics. Umvers~ty of Bucharest),Str Acadcmmi 14, R-70109 Bucurestl (Bucharest), RomaniaIMathemat~cal Analysis. Mathematical Lmgutstics and Poetics,Mathematical Semrottcs of Natural and Social Sctences]

P,u.ss,o Vladimir A. Koptsik, Fiztchesk~l fakultet,Moskovskn gosudarstveonyl umversltet(Physical Faculty, Moscow State Umverstty)117234 Moskva. Russm[C rystalphysics|

Scandmovla" Ture 9,~ster, Sktvelaboratoriel,Ba::rende Konstraktmner, Kongehge DanskeKunstakademi - Ark~tektskole(Laboratory for Plate Slructures, Department of SlructuralScience. Royal Damsh Academy - School of Architecture),Peder Skramsgade 1. DK-1054 Kobenhavn K (Copenhagen),Denmark [Polyhedral Structures. Biomechanics]

Switzerland" Caspar Schwabe, Ars GeometncaRamlstrasse 5, CH-8024 Zurich, Switzerlandtars Geometrical

UK. Mary Harris. Moths in Work Project.Insutute of Education. Umversity of London.20 Bedford Way. London WCIH 0AL. England[Geometry. Ethnomathemaocs. Textde Design]

Anthony Hill. 24 Charlotte Street. London WI, EnglandIV~sual Am. Matbemaucs and An]

Yugoslama. Slavik V. Jablan, Matematd:kl instttut(Mathemattcal Institute), Knez M~hallova 35, pp. 367,YU-II001 Beograd (Belgrade). YugoslaviaIGeometry. Ornamental Art, Anthropulogy]

Chatrpersorts of

Art and Sctence Exhlb,tions" L,~l~zl6 Beke,Magyar Nemzeti Gal6rm (Hungarmn Nat,onal Gallery).Budapest, Buda’.~n Palota, H-IOI4 Hungary

Itsuo Sakane, Faculty of Env,ronrnentallnformat*on, Kelo Umvers,ty at Shonan Fujisawa Campus,5322 Endoh, Fu)~sau,’a 252, J’apan

Cogmttve Science Douglas R. Hofstadter, Center for Researchon Concepts and Cognition, Indiana Umversuy,Bloomington, Indmna 47408, U.S A

Computing and Applied Mathematics: Sergei P. Kurdyumov.lnst~tut prlkladnol matematiki ~m. M V Keldysha RAN(M V Keldysh Institute o[ Apphed Mathematics, RussianAcademy of Sctences), 125047 Mosk~, Mtusskaya pl. 4, Russm

Educanon Peter Klein, FB ErT:iehungswissertschaft,Umverstt~t Hamburg, Von-Metle-Park 8,D-20146 Hamburg 13, ER. Germany

Htstory and Phdosophy of Science. Klaus Mainzer,Lehrstuhl Fur Phllosophm, Universitfi! Augsburg,Umversitatsstr 10, D-W-8900 Augsburg, ER Germany

Project Chairpersons

Archttecture and Mustc" Emanuel Dimas de Mein Pimento,Rua Tterno Galvan, Lore 5B - 2.*C, P-1200 L~sboa, Portugal

Am and Biology: Werner Hahn, Waldweg 8, D-35075Gladenbach. ER. Germany

E~9lutton of the Universe. Jan Mozr-zymas, Instytut Fizykt,Umwersytet Wroci’awski(inslltute of Theoreucal Physics, University of Wroc~’aw),ul. Cybulskaego 36, PL 50-205 Wrocl’aw. Poland

Higher-Dimensional Graphics Koji Miyazaki,Department of Grapfocs, College of Liberal Art&Kyoto Umvers~ty, Yoshida, Sakyo-ku, Kyote 606, Japan

Knowledge Repnvsentation by Maastwacmres: Ted Goranson,Sir,us Incorporated, 1976 Munden Point. Virgima Beach,VA 23457-1227. U.S A

Pattern Mathematics: Bert Zaslow,Department of Chermstry, Arizona State University, Tempe,AZ 85287-1604, U.S.A.

Polyhedral Transformations" Hare~h Lalvani,School of Architecture. Pran Institute, 200 Willoughby Avenue,Brooklyn, NY 11205, U.S.A

Proportion and Harmony in Arts: S. K. Heninger, Jr.Department of English. Umversity of North Carolina at ChapelHill, Chapel Hill. NC 27599-3520, U.S.A.

Shape Grammar: George Stiny, Graduate School of Architectureand Urban Planning, Universtty of Cahfonfia Los Angeles,Los Angeles, CA 90024-1467, U.S.A.

Space Strucnares" Koryo Miura, 3-9-7 "Tsurul~v,-a, Machida,Tokyo 195, Japan

Tibor Tarnai, Technical Untverslty of Budapest,Department of Cwll Engineering Mechamcs,Budapest, Mdegyetem rkp. 3. H-IIII Hungary

Liaison Persons

Andra Akem (International Synergy Institute)Stephen G. Davies (Journal Terrahedron: Assymmetry)Bruno Gruber (Symposia Symmemes in Science)Alajos KAIm~n (International Union of Crystallography)Roger F. Malina (Journal 12onordo and International Society forthe Arts, Scmnces, and Technology)Tohru Ogawa and Ry’uji Takak{ (Journal Forma and Society forScmnce on Form)Dennis Sharp (Comit~ International des Critiquesd’Archttecture)E.rT~bet Tusa (INTAK’T Society)

Eo.’~