THE SQUARE

The square is as high and as wide

as a man with his arms outstretched. In the oldest

writings and in the rock inscriptions of early man, it signifies the idea of enclosure, of home, of settlement. Enigmatic in its simplicity, in the

monotonous repetition of four equal sides and four equal

angles, it creates a series of interesting figures ...

BRUNO MUNARI

The word ing o f the first edition has been kept to high­light the sources used by Munari and lo renew our grati­tude.

Au thor and publishers would like to thank all the pub­lishing houses listed on the last page for their contribu­tions, particularl y Einaudi and Bompiani ; and Arn::tra De Angelis, Fra nco Potenza, the Museum of Natu ral History in Milan, the Pisa Tourist Board , Max Bill and the archi­

tect Katzky Jwabuci.

BRUNO MUNARI THE SQUARE

THE SQUARE

The square is as high and as wide as a man w ith his arms o utstretched. Jn the o ldest w rit­ings and in the rock inscriptions of ea rl y man , it signifies the idea o f enclosure, o f home, o f settlement. Enigmatic in its simplicity, in the monotonous repetition of four equal sides and four equal angles, it creates a series of interesting figures: a w ho le group o f harmonic rectangles, from the J-lemidiagon to the Six ton , generate the Golden Section and the logarithmic spiral found in na­ture in the organic growth o f plants and an imal parts. Its strucrural possibilities have helped arti sts and architecrs of all generations and sty les by giving rhem a harmon ic skeleron to w hich to apply an arti stic construction. Accordingl y, it is presenr in all sty les o f all peoples in all ages, borh as a structural element and as a surface rhat supports and determines a particular deco­ration. It is static if il stands on its side and dynam ic if it sta nds o n an edge. ll is mag ic if fill ed

w ith numbers and it ca n even be diabolic and sa tanic when these numbers retain the same relatio nship even w hen squared or cubed . Ir is found in nature in many minerals. Accord ­ing to Pea no it is a curve. It ca n be !urned into triangles o r rectangles only by making the right cuts and moves. In ancient times it had the power to drive out the plague. ft has given shape to famous ancient cit ies and even mod­ern buildings: Baby lonia, Tell el Amarna, rhe Parthenon , rhe arch of Septimius Severus, the Duomo of Pisa, Pala zzo Farnese, Le Corbusier 's museum of unlimited extendibility ... The bays o f Brunellc.:schi 's Po rti co degli Innocenti are square. In the g round plans o f many churches the square space benea th the hemispherica l dome is the most logica l shap e. In the same w ay, the square of the pho tograph coincides w ith the round lens, without waste or distor­tion . Phidias used a square module fo r his lacu­naria. Al the Acropo lis o f Olympia the Palestra, the Theocoleon, the Leonidaeum and other buildings had square plans .. Many ancient games still played tod ay are based o n the square: chess, checkers, the 15 game w ith its ten trillion possible combinations, dice , puss-in-the-corner. .. and then there are the famo us cowboy Square Dances. Jn the Eastern Chin dy nasty it gave Chinese writing a stable square form . It gave structure to the len ers of our alphabet and to the He­brew o ne and others. A small square o f cloth is a handkerchief the world over. Two square mats are the basic unit for the traditional Japa­nese house. Twenty-eight squa res cover the sur­face o f a brick. According to an ancient Chinese saying the infinite is a square w ithout ang les. "The squa re is the finest expression o f a spat ial idea complete in itself. It represents an o rder o f charged spiritual symbolism. All other rect­angles, with their different sides, deri ve from the square but relax its law by expanding in height o r w idth ".

D Horologion

0 10 20 30 40

7

AGO RA

Portn Nord

N

\

Plan o f the Agora o f Ephesus in the Hellenistic period.

JOSEF ALBERS

Homage to the "twilight" square. 1951.

ALTA I~ OF T H E HOLOCAUSTS

"And thou sha lt make an altar o f shittim wood , five cubi ts long, and five cubits broad ; the altar shall be fou rsquare ... " Second book of Moses - Exodus 27.

8

AAGARD A DERSE

Black and wh ite image on a surface of 48 squares.

ANAGUTA

Pictures fro m America n p etrog lyph s al An a­guta.

ANCIENT EGYPT

In ancient Egypt art was based o n geometry and on the fi xed rules o f a ca non . The square was at the basis o f this ca non . Since the time of the O ld K ingdom , as the third dynasty tomb

9

o f King Zoser at Sakkara show s, the artists w ho were to decorate a wa ll bega n by div iding it up into a network o f squares so as to be able to ca lculate the proportions. Traces o f these grids we re also found in o ther tombs at Sakkara and in the tombs o f the New Kingdom at Thebes. In sculpture the square inspired the so-ca lled cubic statues in w hich the human figure, usu­all y a person o f rank, was arranged so that it remained set w ithin the coord inates of a cube. A typical exa mple o f this is the statue o f the architect Senmut of the X VIII dy nasty, in the Museum of Ca iro. The square is central to the pyramid and, in the Gnostic period , mystica l and symbological interpretations associated w ith it became wide­spread. With its four sides, the square w as seen as the symbo l of the four elements and there­fo re of "materi ality ", subord inate to the triang le w hose three sides, symbo l o f the spirit , also determine a schematic ri sing fl ame: py r, fire, thus pyramid.

A box kite of two simple parts.

ROMAN ARCH

' '

_::)<~---, __ /

// \ - - - 7- -- - - -- - - -\.- - - - -

/ \

/

Proportio ns of Septimius Severu s Arch in Rome.

JO

MAYAN ARCHITECTURE

Detai l of a fri eze at the Temple of Uxmal , Yu­ca tan . The structure o f all Mayan bu ildings is based on mathematica l laws, units and exact proportions.

BABYLONIA

Babylonia is a square city, says Herodotus, and each side measures one hundred and twenty stadia. The city is fu ll o f three o r four-sto rey houses and it is crossed by straight parallel streets perpendicular to the r iver and is divided into two parts. On one side is the pa lace and o n the o ther the Zeus Belo sa nctuary w ith its bronze doors, measuring two stadia square. In the middle o f the sa nctu ary there stands a mas­sive tower w ith a square ba se measuring one stad ium per side. This tower supports o ther towers, o ne on top o f the o ther in decreas­ing o rder. In all there are eight tow ers. O n the last tower there is a g reat temple and in the temple a richly adorned bed and , beside this, a table made o f gold . o deity nor human being lives in this p lace, except one woman chosen by God.

BA-M I OM

LJ Brahmin character, pho netic value: ba. T he sa me sign in Korean script has the phonetic va lue: miom.

BET

[)0[] Egyptian hieroglyphics, phonetic va lue: bet.

II

GREEK BAS-RELIEF

/ /

/

Proportions of a Greek bas-relief, Athens.

---- --

1111 1111

II

.. .. -:. ii!

11 m II

I+~ ·~ ••• ~

-------

BAUHAUS

Ex periments w ith different groupings o f nine squares. The Bauhaus, Weimar.

BELLO LI

the fi eld

square

the square

square

the city

square

the pri son

square

the tomb

square

the tent

square

the ski n

square

the pupil

square

the square

is

society.

From p oem s o n elementa ry geometry, Basie 1959

13

BAAL SHAMIN

r I

•, .I .... ~ •· .;'.it.:·.'.

Axonometric v iew of the m onumental buildings of the temple o f Baa l Sharnin at Palmyra (re­construction).

MAX BILL

Field w ith eight groups, 1956.

14

Monument to the Unknown Politica l Prisoner 1952.

White element, o i l on canvas, 68x68 cm d iago­nal.

MOUTH

Chinese ideogram meaning "mouth ".

HOUSE

The word " house" in Sumeri an ideographic script.

MASUZAWA'S HOUSE

Home of the architect Masuzawa in Tokyo, 1952. 16

GOTHIC CATH EDRAL

Propo rtions o f Chalons sur Marne ca thed ral, cross-section, Q uerschnin.

ELECTRONIC BRAIN

CHURCH OF SANTA SOPHIA

Santa Soph ia, Constantinople. The plan consists of a series of units based on a central square surrounded by rectangles w ith one side mea­suring ~ 2 times the other.

FIVE TIMES

Given a square, execute another one fi ve times larger.

18

CHURCH OF SAN LORE NZO

-· -·-· -··--·-· -· -·-· -· --~

The Church of San Lorenzo, Turin , by Guarino Guarin i . The whole composition is based o n the square which is in turn divided into squares and golden rectangles.

SHELLS

Th e o rga nic growth o f many shells develops according to the curve o f the logarithmic spiral , w hich derives from the square.

COMPOSITIONS

Composition w ith four squares.

Composition w ith eight squares.

20

CONCAVE CONVEX

';,------+----'

Model obtained by bending a square of meta l netting unti l the corners touch at pre-estab­lished points on the surface. The explanatory diagrams show the va rious stages in the devel­opment o f the model. The po ints of anchorage are established o n the surface o f the square

in ha rmo nic proportio ns. The nets, curved in this way, ca n be hung fro m the ceiling so that their transparent mo bile shadows decorate a wall o f the room, g iving it a c ross-hatched effect. Munar i. 1948.

21

SOLAR ENERGY CONCENTRATOR

This concentrator is used in America to har­ness solar energy. It is composed o f 180 adjust­ab le square mirro rs, each about 50 cm square.

22

AGA INST THE PLAG UE

In anc ient times the square was considered to have magica l properties, among w hich the power to wa rd o ff the pl ague, and it was cus­tomary to hang a sil ver pendant, w ith a square engraved on it , around the neck .

ROMA ESQUE CROSS

Notre Dame du Po n , Clermont Ferrand, elev­enth century.

CRETE

t! l2J Characters from a M inoan inscriptio n, Crete.

HOOF COVER ING

23

These hyperbolic paraboloid roofa are produced by disto rting a flat square. T he shape of the roof changes depending on ho w the four basic elements are jo ined.

PEANO'S CUHVE

One o f the ca rdin al princ iples of geometry is that po ints have no dimensions and that a curve has only one dimension and can there­fore never fi ll an area. This is another firm be­l ief that has to be shattered because "Pea no's curve", w hich belongs to the supreme type of pathologica l curves, does indeed fi ll an area. Not only does it occupy the w ho le inside o f a square but it climbs into the space o f a whole cubic box. T he subsequent stages o f this curve are shown in the fo llowing figures:

Take any po int of the square o r the cube. It ca n be demonstrated that, w hen completed , the curve will pass through this point. An I since the same reasoning ca n be extended to any other po int, the logica l conclusion is th at the curve must fill the whole of the square and the whole of the cube. Mathematicians once held that every curve had to have a tangent because, they claimed , this property is sel f-evident. But in 1890 Peano, to the amazement of the math­ematicians, devised a curve that cl id not allow for tangents at any o f its po ints, a curve that filled the whole area of a square, and as such could o nl y be represented by a sq uare b lack

24

mark. Since then mathematic ians have been fa ­miliar with curves w ithout tangents.

CH EC KEHS

Checker players in the open air at Bour­nemouth.

ISLAMIC DECORATION

Thirteenth century Islam ic mural decoration.

D IAGONAL

Given the d iagonal, construct the square.

DIAGONALS

The diagonals o f these two shapes are eq ual.

26

DIATOM

Microscop ic algae are o f va rious fo rms, some of them square.

D ISTORTIONS

Distortions o f a square surface, keeping the two med ians the sa me.

DOUBLE, TRIPLE

c=---=-=--=---=:--:-;-1 I -;') . r ·- - ----- 71· I I ~-------~''. i i . . i I I i

. I . . - '1'7 ' 'i i

lffl Given a square, make one double, triple, qua­druple the size, etc.

27

DUPLICATION

The problem o f duplicat ing the square was easil y solved by the Greeks by constructing a new square on the diagonal of the first. But this raised another problem: duplication of the cube. Working on this problem M enaechmus tried in va in to resolve it w ith a construction o f straight lines and arcs o f circles and w ith o ther curves. After much work, instead of the duplica tion o f the cube, he discovered the el­lipse, the parabo la and hyperbola.

ALBRECHT D ORER

I I

I

i

, I

' \ '

" I

' '

,

/

-/ ,

, 'x./ / '

/ --/ ',,

' ' ' ' \ / '

./<',

/

' /

,)<.

'

/

/

\

' \

I I

I , I

/

Proportions of the chu rch o f Santa Maria degli Angeli , by DC1rer.

INTERNAL DIVISIONS

Divisions of the internal space o f a square sta r t­ing from the princ ipa l combinations between l i nes and curves deriving fro m the measure­ments of the square itself.

28

DUOMO OF PI SA

Pro portio ns o f the fa<;;ade o f the Duomo o f Pi sa.

GOD OF THE TEMPEST

Temple of Goel o f the Tempest at Carchemish. Its sanctuary was a perfect square.

VAN DOESBUHG

Arithmetica l composit ion. 1930.

30

HEX AGON

The projection o f a cube w ith one o f its edges standing on the nat is a hexagon.

IVOHY LABELS

'1· .1

nt{nn nnn ''' . -•.

Ivory labels w ith numerica l mark ings, found in the roya l tomb attributed Lo Menes, founder o f the first Nagacla dynasty, in Egypt.

EUCLI D

The square c ircumscri b ing a g iven circle has tw ice the area o f the square inscribed in the sa me circle.

SQUA RE HAND KERCHI EF

A scene from Verd i 's O thello.

31

FA UTRIEH

The Empty Box, painting, 1954.

IRON

When a mass o f mo lten iron is cooling, the transformation from liquid to solid takes place clue to the formation o f crystallo icl nuclei in the shape of cubes (as ca n be seen in a series of micropho tographs) that rapidly multiply alo ng crystallographic lines, until the w ho le mass has been filled.

DITCH

[J Character used in the w riting o f the Bamon tribe (formerl y the German Ca meroons) to in­dicate "ditch". Similar characters are also found in Proto-Sinaitic w ritings.

FORMS DERfVED FROM THE SQUA RE

Two-dimensional compositions obtained from combining, ro tating o r modify ing square sur­faces.

0

32

FLUORfTE

Right-angled st ructure o f fluo rite cr ys tal s.

WIN DMILL

Fou r cuts along th e diagonal s of a square of ca rdboa rd make up this simple toy.

GARDEN

The ga rdens at Villa Medici on the Pincio hill , Rome. Architect: Annibale Lippi.

34

SUNFLOWER

The arrangement o f the seeds of the sunflower fol lows the curve of the loga rithmic spira I.

GAMES

Some of the games de ri ved from the squ a re: chess, checkers, dominoes, puss- in-the-corne r.

THE FIFTEEN GAME

1 2 3

5 6 7

The 15 game, ca lled the "15 puzzle" by its in­ventor Sam Lloyd in 1878, was an incredible success. Employers had to put up notices forbid­ding their staff to play the ga me during work­ing hours , unde r pa in of dismissa l. A French journalist at the time claimed that it was mo re harmful than tobacco and alcohol , and respon­sible for headaches, ne uralgia a nd neurosis. Fabulous prizes were o ffe red for the solution o f seemingly simple proble ms but no o ne ever won the m. The numbe red squares ca n theore tica lly be moved around the box in 20,922,789,888,000 differe nt positions. But two America n mathe ma­ticians, Johnson and Story, de mo nstrated that, starting from a given position, onl y half of a ll the possible positions can be reached. Thi s mea nt that only te n trillion moves could be made a nd that another te n trillion could not. The fact that there were positions that could

36

not possibly be reached explains w hy the gen­erous pri zes were never won, a nd chance had it th at the prizes were a lways offered for the impossib le positio ns.

GRADATION OF VALUES

•••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• ................ •••••••••••••••• •••••••••••••••• • ••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• •••••••••••••••• Gradat io n o f va lues of a surface obta ined by disto rting a grid pattern. Ulm School. 1957.

JAPANESE TOYS

Numerous toys ca n be made hy the .J apanese art of folding a square of p aper in di ffe rent ways. Some o f t he most fa mo us are a bird that moves its w ings, a fold ing box, an object shaped li ke four empty pyramids turned upside down and attached together so th at you ca n put fou r fi ngers inside.

INCOMMENSURAB LE SIZES

Cont ra ry to the Pythagorea n conce1) t o f the ex­tended point, incommensurable sizes, i.e. sizes th at have no common submultiple, do ex ist. T he d iscover y of the incommensurabil it y be­tween the sizes o f the side and diagonal of the square would seem to have been m ade by Hip­pasus of Metapontum , a fo llower o f Pythagoras, w ho having been the first to make public the

37

theory o f incommensurables, then perished in a shipw reck because, according to legend , the in­expressible and unimaginable should have been kept hidden forever. So the culprit w ho had unw ittingly come across and revea led this truth o f nature, was taken back to his place of origin to be thrashed by the waves fo r ever. The legend probably hides a rea l feeling o f re­sentment about the fact that the results revea led to the general public were unquestionably ex­tremely embarrassing because they showed that Py thagoras was mistaken in his conception o f the po int. This discovery, in fact, led to a revi­sion of the fundamental concepts o f geometry, w hich in turn made way for the famous soph­isms o f Zeno and , eventually, to the Elements o f Eucl id.

GOTHI C

Decorati ve Gothic mo tifs.

MAX HUBER

Composition on the square.

38

I !OMO QUADRATUS

In the theory o f Homo Q uaclratus the number, principle o f the uni ve rse, takes o n a symboli ­cal significance based on a seri<.:s o f numerica l correspo ndences that are also aesthetic corre­spondences. Her<.: too the first concepts of the theory had to do with music: there are eig ht musical tones - no ted an anony mous Ca rthu­sian monk - because four were found by the ancients and fo ur were added by the moderns (he is rele rring to the four authentic modes and th <.: four p laga l modes). The number four, ac­cord ing to other w riters o r to common belief, is a key number, a powerfu l seria l definer. Four are the cardin;tl points, the principal w inds. the phases o f the moon , the seasons, four is the constituti ve numb<.: r of Timaeus' tetrahedra l o f fire, four are the letters o f the name ADAM. A nd , as Vi tru v ius teach<.:s , four is the number o f man hecaus<.: the w idth of a man wi th hi s arms outstretched corresponds to his height . thus forming the base and height of an ideal square; and fou r is the number o f mo ral per­fection , since the mQra ll y battle- traim:d man stands four-square.

Pl IOTOG HAPH

Photograph of th <.: elements of a bridge, by Ma x Huber.

OPTICAL ILLUSIONS

Optica l illusions o n square spaces by Hering (or Helmholtz).

CUB IC LAMP

The shape o f this lamp, contained w ithi n a cube, makes it simple to assemble and d is­mount th anks to two diagona l meta l frames. Its simplicity makes it very inexpensive to pro­duce.

HOMANESQUE IN ITI ALS

From a twelfth century manuscript.

KAHL GEHSTNEH

Blue-yellow sequence on red. 1958-59.

40

INFINITE

·The infinite is a square .. w ithout corners". O ld Chinese proverb.

THE CHI NESE AND PYTHAGORAS

Ahout five centuries before Pythago ras, the Ch inese knew the relationship between the sides o f a right-angled triang le. The demonstra­tion found in the "Book of K ing Ciu-Pei-Suan" is not unlike th at of Py th agoras. Let a and h be two sides of a r ight-angled tri ­angle (the sides of the right angle), place them so as to have a sing le straight line. Construct the square ABCD. Mark the meeting po int o f the two segments a and h o n all four sides of the square at E, G , H , F. j oi n E w ith F and G w ith 1-1 to divide the square ABCD into an area that is the square of a, an area that is the square of b and two rectangles each o f area ab. Div ide each o f these two rectang les diagonall y and we have four equa l right-a ngled triang les with hypotenuse c. Now place the four right-a ng led tri ang les o n the area ABCD so that the right ang les o f the triangles coincide w ith the right angles o f the squ are, and we obtain in the midd le an area that is the square o f hypotenuse c.

A

G

D

4 I

a E h B

-- -- - --- H

F c

c

KLEE

Studies by K lee on squares.

Dynamic transience o f a g iven static.

42

LE CORB USIER

Museum o f unli m ited extendabili ty seen fro m above. Construction on pi les, entrance in the centra l square. Infinite number of different in­ternal arrangements with prefabricated units.

MAZES

Layout o f mazes in seventeenth cernury Italian v i lla ga rdens.

~ .._ ,.. .... ...

• -I 1 -

I

~ ...

43

LEONARDO

Human figure set within a square, for a study o f proportions .

In the world of Egyptian priesthood the initi­ates conside red the square as the expression of man, in that when man stands upright with his arms outstretched, height and width are of equal measure.

LEONARDO

VERENA LOEWENSBERG

Proportions of a profi le .

THE GREEK PATTERN

Composition 257. 1956-57.

44

THE EIG HT QUEENS

• •

• • •

• •

• •

Problem : p lace eight queens on an ord inary chessboard w ith 64 squares in such a way that none o f them can be taken by the one of the others; this was proposed by Nauck in 1850 and resolved by him , and later by others. There are 92 possib le solutions. Here are two o f them, the second o f which is symmetrical , i .e. it does not change even w hen the chessboard is ro ­tated by 180 degrees .

• •

• •

• •

• •

RI CHARD P. LO HSE

Hori zontal divisions. 1949-53. 45

KASIMIR MALEVlC

"When in 1913, in my desperate attempt to free art from the useless burden o f the object, I sought refuge in the form of the square and exhibited a painting that represented nothing o ther th an a black squ are on a w hite back­ground . The criti cs complained and w ith them the public: "Everything we have loved has been lost; we are now in a desert. Before us there is nothing o ther than a bl ack square on a w hite background". The p erfect square seemed to the critics and to the public incomprehensible and threatening .. this was however to be ex­pected. The climb to the top o f non-fi gurati ve an is hard and full o f torment... but sa tisfying none­theless. Rea l things move further and further back , objects disappear w ith each step back, until at last the world of ordinary ideas, in w hich we however li ve, va nishes altogeth er. Away w ith pictures of rea li ty, away w ith idea l representation - nothing but a desert' But this desert is fu ll o f the spirit of non-objec­tive sensibility, wh ich penetrates everywhere. I too felt a sort o f shyness and I hesitated to the point of anguish because it mea nt aba n­don ing the "world o f w ill and representation" in w hich I had lived and created , believing in its authenticity. But the sense of sa tisfaction I felt at my lib ­eratio n o f the object ca rried me even further into the desert , to t he po int w here noth ing was authentic except fo r sensibility ... and that is how sensib i l ity became the ve ry substance of my life. The square I had exhibited was not an empty square, but sensibility in absence of an object. Painting has been surpassed for a lo ng time, and the pa inter is a prejudice o f the past ...

Suprematist composition, painting, 1915.

46

MONUMENT IN UDINE

Monument to the Resistance, in Udi ne. Mod­el by arch itects Va lle and Ma rcori , seen fro m above.

ENZO MARI

Experim ental swcly o f co lour on cuhic space.-;.

CHINESE COINS

Chinese coins with square ho les.

48

USE LESS MAC HI NE

II • • .. • ._

Harmo nic sketch o f si x elements that make up Munari 's "Useless machine 1956''. Th e six elements placed together on one level occupy a space of two and a hal f squares. Hung on ny lon threads they move in infinite com b ina­tio ns. The object is made o f natu ra l anod ised aluminium.

MODULOR

\

Le Corbusier 's modu !or is a unit o f measure­ment based on the human figure and on math­ematics.

MOND RI AN

Composit ion wi th reel , painting, 1936.

50

TRADE MARKS

g

51

HALF

", " ' .· I . , '.

/ i ".\ \ ./ · ! j '

Give n a square , draw o ne that is half or three­q uarters smaller.

NICAEA

The city of Nicaea, in Bithynia (around 300 BC) was formed of a square measuring about 700 metres a side, with two large roads that inter­sected at rig ht angles and joined the four gates of the city.

MUSICAL NOTES

A line o f mus ic written in the e leventh cen­tury.

SPIRAL NEBULA

The loga rithmic sp iral is found in some nebu lae.

PIER LU IGI AND A TONIO ERVI

"Palace o f Work " in Turin .

52

NEGATIVE POSITIVE

Pictorial com postt1o n w itho ut background , in w hite, black and red. Munari 1951.

NINTH ORDER

Magic square o f the ninth o rder arranged ac­cording to Bachet's rule. The sum of the numbers in each vertica l column or each hori zontal line is always the sa me.

77 28 69 20 61 12 53 4 45

36 68 19 60 11 52 3 44 76

67 27 59 10 51 2 43 75 35

26 58 18 50 42 74 34 66

57 17 49 9 41 73 33 65 25

16 48 8 40 81 32 64 24 56

47 7 39 80 31 72 23 55 15

6 38 79 30 71 22 63 14 46

37 78 29 70 21 62 13 54 5

NOTRE DAME

J ,,

" I

' ' )

,'

~l--'--~------>------'

Scheme of the fac;:ade of Notre-Dame.

OPERA QUADRATA

Ancient system o f maso nry in Etruria and Lazio whereby square blocks of marble or tufa were placed on top of one another.

COUNTRY

l~ Japanese ideogram for "country": an enclosure (the frontiers) contains a mouth (the people) and a bow (the army).

54

PAVILION

CM. 260 11 2 60

CM. 65 x 65

---·-1

..... I '---+=--=+- ________ _ . .J

Textile pavi l ion at the eleventh Milan Triennal ex hibition , designed by the architect Edoardo Sianesi. Ground plan and one of the four equal sides.

SQUARE PYRAM ID

As shown in the pho tograph , a square of ca rd­boa rd with many alternat ing cuts, suspended from the centre, spontaneously forms this three dimensiona l figure.

PALACE O F TH E ACHAEMENIDA E

General p lan of the palace o f the Achaemeni­dae at Persepo lis (reconstruction).

PALLADI O

In Palladio 's architecture, the use of the square often determines the overall appea rance o f the bui ldings. Villa Foscarini at Stra has a square g round plan, w hile other architectural square

56

areas ca n be found at the Villa Thiene at Quin­to Vicentino, at th e Pa lazzo lseppo de' Porti in Vicenza , and at Palazzo Thiene also in Vi­cenza.

PARTHENON

Propo rtio ns o f the fa<;:ade of the Part he no n.

57

PLATO

I

I I

I

I

I I

I I

I

I I

.'

' '

' . I •

I / /1 ,.

I I

I

I I

/ I , .

I , /

I

, • .1'

, ' / ' / , " ,'

"We have div ided the w ho le series o f num­bers into two classes: each numbe r tha t can be fo rmed by multi p lying two eq ua l factors we have re p resented in a figure with a sq uare a nd we have ca lled it a square and equilate ra l numbe r".

PALAZZO FARNESE

G i,,(lYjt1

17) f.t. ti•

l7J 1:3 Hill f fJ

17) f.t.11•

0BH1Yf1J

17) f.t. tJ>

17JB H1Y f11

(j) t.r.11•

0 LJH1Yf!J

58

JAPA ESE POETRY

w hite square o f inside the w hite square o f inside the w hite square o f inside the w hite square of inside the wh ile square

Katue K itasono

PROVERB

Those born square w ill no t die rou nd. Neapoli ­tan proverb.

P RSE

Leather purse. It is shut by folding the diago­nals of che vertica l wa lls o f the cube.

59

PYRJTE

The shape of the cube in nature. In the pyrite cubes on Elba.

PROPORTIONS IN GRAPHI CS

Uonard de Vinci 1509 Geoffroy Tcry lri2A

'uan Y<'iar lMO Serlio 1549

Studi es in proportions fo r th e lette rs of th e alphabet: Leonardo 1509, Geoffroy Tory 1529, Alb recht Dlire r 1525, Serlio 1549, Juan Ycia r 1550, Pierre La Be 1601.

Albert Durer 152.)

Pierrt' La Bi 1601

60

PYTHAGORAS

De monstrations of Pythagoras ' theorem.

61

SQUARE - WINDOW

"A woman suffe ring fro m brain disease was un­able to copy a sq uare, and it was clea r that the figure mea nt nothing to he r. When asked what it was, she expl a ined that it was a window. Afte r many exa mples, it became clear that she could o nly ever copy models that seemed to he r to be concrete objects. Whe n she was un­able to copy a mode l because it had no mea n­ing for he r, she someti mes altered it so that it took o n the appea rance of a concrete object. O nly the n was she able to copy it. When g iven a sq uare she transformed it into the fo llowing:

When asked what these figures mea nt , she re­plied: 'The windows of a church'. She did not draw squa res with no meaning, but two church w indows in a positio n th at could really have ex isted . Evidently where we see a n abstract geometric fi g ure, the patie nt saw a concrete object".

MAGIC SQUARES

A mag ic square o f the nth o rder is a square div ided , like a chessboa rd, into n2 square cells w here n is the number o f cells in any side. A number is then inserted so that the sum of the numbers in any line, ho ri zontal, vertical o r d i­agonal is always the same. The sum is ca lled the magic constant. A magic square remains such if one o f the fol­lowing transfo rmations, ca lled simple transfor­mations, is applied to it:

a - rotation around the centre of one, two or three right angles, clockwise for example.

b - symmetry to the horizontal o r vertica l medians.

c - symmetry to either diagona l.

d - the substitution o f each number w ith its complement in respect to n2 + 1.

Eight order 3 mag ic squares:

4 3

9 5

8

2 7 6

2 9 4

7 5 3

6 8

6 7 2

5 9

8 3 4

(I 6

3 5 7

4 9 2

62

2 7 6

9 5

4 3 8

8 3 4

5 9

6 7 2

4 9 2

3 5 7

8 6

6 8

7 5 3

2 9 4

Mag ic squares, wh ich the o ld mathematic ians o f India already knew how to construct, ca me to Europe in around 1420. In the Middle Ages magic squares were considered to have super­natural qua lities and they were used as amu lets to ward o ff the evil eye, the p lague and other i llnesses. They had special significa nce for as­tro logers, and to Cornelius Agrippa the o rder 1 magic square symbolised unit y and eternity, w hile the impossibility of the o rder 2 mag ic square was a sign o f the imperfection o f the four elements: air, wa ter, ea rth , fire. The sev­en mag ic squares o f the third to ninth o rders represented the seven p laners then known to n1an .

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14

The fo llowing o rde r 4 magic sq uare is found in a grotesq ue, e ntitled Mela ncho lia, e ngraved by Albrecht Dlirer in 1514; the yea r appea rs in the middle of the botto m line . In 1599, Diego Palomino published a work in Madrid abo ut magic squares but without pro­viding any indicatio ns o n how they should be constructed. In 1612 C. G. Bache t publishe d a method fo r const ructing o dd-orde r o nes. In 1691 , De la Loubere published another o ne and in 1693, Philippe De la Hire published a metho d for constructing magic squ ares of even orde rs, ca lled the border method, found by B. Frenicle De Bessy, w ho discovered 880 exa m­ples of magic sq ua res.

DIABOLIC SQUARES

Eul e r then discovered pa nmag ic squ a res in w hich the sum of the numbers situated in any pair of comp le me nta ry lines is a lso equa l to the magic constant , mea ning two para lle l lines to a diagonal , o ne of k (k = 1, 2,.. n-l) a nd the othe r o f n-k cell s. The following o rde r 4 sq uare , for exa mple , is panmagic.

12 7 14

8 13 2 11

10 3 16 5

15 6 9 4

Between 1866 a nd 1886, severa l studies ap­peared o n w hat E. Lu cas ca ll e d di abolic squares, such as those by A. H. Frost a nd M.

63

Frolow. Diabo lic squares have the prope rty to remain magic w he n they a re broke n into two recta ngles by a ny line paralle l to the sides, and these rectangles are the n transposed. The fol­lowing is an o rde r 4 diabolic square:

14 4 15

12 7 9 6

13 2 16 3

8 11 5 10

In 1894 E. Maille t published his stud ies o n a genera l theory o f magic squares based on the general theory of the substitutio n of n lette rs. G. Arnoux published his "Les es paces arithrne­tiq ues hyperm agiques" in Paris in 1894, where he dev ised a re ma rkable metho d for the con­structio n o f o rder 1 mag ic squares. This meth­od was late r extended by A. Margossian in 1908 to the case o f magic squa res of any composite o rde r.

SATA NIC SQUAR ES

Inte resting s tudies were also published by A. Aubry in 1926 in "Spinx-Oedipe". lf re pl aci ng each numbe r by its square in a magic square produces anothe r magic square, it is sa id to be a bimagic square; if replacing each number by its square or cube in a magic square produces anothe r magic square, the squa re is sa id to be trimagic. These squares are also ca lled satanic o r cabbalistic.

MILITARY SQUARE

Ranks of soldiers in square fo rm atio n .

MILITARY QUADRJLATERAL

Quadrangular area o f land defended by four strongholds.

SQUARJNG THE CIRCLE

The most famous problem in the hi story of mathematics is that o f squaring the circle. The earliest elements o f geometry w ere concerned with the possibility of measuring a figure delim­ited by straight lines. In the Nile Valley, where every year flooding erased all the boundaries made by the fa rmers to indica te ownership o f their fields, geometry helped find them aga in. The areas delimited by curved lines w ere hard to calculate and so they tried to conta in the problem by using straight li nes on ly to mark out the areas. If a square could be constructed that had the same surface area as a circle, by measuring the square area one would also ob­tain the area of the circle. This was the o rig in o f the expression: squaring the circle. But the difficulty of squaring the circle lies in the na­ture of rr.

64

SQUARE - TR IANG LE

A square cut as shown and hinged at the points indica ted , is transformed into an equilateral tri ­ang le by rotating it on the fl at.

FOUR PO INTS IN SPACE

The four corner po ints are all you need to cre­ate the impression o f a squa re.

•

• •

ANCIENT RO ME

According to an ancient tradition recorded by several Greek authors, there appea rs to have been a Roma Quadrata distinct from the one founded by Romulus, and o lder than it. By comparing the accounts o f va rious historians and archaeologists this Roma Q uadrata would seem to have been a square storeroom in front of the Temple of Apollo , on the Palatine hi ll. It served to house all the objects o f good omen for the founding of the city before the city was actu ally founded , and it is said 10 have been kept closed w ith a square stone. Hence the tradition of a Roma Q uadrata w hich precedes that of Romulus. What was initiall y nothing more than a storeroom later gave its name to the whole city, called either Quaclrata or Homulea, without distinction.

Figure deri ving from the rotation o f a cube.

66

REN AISSANCE

Painted decoration.

RAMSES IV

Proportional analysis o f the plan o f Ramses I V's tomb.

RI NG

PROPORTIONAL RECTANGLES

Lowering the diagonal of a square o nto the extension of the base, w e get a rectang le in which, whatever the unit o f measurement, the ratio between the sides is 1: ~2. Continuing the process, we get other rectangles with the ratios: 1: ~3, 1: ~4, i.e. an area o f two squares, 1: ~5, and so on.

68

CHESS IN MOSCOW

Opening of the Twelfth World Chess Cham­pio nship at the Central Theatre of the Soviet Army in Moscow.

KNOWLEDGE

The ancients represented Fortune seated on a round stone and Knowledge on a square one, showing that the fo rmer is changeab le, whi le the seat of the other is steadfast.

69

DERIVED RECTANGLES

Hectangles obtained from the square by project­ing its own measurements.

• Raphael - The Marriage of the Virgin.

Signs pa in ted in black-blue a nd linear-red, of the oldest series, found at Cachao da Rapa o r Curra! das Letras, Braganza. Simil ar signs are also found in pa leo-Babylo nian, north Se mitic and America n proto-lndi an writings.

THE KNIGHT'S TOUR

The puzzle in w hich the object is to move the knight around a ll 64 squares o f the chessboa rd in 63 moves, known to the o ld Ind ian mathe ma­ticians, has an as yet unknown number o f solu­tions. Euler and various o the r mathematicians indi­cated ways of solvi ng the puzzle . He re are two solutions (the numbers ind icate the fo llowing positions of the knig ht) of w hich the second is "closed", i.e . fro m the final positio n the knight can , in o ne move, return to his ini tial posi tion:

22 25 50 39 52 35 60 51

27 40 23 36 49 58 53 34

24 21 26 51 38 61 56 59

41 28 37 48 3 54 33 62

20 47 42 13 32 63 4 55

29 16 19 46 43 2 1 10

18 45 14 31 12 9 64 5

15 30 17 44 1 6 11 8

70

43 40 15 26 13 30 5 8

38 25 42 29 16 7 12 31

41 44 39 14 27 4 9 6

24 37 28 17 54 11 32 3

4 5 18 23 64 33 2 55 10

36 49 46 19 22 53 58 61

47 20 51 34 63 60 1 56

50 35 48 21 52 57 62 59

BRUNO SERVI

The sq ua re is a number which has reached per­fection, because in it the number o f times its base is repeated is the sa me as the number of times the unit is repeated in the base. In squ a ring, in fact, simply e nunciating the base expresses in itself the times it is repeated; in th is way the e le me nts of a rbitra riness and exte­riority a re removed that, in the determination of the numbe r o f times, were p resent in any othe r no n-potentia l operatio n.

ROCK SALT

The squ are in nature: a crysta l of rock sa lt.

ROAD SJGN

Road with right of way sign and encl of right.

GOLDEN SECTION

A graphic show ing the golden section sta rting

from the square.

"

72

LYBIAN WRJTING

Characters in Libyan w riting.

SQUARE WR!TlNG

£:] u z

n ir N

---

Almost every country has w hat is ca lled a square form for the letters o f its alphabet o r ideograms. In Home, the o ld monumental sty le lapidary capital letters were defined as "square". H ebrew, Chinese and Japanese writing is square. Clea rl y, a square shape is best for composing a text verti­ca ll y or hori zontall y o n a fl at surface. Characters in Balti writing, in the state of Kash­mir, corresponding to their phonetic va lues, in

ttJl-PP db F6

order: cl, ts, b , h, k. Many characters of this w rit­ing, which is presumed to have been invented at the time o f the conversio n of the popu lat ion to Islam in around 1400, appear to derive from a square grid structure. It is likely that a square grid provided an o rder for each letter that made it easy to remember.

SQUARE DANCES

• • •••

•••••• Jnstructions fo r the fi gures of square dances, suggested by Ca rson Ro bison. Formatio n: four couples to each gro up. Form a sq uare w ith a couple each side standing face to face. A group sho uld occupy the space o f a sq uare nine feet e ithe r side . Ladies stand to the right o f the ir partne r. The lad y to th e le ft of each gentle man stands in the lady corne r. The ma in coupl e, couple number o ne, moves backwards towa rds the part o f the hall where the o rchestra is. The couple o n the right of the nia in couple is couple numbe r two, and then there is the third and fourth , o r last coupl e. Honor: the gentleme n bow, the ladies curtsy. Sw ing: the gentlema n ta kes the right ha nd o f the lady in his left hand and puts his arm ro und he r wa ist, as fo r a wa ltz. He p laces his rig ht foot on the o utside of the rig ht foot o f the lad y and

73

togethe r they dance around in a clockwise di­rectio n , spinning on the right foot. Promenade: the couples cross hands and dance around the group in an anti-clockwise direc­tion . Do-si-do: the lady and gentleman wa lk towards and past each other, right shou lder to right sho ulder. Then each takes a ste p to the right, passing back to back , and the n they return to

the ir o rig inal position. Alle mande left: the gentle man stands facing his lady in the corne r. They take each other by the left hand and reel ro und each other clockwise, then return to the ir original positio ns. Grand right and left: the partners stand face to face and take each othe r by the right hand. They move towards and past each othe r. The gentle­me n reel anti -cl ockw ise and the ladies clock­w ise. Take your ne ighbour's left hand, pass, take your ne ighbour's right hand, pass, and in this way go forward in a circle. Some names of sq uare dances: The square waltz, Square Dance Po lka , Promenade Indian Style, Around the Out­side, Birdie in the Cage.

RUNlC CHARACTERS

From a Runic alphabet, phonetic value: Oe, ae, e, o. The sq uare has the phonetic value: ng.

LOGARITHMI C SPIRAL

From a square we can obtain a golde n rectangle by lowering the diagona l to the exte nsio n o f the base. If, to this rectangle, we attach another square having a side equal to the lo nger side o f the rectangle, we o bta in anothe r golden rect­angle . Co ntinuing to construct squares o n the lo nge r sides of the rectangles we w ill o btain a figure composed of squares arranged in a spiral around po le 0 , which is the point o f intersection of the ma in diagonals of the subseque nt rect­angles. If we draw a curve thro ugh the po ints o f intersectio n of these diagonals with the subse­quent squares, this will be a loga rithmic spira l. This curve re presents the law of visible o rga nic growth, particularly in shells, in the arra ngement of sunflowe r seeds, in some spiral nebulae and in many othe r cases .

74

A LOCK

(;:' .'II').ic> ... "; . . ._.,,....-....... . : .

. ·("

Twe lfth century Sca ndinav ian lock.

TATAMI

A tatami is a thick mat that the Ja panese have been covering the floors of the ir ho uses w ith for ove r a tho usa nd yea rs . The tata mi measures two squa res in size and it serves as a bas ic unit o f measure me nt in the design o f these ho uses . All the roo ms a re conceived o n the bas is o f the numbe r o f tata mi to be used , a nd it is suffi cie nt to just state the numbe r o f tatami to be used in a room to get a n idea o f its shape and d ime nsio ns, and , as every part o f the ho use is formul ated o n the bas is o f the tatami , the re is no need to take measurements.

FOSSIL SPONGE

Structure of a fossil sponge.

HITTITE CHARACTERS

Hittite characters from a hieroglyph. Inscriptions found at Hama , the second character has the phonetic value : k, g.

SEAL

Sign made from a proto-Inclian sea l.

EMPIRE STYLE

Detai ls of an Empire style ba lustrade.

ANTON STANKOWSKI

Graphic design.

76

CUTT! G AND RECOMPOSING

-- A square cut in this way ca n be recomposed in many other ways .

•• EE sa rn rn ~ w ~ tarn ~ rn ~ rnru rn mm EB tE EE til EB rn EE EE rn Eli EE tE

CE 00 EE ~ 89 l~HE m mm ta ~ EE EE ~ ~ ~ B~'.H~E tE ~ ~ ~EE tfHE t'E EE EE 58 EE ~UHtJ E\tJ ~ rn EB ~ ~ tt? mOOJ~~~~~f:E~EE

TETRAHEDRON

A letrahedon is cul into two equal parts by a square.

TELL EL AMAR NA

A model v illage at Tell el Amarna (1 370 BC) where the workers and arti sa ns l ived who were employed on the constructio n of the tombs in the rock in the higher part o f the desert . This vi l­lage, situated in a desolate va lley, fa r fro m w ater but close to the cl iffs w here the toml s were to be cut out of the rock , was a square surrounded by a high wall in w hich there was only one en­trance. In the v illage itself there were five narrow streets w hich ran parallel from north to south , through rows of small , rentable houses, all iden­tica l in shape. On ly in the southeast corner, nea r the gate, was there a bigger and more elaborate bui ld ing, w hich w as probably the house o f the supervisor. The monotony of all these houses

78

was compensa ted for by the va riety o f their contents. The men w ho worked in the tombs as painters and decorato rs wou ld b ring hom e some o f the colo urs, so that the wa lls o f their ho uses were all decorated w ith ro ugh d rawings. During the excava tions, tools of every kind were found: clay crucibles, uncut precio us sto nes, parts of looms, and r ings w ith unfinished cut stones. The well -protected town houses of the members of the court seemed lifeless and inhuman in com­parison with this humble workers' v illage.

TR IGON

A Swiss game composed o f squares cut-up so as to make different figures.

" == =u " " " "

... " II II

:: {}=:::: == .;;:: d ~ "

" " ... " "

" " " " "

TELL EL AMARNA

PIAZZA SUD

THE TEMPLE OF SOLO MO N

The Te mple of Solo mo n in Je rusa le m. The Sa nc­ta Sa ncto rum was a pe rfect squ a re, w ith mea­sure me nts almost equ al to those at Ca rche mish. It was comple te ly pl a in , e xcept fo r th e fa<;:a de ado rned w ith a curta in e mbroide red in va rio us colo urs.

ONE QUARTER G REATER

Give n a squ a re, draw a no th e r g rea te r by a qu a rte r.

A FIFTH

Given a square, draw a nothe r o ne fifth , two thirds, e tc., sma ller.

80

TETRA FLEX

Three-dime nsio na l constructio n made w ith fo ur sq uares fi xed togethe r at the ir pa ra lle l sides and creased at every d iagona l. This o bject can be fo lded in many diffe re nt ways if you a lways fo l­low the lines o f the creases.

TWTD D LE

Cubic o bject cut into sixteen eq ua l prismatic pa rts. These pa rts a re hinged togethe r so that the cubic o bject ca n be tra nsformed into othe r pa ra lle le pipeda l o r prismatic objects, a lways keeping all the pieces attached to each o the r.

PH YS IONOMIC TYP ES O F LEIDOS

The squa re type indicates a n e ne rget ic brusqu e nature, a firmn ess o f characte r bo rde ring o n in­fl exibility a nd that ca n eas il y degene rate into stubbo rnness. An a b le thinke r, rapidly a nd pow­e rful ly logica l, w ith a well-developed pract ica l sense a nd pa rticula rl y we ll suited to the exact scie nces, to phi losophy a nd mathe matics, but lacking in imaginatio n , w hich ma kes him ill suited to the arts. A lover o f constructio n , he has a be nt for a rchitecture. His nature, sceptica l a nd doubting, leads him to mate ria lism; but if fa ith illuminates his soul , it w ill be built o n unsha ke­ab le fo undatio ns; his head w ill a lways cont ro l his hea rt. This type is gene rall y lo ng-lived.

EQUILATERAL TR IANGLES

Equilatera l triangles and models based on equi­lateral triangles in cubic spaces.

81

uo

Maya n chara cter, phonetic va lue: uo.

URA !UM AND GRAPH ITE

A large cube o f uranium and graphite for the first nuclea r reactor.

82

l x5-2x2

rs:rsJ I Transformation of a rectang le l x5 into a square 2x2.

VASA RELY

Detail o f a large composition in black and w hite.

----··--·····-·---­••••••••••••••••••• ············-······ ············-······ ••••••••••••••••••• ••••••••••••••••••• ··---··---········· ··---··---········· ··---··----·····--· --··········---···· --··········---···· ············---···· ··········---······ ····-·····---······ ··········---······ ••••••••••••••••••• ••••••••••••••••••• ······--··········· •••••••••••••••••••

MIES VAN DE RO H E

Plan and front eleva tion o f the "50x50" house.

84

MA RY VIEIRA

From square to cube. Model in stainless steel 35x35x35 cm . Basie, 1959. Collectio n o f Dr. Markus Kutter, Basie.

WANG HSl-C HJH

The true father of Chinese ca lligraphy is consid­ered to be Wang Hsi-Chih (AD 316-420, Eastern Chin dynasty). He perfected the " Ii ssu " cha rac­ters by giv ing them their beautiful square form , as well as the script we nowadays ca ll cursive, in his time rapidl y undergoing develo pment, and he invented an intermediary fo rm between the two. Present-clay Chinese characters therefore belo ng to one o r other of the three forms he established: square, medium and cursive. H e is respo nsible fo r hav ing perfected them in such a highly arti stic and admirable manner.

- iii - iii tv.J ·H: • ~ /pJ .=f ii 611, ~ HJJ B ~ /pJ .:t- tv.J • ,11 T o /f' m * t1! JJJ ie , ~ ~ m ,tj~ ~ /pJ -f-~ IJ! I! • ,~ ~ JJ! JR tr I! tt r& 10illJJ'lil8\~ liJ ~'.If'* fl Jft JJJ' flJ f~ -~-0 mm*-lftv.J~i~,~re ~ ~~~!*~

~f,fll~ftre ~'~ ~tJFi~To:tE .mft ~ re 9t lil t8R nx: rfl lil ~, .R ;ft" ~Jm * J! m ft tv.Jo ~ --- ~ m §I. Jt re. J: t(.J ~ tt ~f,F-, ft-/pJ ~ -W\,. :I! f,F ~f '~. fto mi * ~ r:J ~}fl m' ii ~ m ~ m §. Jt, JI .if~* i! * JfJ ~,~:fr T 1£ ::f 1fi: ~~*fto 4ij * PJq ~ iii-!Jl ~ M« ~ii flj ~ ~~ ru fill J: t(.J t! /pJ ~ • .d:o ~ /pJ -1-t(.j ~~ .s11wa9E ae; ~ ~ 5e m ~ ~ ~-

INDEX

As all the subjects in this book have been ar­ranged in alphabetica l order (as far as make-up allows) no index is necessa ry.

Books and publications consulted:

Henri Poinca re LA SC IENCE ET L'HIPOTESE, Paris 190 1. Luigi Dami I G IARDIN I ITALI AN ! Bes tetti Tuminelli , Milano 1925. Hornun g's Handbook of DESIGN AND DEVI CES Dover Publica tion , New York 1946. Cesa re Bairat i LA SIMMETRIA DINAMI CA Tamburini , Milano 1952. Alexa ndre Speltz LES STYLES DE L'ORNEM ENT Hoepli , Milano 1930. Ga rdner Murphy SOMMAR IO DI PSICO LOG IA Ei naudi, Torino 1957. Egmont Coleru s IL ROMAN ZO DELLA GEOMETR IA Garzant i, Mil ano 1937. Bruno Zev i PO ET ICA DELL'ARC HIT ETTURA N EO PLASTICA Tamburini , Milano 1953. Maurice Kraitchik LA MATHEMAT IQUE DES JEUX, Paris 1930. BAUHAUS 1919-1928, The Museum of Modern Arc, New York 1938. Marcel Poete LA CITTA ANTI CA Ei naudi , To rino 1928. Dav id Diringer L'ALFABETO NELLA STO RIA DELLA C IVILTA Barbera, Fi renze 1937. Arthur Drexler THE ARC HIT ECTURE OF JAPAN The Museum of Modern Arc, New York 1955. Ernst Mossel VOM GE HEIMNI S DER FORM UNO DER URFORM DES SE INS , Stuccgarc 1938. Gyorgy Kepes LANGUAGE OF VI SION Theobald, C hicago 1944. Gyorgy Kepes THE NEW LANDSCAPE Th eobald , Chicago 1956 . Karl Gerscner KALTE KUNST? Ni ggli , Teufen AR, Svizzera 1957. Wolfgang von Wersin DAS BUCH VOM REC HTEC K Maier, Raven sburg 1956. Hugo Steinhaus MATHEMATI CAL SNAPSHOTS Oxford University Press, New York 1937. Dav id Katz LA PSICO LOG IA DELLA FORMA Einaudi , Torino 1948. Giuseppe Ronchetti DI Z IONAR IO ILLUST RATO DEi SIMBOLI Hoepli, Milano 1922. Edward Kasner I James Newma nn MAT EMAT ICA E IMMAGINAZ IONE Bompiani, Mi lano 1948. ENCIC LOPEDIA DELLE MATEMATI C HE ELEMENTAR J Hoepli , Milano 1930 -195 0. Pio Emanuelli IL C IELO E LE SUE MERAVI G LIE Hoepli , Milano 1934 . Umberto Fo rri GEO METRIA PIANA Parav ia, Milano 1947. Leonard Wolley IL MEST IERE DELL'ARCHEOLOGO Einaudi , Torino 1957. DE ST IJ L Stedel ijk Moseum , Amsterdam 195 1. KONKRETE KUNST Kunsthalle , Basel 1944. ARTE ASTRATTA E CONC RETA Alfie ri e Lacroix, Mil ano 1947. AUJOURD' H UI, Paris, n. 3-4-9. Cl VILTA DELLE MACCHINE, Rom a 1954- 1958. SC IEN T IFI C AMERI CAN , New York, 5/ 1958, I 111 958 . ILLUST RA Z IONE SC IENT IFICA, Mil ano 1955 . ART, Paris. ENCICLOPED IA DELLA CIVILTA ATOMI C A, II Saggiatore, Milano 1959-60. Boris de Rachew il tz INCONTRO CON L'ARTE EG IZlANA , Marce llo, Milano, 1958.

THE SQUARE DISCO VERY OF TH E SQ UARE

Bruno Munari

© 1960 Bruno Munari All ri ghts reserved by Mauriz io Corra ini s. r.I.

No part of this book may be reproduced o r 1ransmi1tcd in any

fo rm or by any mea ns (deciron ic o r mechanical, incl ud ing

photocopying, reco rdin g or any in fo rmation rc1ricval sys1cm) without permi ss io n in w riting fro m th e publisher.

Ori ginal firs t editio n published in 1960 Schciwillcr, Mila no

M auri zio Corraini firsr edition 200 6 M a uri zio Corra ini second rcprinr 2011 l"hc publis her will be at complete disposal 10 whom migh1 be rclaicd co chc un identified sources printed in this book.

Tra nslati o n co rra ini Studio Printed in lra ly by Srilg raf, Viada na, M a nrova November 201 1

Maurizio Corra ini s. r.I.

via Ippo li to N ievo, 7/A

46 100 Mantova

tel. 0039 0376 322753 fax 0039 0376 365566 e- mail: sito@corrain i.com

www.corrain i.com