Original Article Klein Bottle Logophysics, Self-reference ......biology, astrophysics and chemistry....

Quantum Biosystems | November 2016 | Vol 7 | Issue 1 | Page74- 106 74 Diego L. Rapoport

ISSN 1970-223X www.quantumbiosystems.org

Original Article

Klein Bottle Logophysics, Self-reference, Heterarchies, Genomic Topologies, Harmonics and Evolution.

Part II: Non-orientability,Cognition, Chemical Topology and Eversions in

Nature

Diego L. Rapoport*

Abstract

In this second part of a series of articles, we discuss the fusion of analogue and digital decurring from the Klein Bottle, the Inside/Outside image-schema of Cognitive Semantics and its role in the dualistic organization of knowledge, presenting several examples in biology, astrophysics and chemistry. We present the Möbius strips and Klein Bottle (KB) and HyperKlein Bottles non-orientable surfaces, the non-dual logic and logophysics of the KB, that surmount this dualism, and their elementary harmonics, and the relation with palindromes .We present a corresponding topological protoform of Newton’s Third Law surmounting its dualistic character. We discuss Chemical Topology as a paradigm incorporating at its foundations the Klein Bottle ontology and logophysics, and particularly the characterization of life as artifact-making, of molecular complementarity and of semiotic agency as related to self-reference. We discuss the relation between these surfaces and enantiomerism and the existence of a bodyplan for humans related to them and give several examples of its manifestation. We discuss the relations between the Inside/Outside image-schema and its surmountal by the Klein Bottle logophysics in biology, chemistry, astrophysics, holography, metamathematics, and of the location of the real world. We introduce the non-orientable topology of the action/perception cycle and discuss the KB topology of the visual and somatosensory cortical mappings, music perception and of electromagnetic and sound vortices. We discuss the Inside/Outside image-schema and its relations to the topologies of genomes. Key Words: biocomputation, biophysics, blown-up systems, chemical identity, cognition,complexity, harmonics, heterarchies, neural networks, image-schemas, morphomechanics, non-classical logic, non-linearity, ontology, palindromes, pattern recognition, phenomenology, physiology, quantum holography, torsion geometries, systems theory, vortices.

Quantum Biosystems 2016; 7 (1): 73-105 1

In Memoriam and Gratitude, to George Spencer-Brown (April,2,1923– 25 August 2016)

*Corresponding author: Diego L. Rapoport Address: (Retiring from) Departamento de Ciencia y Tecnologia, Universidad Nacional de Quilmes, Bernal, Buenos Aires, Argentina. e-mail: [email protected]

mailto:[email protected]



1. The Klein Bottle and the Möbius strip non-dual logophysics.

1.1 Introduction: Non-dual

logophysics, the Fusion of Analogue and Digital, and Genomes

The topologies of the Klein Bottle and the Möbius strip are related to a non-dual logophysics, which in a quite elementary sense, means that it is not possible to divide categorically the world produced by a boundary, in terms of Inside and Outside, since there is a continuity between them, which is basic to physics and to cognition (Rapoport, 2005, 2009, 2010 a, 2010b, 2011a, 2011d).

In breaking or disconsidering this continuity, we obtain as a particular reduced case the usual dual logic of Aristotle as formalized by Boole which is deemed to be the usual basis for informatics and in particular its bioinformatics application to genomes (Rapoport, 2011b).

These topologies which integrate Outside and Inside are associated to self-reference, as a principle of self-organization, and cognition (Rapoport, 2009, 2011a, 2011b, 2011c, 2012, 2013, 2014).

Specifically we shall consider, as we already introduced them, the non-orientable Möbius strip and Klein Bottle surface. Despite the non-dual self-referential nature of the Klein Bottle, it is naturally amenable to produce a binary codification; in fact, this discrete codification is mandated by the self-penetration of the Klein Bottle, which conflates the continuity of the surface with the discontinuity produced by its self-penetration.

The binary codification, in turn, will produce through a very simple algorithm, the basis for numerics in terms of hidden structures of genomes, which can be elicited by the study of the databases provided by the Encode Program.

Remarkably, the frequency of these numerics will appear in the very vein that Pythagoras originally proposed, in terms of harmonics and thus amenable to a musical transcription.

In short, bioinformatics as the study of the Genetic Code will be found to stem

from a logophysics associated to topologies of self-reference, rather than being inherently about digital data bases.

As we said before, data possess shape; already Fisher Information Theory, as derived from Bayesian statistics, shows this to be the case though topology was not addressed in this setting (Frieden, 2004).

The digital contents are embodied by the non-orientable topologies of genomes, which have an inherent wholeness as their signature.

The underlying conception is that rather than studying genes as singular events, we must consider the genome as a whole in which groups of codons are crucial to the formation of genomes as algorithmic structures derived from the topology.

These natural groups of genomes span a harmonics which furthermore is related to the topology of the genome and yet, these algorithmic structures appear to be amenable to generation as if independent of the topological structures of genomes, say as a finite automata (Perez, 2009).

Topology was understatedly introdu-ced with the double helix model (DH) of DNA by Watson and Crick, as an orientable topology, which means that DNA, in being left or right handed, its helixes have a well defined normal vector –i.e. a vector perpendicular to the surface- defining an Outside and Inside, which is crucial to the structure of DNA, as we shall discuss below.

These two normal vectors are different states proper to the two-sided character of the DH orientable geometries, rather than the one-sided character of non-orientable surfaces.

For the latter kind of surfaces, due to the 180° twist of the Möbius strip or the Klein Bottle, there is no globally defined Interior nor Exterior; a normal vector which locally points to what appears to be Inside, can be transported to a normal vector pointing Outside, as supported at the homologous point at the “other” side, but now pointing Inside; see Figs. 1.II to 4.II below.

Subsequent discussions on the validity of this model and its mathematical-physics aspects, as well as alternative geometries for the DH, have been related to topology



(Swigon, 2009; Benham, 1995) and remarkably to the statistics of codon frequencies, in relation with the palin-dromic structures which are recognized in the palindromic structures of DNA. We recall that in topological terms, palin-dromes can also be thought as Möbius strips, whereby introducing a 180° torsion on the sequence, an identification takes place.

This can be nicely illustrated by con-sidering a musical example, Johann Sebastian Bach’s Toccata and Fugue in D Minor, the Crab Canon (Bach); see note no. 15.

Yet, as we shall see, the palindromic structures which are so ubiquitous to genomes, and topologically can be concei-ved as Möbius strips, are related to an archaic genome.

Genomic palindromes are assumed by default to be two-sided as DNA is usually assumed to be in the Double Helix model; more of this below. Basically, we shall demonstrate that the remarkable existence of a mirror-codon in single strand BUILD 34 genome reveals the existence of higher order structures that are related to non-orientable topologies of the genome, and further related to the existence of harmonics.

Closed topologies are known to be the case of the mitochondrial DNA (mtDNA), a special kind of DNA that does not reside in the nucleus of the cell, but in an organelle named Mitochondrion.

The Mitochondria are structures that reside in cells that convert the food into energy for the use of the cell.

The Mitochondria have the special ability to replicate themselves independently of the gene information in the DNA, unlike all the other cells in the organism.

Also, unlike normal DNA, mtDNA is in most species solely inherited from the mother. The mitochondrial DNA is a circular structure of approximately 16,500 base pairs, in humans (Sykes, 2003).

As for bacterial genomes as well as those of eukaryotic viruses, such as SV40, they are known to be closed Möbius loops (Prunell, 1998).

The organization of chromatin into closed loops is believed to be crucial to

DNA compactation and gene expression; each such loop may act as an independent unit of gene activity (Van Driel and Otte, 1997).

Since biological evolution is presently conceived as having arisen from bacteria and viruses (Koonin, 2012), this identifies a genomic metashape for it, in this non-orientable surface.

Indeed, the consideration of the palindromic structure of an evolutionary novelty in prokaryotes, given by a system of adaptive immunity common to most bacteria and archea (Koonin and Wolf, 2012), has served for its promotion as a basis for a Lamarckian punctuated evolutionary theory, in the framework of comparative genomics (Koonin, 2012).

Let us state what we believe to be the ultimate basis for genomes: They are structured and processed by a non-dual logophysics which is keenly associated to the torsion geometries that are basic and pervasive to physics and nature, as primeval vortices (Rapoport, 2013).

Their ubiquitous and basic role as a geometry of self-reference, and its instantiations as a unifying principle in anatomy and physiology, perception, physics, geophysics, chemistry, biomecha-nics and cognition was discussed in (Rapoport, 2009, 2010a, 2010b, 2011a, 2011b, 2011c, 2012, 2013, 2014).

The role of vortices to biology and the unity of nature was already stressed by Bell Pettigrew (1873, 1908), while most material systems in Nature are viscous liquids and thus vortical torsion geometry systems (Rapoport, 2005).

This pervasive geometry was subse-quently neglected, but for Edgar Morin’s theory of complexity (Morin, 1992) and also by the theory of chaos and blow-ups of non-linear systems due to Yi Lin Forrest and Shoucheng Ou Yang (Lin, 2002; Wu and Lin 2002, Lin, 1998, 2008); see note no. 16.

The latter ultimately rests on the non-orientable structure of the compactified complex number system which signifies a novel cycle of such systems, as a generic renewal principle (Rapoport, 2013); see Fig. 2.I.

The Möbius strip and the Klein Bottle are two elementary examples of vortical

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structures, yet in which self-reference is embodied.

As for the non-duality which is organized in terms of non-orientable topo-logies for which there is no global Outside nor Inside, it fails to abide to the principle of the excluded middle: namely that a proposition may be true and not be true simultaneously; see note no. 3 in Part I.

Or still, they do not abide to the principle of non-contradiction (Rapoport, 2009, 2011a, 2011b, 2011c, 2012).

It is known that organic chemistry may be conceived in terms of topology which does not abide to the usual dualism of Outside and Inside (Rapoport, 2013; Sokolov, 1973) and that biological water, as the non-inert liquid crystal context for chemical reactions and most living systems, also appears to operate in terms of this nondual logic (Rapoport, 2011b, 2014).

The most prominent example to biology is that of liquid crystals, that having dislocations, they self-organize as elastic materials, and fold into Möbius strips configurations (Bouligand, 1978, 1999) or still, can be caged into ordered water domains or conversely, be a cage for ordered water (Chaplin, 2015).

Especially, this is the case of carbon molecules, which may form tensegrity structures which as we said, may incorporate water of rather be contained in it (Chaplin, 2015).

Thus, in the relations between the carbon molecules crucial to life, with water, a principle of literally exchanging Inside and Outside appears to be at work.

This principle is patent in topological chemistry –extending the case of Möbius molecules, catenanes and knots- of endohedral fullerenes (Dodziuk, 2011), following the discovery of C60 after Fuller’s theory of Synergetics (Fuller, 1975), which represent a remarkable expansion of chemistry and materials science.

While C60 stands for a spherical shaped multi-carbon molecule, other “exotic” possibilities such as tori or Klein Bottles have been considered for its stable alternatives (Kirby, 2002).

The latter is remarkably natural due to the relation of the Klein Bottle with the

torsion geometry of the pentagon which in the regular case yields the Golden Mean.

1.2 Topological Chemistry,

Turning Inside-Out & Outside-In, and Contextuality in Chemistry. The Problem of the Origin of Life as Artifact-Making.

It is important to stress the role of the tetrahedron, among all Platonic polyhedra, as a geometrical model for molecules.

Aside from being the most common arrangement of water, virtually all satu-rated organic compounds, most com-pounds of Si, Ge, and Sn are tetrahedral.

Yet, as repeatedly observed by Buckminster Fuller in his opus magnum (Fuller, 1975) the tetrahedron, upon construing it with elastic sides, is unique among all Platonic polyhedra in that it can be turned Inside-Out resulting in another (the ‘negative’) tetrahedron, and by a second motion return to the previous tetrahedron.

Fuller conceived the tetrahedron as the minimal conceptual system, the “topological minimum” of which all form and structure are made of.

This notion of chemical shape as a ‘resonating structure’ whose stability is produced due to its oscillation between two –or more- configurations, was early suggested to be the case of chemical bonds by Pauling (Pauling and Bright, 1935; Pauling, 1940).

Nowadays, in the usual geometrical approach to chemical configuration, these intermediate or alternative structures (mesomerisms) are considered to be purely imaginal and related to electron delocalization (Kerber, 2006).

The issue is the characterization of a single ‘substance’ having multiple struc-tures –which coincides, as we said before, with the present ontology in which the principle of identity may no longer be the case.

In this setting, it is actually impossible to determine in practice which configurations are ‘real’, and which are merely graphical (Lewis dot diagrams) decurring from their rules of formation. This has prompted the notion of hybrid intermediary structures as being the ‘real’



configurations, rather than the graphical ones. The latter problem is the chemistry cognate of the ‘reality’ of the solutions of xx= -1, none for the real numbers system or in Boolean logic.

As we shall see, in the topological chemistry paradigm, they may be associated to the Klein Bottle topology of the oscillation of the configuration of the electron distribution near to or in the minimal energy state of the molecular wave function, and they are quite pervasive.

It is this oscillation of the electron distribution which will turn out to be crucial to the coexistence of multiple configurations of a molecule (Fowler, 2002), which in dualistic terms is phrased as a ‘split’ identity (Stapien, 2007), and produces an orientable and non-orientable aromatics, to be discussed below.

In distinction of the current imperative course towards the primacy of syntax jointly with a rejection of both the imaginal and the conceptual domains which is embodied in the setting of Abstract Set Theory for the cornerstone of mathematics -which triggered the unresolved crisis about the foundations of mathematics (Kleene, 2009), Fuller insisted on the topological foundation for a conceptual and hands-on modelling of the Universe through synergetics.

He based it upon the remarkable property of the tetrahedron previously alluded.

Thus Fuller further noticed that at the level of the tetrahedron, the Universe conceived as a tensegrity structure made of polytopes, can turn pulsating Inside-Out and Outside-In maintaining equilibrium. Fuller made of this uniqueness the signature of stability under Exterior perturbations together with the outbound stabilization of the former pulsation, upholded as an elementary property of the Universe.

Yet, inward and outward motions are qualitatively different: inward motion is unidirectional while outward motion is omnidirectional (i.e. anydirectional), the latter representing regeneration.

The twoness in oneness of this reversible motion reaching for the identity

through its recursion, is already embodied by the Klein Bottle and the Möbius strip.

It is phenomenologically identifiable to the bistable perception of the (Necker, or regular) cube (Logothetis, 1988), of music audition as in the Tritone Paradox –to be discussed below, and in olfactory experience (Zhou and Chen, 2009).

The turning Inside-Out of geome-trical configurations appears also as the so-called Walden inversion, first observed by Walden in 1896.

This proceeds as: “…the inversion of a chiral center in a molecule in a chemical reaction. Since a molecule can form two enantiomers around a chiral center, the Walden inversion converts the configuration of the molecule from one enantiomeric form to the other [which is supported by the Möbius strip configuration]. For example, in a SN2 reaction, Walden inversion occurs at a tetrahedral carbon atom. It can be visualized by imagining an umbrella turned inside-out in a gale” (Wikipedia, 2015).

The dynamical intertransformation of Outside and Inside as already explained in relation with the tetrahedron, as the unique polyhedron that everts Inside-Outside and viceversa, is the nature of the Klein Bottle as it transpires in bistable visual perception.

Furthermore, the reductive CONTAIN image-schema must fail in chemical phenomena, and particularly to clusters where containment is important. Indeed, while usually a molecule must be either Outside or Inside the cavity at any given time, but when a molecule with both cationic guest and repulsive negative terminus components is used, one end of the molecule is encapsulated by the host, while the other end remains Outside the capsule (Tiedemann and Raymond, 2006).

Furthermore, this Inside-Outside exchangeability not only comprises the geometrical configuration of molecules but also their hybridization.

We recall that hybridization is the concept used for explaining molecular

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shapes in the usual approach, of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory.

As already discussed, molecules may appear as having multiple conformations, or a “split personality” in a colloquial dualistic description: on the one hand orientable conformations may turn to be non-orientable, and the converse can also be the case (Rapoport, 2011, 2013; Sokolov, 1973; Stapien, 2007).

It is the hybrid orbitals which are considered to support the “split personality” whereby , say graphene, co-exists as in the normal and the Möbius strip configurations (Wang, 2013).

Graphene is considered to be a peculiar form of aromatic molecule (Popov, 2012).

This double configuration is more generally encountered in aromatics (which started with Kekule’s model of benzene, which is a case of ‘Möbius aromatics’ as well as ‘regular’ aromatics, to be defined below), i.e. the cyclic ring shaped planar molecules.

Despite their unusual stability, as compared to other geometric or connective arrangements of the same set of atom, they admit a Möbius strip configuration, first synthetized by Herges (2006), hence the ‘Mobius aromatics’.

In terms of molecular orbital theory these compounds have in common a monocyclic array of molecular orbitals in which there is an odd number of out-of-phase overlaps, the opposite pattern compared to the aromatic character to Hückel systems molecule of ‘regular’ aromatics. Indeed, the existence of multiple configurations as stereoisomers lays at the very basis of Topological Chemistry, and the synthesis of catenanes, rotaxanes, knotted molecules such as the Borromean ring, etc., of molecular motors in short, for which Möbius strips may be used to that effect (Breault, 1999); this paradigm led to the bestowal of the 2016 Nobel Prize in Chemistry to J.P. Sauvage, J. Fraser Stoddart and B. Feringa.

With regards to proteins in which palindromic sequences are abundant, stu-

dies appear to indicate that the reverse sequence leads to the production of the same 3d protein fold or still its enantiomer (Sheari, 2008).

This is most remarkable since it indicates that the 3d structure of proteins may be related to a lower dimensional -actually a 2d Möbius strip- metaconfi-guration, the untwisted orientable case corresponding to the symmetric former case; see note.no.15.

Thus, the principle of “multiple identity” appears also to be the case of proteins, contradicting thus the principle of identity basic to dual logic.

This is relevant in regards to bio-semiotics and the characterization of life as related to artifact-making, for which molecular motors and DNA may play the role of paradigmatic examples.

According to Barbieri (2015) this is setup in terms of the Inside/Outside divide and still in the alledgedly linear character of the geometry of the biological digital codes, and still its digital character vis. Barbieri:

“…molecular biology has discovered that the production of genes and proteins requires not only catalysts but also templates. The catalysts join the subunits together by chemical bonds, and the templates provide the order in which the subunits are assembled. It is precisely that order that determines biological specificity, the most important characteristic of life, and that order comes from a molecule that is outside the assembled molecule. This is precisely the characteristic that divides spontaneous objects from artifacts. In spontaneous and in catalyzed processes, the order of the components comes from within the molecules, i.e., is determined by internal factors, whereas in genes and proteins it comes from without, from an external template” (Barbieri, 2015).

Furthermore: “It is equally a fact that linear and digital sequences that direct the synthesis of molecules do not exist in the inanimate world, so it is

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beyond dispute that a divide does exist between life and matter. It is the divide between the analogue world of chemistry and the digital world of life, and it is not a fiction” (Barbieri, 2015).

However, as we have just seen and as it will turn out to be the case of genomes, rather than linearity we have that Nature rather develops its codification in palindromic sequences related to non-linear non-orientable surfaces, and the digital character of the codifications is derived from the Klein Bottle logic.

Furthermore, what chemical topology has shown is that these molecular motors can be synthetized from the application of Möbius-like twists that anticipate life, so that Barbieri’s account appears to be amenable to be understood in terms of the non-orientable surfaces of self-reference, rather than requiring an elaborate distinction between life and non-life which as discussed in Rapoport (2014b) the dualistic ontology cannot actually sustain this divide.

If life is identified as artifact-making, clearly chemical topology shows that this identification is unprecise –or still, ontologically mistaken- with regards to the actual non-dual ontology and logophysics that operates, an ontopoiesis (Rapoport, 2014b) for which the semiotic agency that develops as complex structures required for life exist in the chemical domain in terms of non-orientability, as already introduced in Part I of this article.

Chemical topology dispels the pledged divide between the “analogue world of chemistry and the digital world of life.” Still, instead of envisaging the Interior/Exterior duality, the interplay of templates, genes, proteins and catalysts is to be understood in terms of a HyperKlein Bottle cybernetics; see Part I.

We shall state without further elabo-ration that the long sought problem of the origin of consciousness is framed in the same dualistic framing which as in the problem of the origin of life does not avail for further understanding in this dual ontology.

Returning to our general discussion, the Inside/Outside divide does not gene-rally apply to molecular configurations.

This is also considered to be the case of the interaction of dynamically coupled enzymes (dyazimes).

Would dyazimes be frequent enough, then they “may be able to efficiently exchange common intermediates without the release of those intermediates into the bulk solution.

This exchange could be thermo-dynamically and statistically favorable. The intermediate would pass through, i.e. sequentially interact with the protein pair without being released from these dynamically coupled enzymes” (Dillon and Clarke, 1990).

Remarkably, the molecular comple-mentarity theory of the origin of life and evolution through self-organization (Root-Bernstein and Dillon, 1997) was proposed with the purport of surmounting the modelling of emergent self-organization by Boolean networks connected randomly (Kauffman, 1994).

The issue of shape recognition of macromolecules is crucial to their integration of diverse processes as coherent wholes.

The disymmetries of macromolecules is necessary for non-local long-distance interactions and the ensuing transfer of electrons and protons along the metabolic network; here, again, we retrieve the Pasteur-Curie principle of synsymmetry, whose metaform we identified as the Klein Bottle, as already discussed.

The idea is that macromolecules and particularly enzymes perform quantum non-demolition measurements of the environment (Igamberdiev, 1993).

Hence since measurements in quan-tum theory are represented by Boolean logic, Kauffman’s metabolic networks assume quantum states amenable to Boolean logic.

However, as noticed in the molecular complementarity theory, in doing so it fails thus to grasp the actual continuity of the changes of shapes.

Thus, in the latter setting it was proposed that the chemical shape that embodies this plural continuity of shapes in the interaction of dyazimes is the Möbius strip (Root-Bernstein, 1993).

This is consistent with the notion that in distinction with the classical conception



of molecular structure as inherent to the molecule for which Boolean logic applies to its modellization and understanding of its operations (Primas, 1981), contextuality is the case of chemical structure reactions, already in relation to vacuum quantum fluctuations (Boeyens, 2005); see note no. 17.

1.3 The Surmountal of the

Dualistic Inside/Outside Divide and Semiotic Agency in the Origin of Life

We have already discussed that the primal distinction establishes a boundary for systems in terms of the Inside/Outside categorization which is a reduction of the Klein Bottle logophysics. The latter operates as a semiotic agency creating a world, yet one which appears as not devoided of meaning.

According to Hoffmeyer: “[F]rom a semiotic point of view the decisive step in the process that led to the origin of life was the appearance in the world of a new type of asymmetry, ‘an asymmetry between insides and outsides’…. The formation of a closed membrane around an autocatalytic closed system of compounds… Such a stable integration of a self-referential digital coded system into other-referential [i.e. hetero-referential] analogical coded system may perhaps be seen as a definition of life” (Hoffmeyer, 2008, 2010). He further identified this origin in

“the never-ending interest of the insides into their outsides or, in other words, cellular aboutness. I have suggested that this “interest” should be understood as a property that ultimately was derived from the primordial membrane itself”. In other words, according to Hoffmeyer, given the membrane as a primal distinction produced by autocatalytic self-closure, semiotic agency is created and life appears. Yet, this primal distinction is no longer symmetric Inside/Outside-wise, but rather surmounts the dualistic Boolean reflexive-ty, still retrieving it as a particular case by disconsidering self-penetration.

It further instaurs a codification which is digital, though being positional and non-

dual, still requiring its translation to other analogical codes. We shall see upon presenting the Klein Bottle logic genera-tion of genomes, that indeed while it embodies an analogical positional-valued logic related to its surface as a self-penetrating membrane, it produces a digital codification of genomes, and still its translation to RNA coding, basic to biochemical organization of the cell.

We have argued that in the setting of the present non-dual ontology, the pro-blem of characterizing what life is, actually is but a transparent by-product –in fact a pseudoproblem- of the likewise trans-parent assumption of the dual ontology.

The non-dual ontology and the associated logophysics operates as a semiotic agency, actually an ontopoietic agency of self-reference as a principle (Rapoport, 2014b); in particular, the membrane is an instrument of it (Rapoport, 2011b, 2011c, 2014b).

Rather than the membrane being a dual ‘interface’ –disproved by Ling (2006) in relation to the notion of cellular ordered water domains (Pollack, 2013) - as Hoffmeyer puts it , actually it is a non-dual agency (Rapoport, 2014b) which operates by locally changing from distinguishing between Inside and Outside, to a non-orientable configuration which integrates them.

As already argued, this kind of metamorphosis is very much the case of molecules. By the same principle, self-reference is unseparable of time which is not a mere descriptive parameter –as in the dualistic ontology that conceives self-referentiality as descriptive rather than generative (Kull, 1997), but entwined with the Klein Bottle self- penetration.

Thus, time is actually timing, an operator rather than a parameter, which is pervasive to biological systems, as the dynamics of this self-penetration which subverts the boundary as a dual operator.

We shall later see this in the very coding that generates genomes, departing from the Klein Bottle logic.

But already chromosomes, of which genomes are made of, simply fail to behave as respecting the Inside/Outside principle of dualism. Indeed, chromosomes are both highly plastic being able to translocate



whole parts of its bodies to other parts or to other chromosomes without affecting its survival.

And all in all, together with the multifarious changes, the overall result is the extreme rigidity of chromosomes taken as wholes.

This coexistence of seemingly antagonic features was noted by the eminent cytogenetist Lima de Faria, for which he coined the term as the “folly of chromosomes”.

Chromosomes are largely available to change, and still to preservation, but rather than this paradox being the signature of the “irrationality” of chromosomes, this possibility of change coexisting with that of extreme conservation is the very character of chromosomes (Lima de Faria, 2008).

They are capable of and operate through change coexisting with overall preservation using constraints.

At times this is carried out irrespectibly of chromosomes being coextensive to the environment super-posed with the possibility of contextual-lization with respect to it, producing changes which will ensure its overall survival.

The bottomline is that in living organisms, chromosomes are embedded in water, which rather than being a mere solvent, is crucial to the whole bio-chemistry of the cell, and particularly the nucleus, where chromosomes are placed.

Water in living organisms is highly organized as a liquid crystal which may have defects whose topology is that of the nonorientable Möbius strip topology as local sections of the non-orientable real projective space (Rapoport, 2013) topology of liquid crystals (Machon and Alexander, 2013).

Cytogenetist Lima de Faria noticed in his theory of evolution through self-organization (Lima de Faria, 1988), that symmetries of structures which already pertain to physics and reappear in crystals anticipate the symmetries which will later appear in biology -an observation predated and amply illustrated by Pettigrew (Bell Pettigrew, 1908) and by Haeckel-, and whose appearance required no genomic

system at all. We shall later retake this issue more extensively.

1.4 Introducing the Mobius

Strip, Klein & Hyper Klein Bottles This article will be devoted to the non-

orientable topologies of DNA and its bioinformatics, and its relation to the evolution of genomes.

These topologies surge from vortical geometries; they are not related to the notion of a metric which is pervasive to theoretical physics.

However, already Pettigrew identified them as the geometry of motion and nature, and particularly of biology (Bell Pettigrew, 1873, 1908). Particularly, they are related to the 5-fold pentagonal symmetry of torsion geometry which is basic as much as to spacetime, biological and chemical structures, crystals, the anatomy and physiology of the human body and to cognition (Rapoport, 2009, 2010a, 2010b, 2011a, 2011b, 2011c, 2012, 2013, 2014). This vortical geometry is related to the Golden Ratio, Φ, the latter discovered by Jean Claude Perez in genomics and proteomics, who after 30 years of research showed that Φ together with the numbers 1 and 2 conform the harmonics of the bioinformatics of genomes, particularly the mitochondrial human genome (Perez, 2009, 2010, 2013).

We shall place in evidence following Rapoport (2011c) that non-orientable topologies are at the very basis of genomes as wholes, and its bioinformatics.

As we shall see, this appears to be related to a deep coherence of genomes in which whole and parts are unseparable to its constitution and equilibrium and stability properties.

This introduces from the very outset for the generation of genomes their dynamical character, embodied as topological non-orientable foldings and discontinuities, identified as the basic genomic operations.

We shall link them both to conservation and novelty, as we shall dis-cuss below.

To do this, we retake the work in Rapoport (2011a) to construct genomes through a very simple self-referential albeit topological algorithm, alike to a



cellular automata (Wolfram, 2002; Perez, 2009).

Our departure point will be a digital numerical representation of the Klein Bottle logic, and the identification of its four states with the four letters of DNA

(alternatively, RNA) for which we need to introduce first the Möbius strip and the Klein Bottle.

Later on, we shall see that the latter can be thought as a nondual logic.

[B] [C] [D]

Figure 1.II. [A] : To turn a rectangle into a Möbius strip (presented at the centre) –in this case a left handed one though right-handed Möbius strips are also the case, join the edges labelled A so that the directions of the arrows match. We note that due to the 180 ⁰ twist of the red line in [A], the Möbius strip can be conceived as a dimensionalization producing process: namely, a two-dimensional surface which is contained in a one-dimensional single closed curve (now painted in

yellow in [B]); (JoshDip BY-SA 30; MobiusJoshDif.jpg in Wikipedia) The Möbius strip as a surface is contained in three dimensional Euclidean space. [C]: To construct the Klein Bottle glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, you must pass one end through the side of the cylinder. Note that this creates a circle of self-intersection in which the surface self-penetrates; this is an immersion of the Klein bottle in three dimensions. But in distinction with the Möbius strip, the dimensionalization is such that two opposite lines (depicted in red and blue in [C]) gives rise to a surface which due to the self-penetration, is still embedded in 3d-space but rather than contained in it without self-intersections as for the Möbius strip, it is self-contained, while still being able to act as a container, albeit an imperfect one; it may leak. [D]: Fundamental polygon of the real projective plane; it is non-orientable and non-self-intersecting, in distinction with the self-penetrating Klein Bottle below (Fig. Wikipedia; Public Domain)

Figure 2.II. Sequence of topological transformations leading to produce the Klein Bottle - with slight transparency, at the rhs; rendered with Mathematica 8 using the parametrisation provided by Robert Israel; uploaded by Wridgers . Creative Commons SA BY 3.0 File:Klein bottle translucent.png, Wikipedia.

http://en.wikipedia.org/wiki/Rectangle

http://en.wikipedia.org/wiki/Immersion_%28mathematics%29



[A] [B] [C]

Figure 3.II. The Klein Bottle and a HyperKlein Bottle. In [A]–courtesy of Theon (File:Bouteille_Klein_2Mobius.png, Wikipedia)- we see two oppositely twisted Möbius strips produced by cutting the Klein Bottle along the longitudinal section; conversely, zipping them we obtain the Klein Bottle. Thus, in distinction with the Möbius strips which can be either left or right handed, the Klein Bottle is neither, yet possess inherent to it both chiralities. [B]– Image of a HyperKlein Bottle, created by Alan Bennett (Science Museum, London); photo by Nevit Dilmen CC BY-SA 3.0 (https://en.wikipedia.org/wiki/Klein_bottle#/media/File:Science_Museum_London_1110529_nevit.jpg). More and very diverse examples of these self and hetero-penetrating surfaces, with several boundaries rather than a single self-penetrating one, which we generally call as the Hyper Klein Bottle, were produced by Alan Bennett, for the Museum of Science, London, see http://www.scienceandsociety.co.uk/ (search Klein Bottle, see those having multiple reentries). In particular see D) www.sciencemuseum.org.uk/images/I065/10328078.aspx, E) www.sciencemuseum.org.uk/images/I065/10328078.aspx F) www.sciencemuseum.org.uk/images/I046/10314766.aspx [C]- the topological identifications which produce the non-orientable real projective space (Real Projective Space, Wikipedia). It is a compact non-orientable two-dimensional manifold, that is, a one-sided surface. Liquid crystals, due to their uniaxial symmetry have this topology through their director vector field. .In distinction with the Klein Bottle, the fundamental figure which shows the topological identifications, has the two sides, both oriented oppositely, rather than one, as in Fig. 1.II [C] versus Fig. 3.II [C]. The Möbius strip can be conceived as a real projective space which one of the pairs of opposite sides identification is frozen. Yet, common to all three, they are all non-orientable two dimensional surfaces. Local section of the real projective space, which is the usual case of bounded structures, say liquid crystals in a cell, are Möbius strips. However, both the Klein Bottle and the real projective space are not contained, but self-contained, realizable by self-penetration, both conditions not being the case of the Möbius strip. The latter has a more restricted self-referential nature, as an embedded surface in ambient three dimensional Euclidean space.

The Klein Bottle through its singly self-penetrating boundary enacts a non-dual integration of Outside and Inside, environment and system. It does not comply with Aristotelian-Boolean (i.e. classical) logic which is the standard methodological setting of systemics and first-order cybernetics: controller detached of the controlled system, enacting instead a second-order cybernetics in which they unite as the system (Rapoport, 2011a).

Yet the classical logic is a reduction of the Klein Bottle logic by discon-sidering its self-penetration, and can also be seen in relation with the Hada-mard matrix (Stern, 2001; Rapoport, 2011a).

As already argued, it is with the work on trans-classical (i.e. non-dual) logics by Günther (1965) that cyber-netics came to consider that dual logic stands for a reduction of more general non-dual logics which systems enact: Logic as a creative agency –an ontopoiesis, both ontological and epistemological, i.e. logophysics, as upholded and developed by the author

(Rapoport, 2014b); in note no.10, we recall, we discussed the contrast with the dualistic reduction.

3. The Harmonics of the Möbius strip and the Klein Bottle and the Eversion of Organisms

In this article we shall deal with palindromic structures in genomes, namely, lexical structures that admit the same interpretation independently of the order in which they are encoded or decoded. For example: ‘ana’; ‘so many dynamos’ –in omitting the blanks, of course. As already discussed and visually shown, generic palindromes can be made topologically equivalent to Möbius strips by a 180° twist at the symmetry point and joining the ends. This inverted repeat of a lexical structure that appears to be as identical to the original transcript, either be a music script, parts of a genome or whatever, might be understood in terms of a musical metaphor, that of harmonics. Let us explain this. Consider the following figures, following the discussion in (Rapoport, 2013).

http://www.sciencemuseum.org.uk/images/I065/10328078.aspx



http://en.wikipedia.org/wiki/Orientability

http://en.wikipedia.org/wiki/Manifold

http://en.wikipedia.org/wiki/Surface



Figure 4.II. The Möbius strip, the 2:1 harmonics and the protoform of Newton’s Third Law (Rapoport 2013). [1]: The green line is a center line. Suppose the red line extends counterclockwise from point a. And it turns back and arrives at point b on the opposite side following a 360⁰ turn –yet it does so without crossing the green line; so to return to point a again another complete 360° turn is required. This is the topological origin of the 2:1 harmonic (–and of the topological protoform of Newton’s Third Law, as we shall see below): We need two complete turns to return to the original departure point. The red line never went across the center green line. Nevertheless the red line runs at both sides of the center line. The red line looks like parallel lines when we see them partially. But it is not true. Route A and route B are actually a single route. The green line looks a median or a central reservation on a road but it does not divide Möbius strip into two. There is no opposite lane on Möbius strip. It is also the origin of a protoform of Newton’s Third Law, albeit it does not require, in distinction with Newton’s formulation, a dualistic assumption. Indeed, consider a normal vector to Möbius strip say surging from point P; if we move the vector along any curve in a 360° turn as before to stand on P but on the “other” side, we would find it pointing in the opposite direction (as depicted for other vectors), and equal in its length, without the need of an assumption as in the Third Law. Another 360° turn will return the vector to the same point and to coincide with its original configuration. Rather than having an action and a reaction, in the Möbius strip and in the Klein Bottle, the opposite and equal modulus of normal vectors is a resultant of the 2:1 harmonic, not an hypothesis for the foundations of physics at large. So it appears that this harmonic is more fundamental to physics than Newton’s Third Law, which invokes an instantaneous symmetric causality. (r.h.s. Third Law figure, copyright Tania Rapoport).

To visualize how the Möbius strip may arise in terms of discrete elements and their identifications, namely musical tones –later codons will be the case- instead, consider as an example the phenome-nology of the perception of the pitch of musical tones lying on an octave, actually the perception of a tritone (half an octave) (Shepard, 1968; Deutsch 1992).

3.1 Octaves, 2:1 harmonics and

genomics arrangements of Mobius strip and Klein Bottle.

We still remark for future developments on addressing the harmonics of genomes, that the 2:1 harmonics of the Möbius strip, or still of the Klein Bottle as two such strips fused together, arises from giving two complete revolutions to return to a point, an octave above, as is already the case of the tritone paradox.

Yet, in this development to one upper octave, what is crucial is that we have in its

motion to the octave above, an equal number of steps which are conceived either as in the “opposite” side –which is actually only a single side- or as lying on both local sides of the surface, or, in other terms, a 1:1 resonance.

In examining genomes, we shall identify these elements that constitute the first octave as the 64 codons, paired as codon and mirror-codons, which shall be defined, and correspond to the palindrome given by the perception of the octave, as explained above.

If wished, the 32 codons will correspond to the first half of the octave, the 32 mirror codons, to the second half of the octave, which in the experience of the tritone, are identified; see Fig. 5. II.B.

This leads to conceive the Möbius strip and the Klein Bottle as having a unison 1:1 that makes up the octave 2:1, alike to the case of the perception of the tritone.



Figure 5.II. Left: The Tritone Paradox of Perception of Music, represented as a disk-shape real projective plane, with antipodes being identified, and tones further synesthesically associated to colours (From (Merrick, 2011) ©, courtesy of R. Merrick). Right: The Möbius strip and Klein Bottle structure of music perception of the Tritone Paradox: In the perception of a tritone (half an octave) appear the perceptual pairings B-F, C-F#, C#-G, D-G#, D#-A and E-A# which are naturally represented along the edge of a Möbius strip with the opposite points joined by lines representing the tritone perceptual identification. This represents the fundamental 2:1 resonance: A complete rotation on the circular pitch space of an octave perceptually translates into two complete rotations on the Möbius strip, say B-C-C#-D-D#--E-F- F# followed by F#-G-G#-A-A#-B completing the single edge of the Möbius strip. The perceptual space turns to be the Klein Bottle surface, on identifying the antipodal points –depicted by lines on joining the antipodal pitches on the edge of the Möbius strip- as perceptual identities. Therefore, perceptual space -according to the Tritone Paradox - is a Klein Bottle surface (Rapoport, 2013). Music perception assents to the 2:1 resonance of a lived world. (Background figure by Fropuff, MobiusStrip-01.png, Wikipedia, Public Domain)

3.2 The relation with

enantiomerism and the underlying non-orientable topology of the human body

Let us further consider the 2:1 harmonics with respect to the issue of chirality of molecules. The main point is that the rotation of a three dimensional structure on the Möbius strip precisely describes enantiomers (Rapoport, 2013).

Thus, it is this non-orientability which allows to embody the fact that chirality is a dimension-dependent phenomena (Mis-low,1999), and what on a planar surface cannot be reflected to pass from one chirality to the other, it only takes a 180° twist in a Möbius strip as embedded in three-dimensional space, to be able to intertransform both chiralities.

For instance, the movement of L-lactic acid around the Möbius strip gives upside-down D-lactic acid.

The principle is general. Indeed, for the genomatrix to be introduced will display a recursive similar identification between codons and anticodons which are naturally located on a Möbius strip or Klein Bottle.

For the Mendeleev table we have instead a superposition of matter/antimatter-atom, in each local side of the Klein Bottle (Boeyens 2005, 2010).

Remarkably, Möbius himself, thinking on molecules as three-dimensional geometrical conformations, rather than topological two-dimensional structures –to be discussed below, remarked that a fourth-dimension would suffice to transform one chirality to the other one (Mislow, 1999).

This is crucial to the biochemical activity of molecules, and in particular, of proteins. We recall that Pasteur discovered the chirality of molecules, which lead him to propose the complementarity of form and function in biology, which is known today as the Pasteur-Curie Principle, in recognition of Pierre Curie’s work on the subject.

Yet, what this examples teach us is that the issue of chirality, in general, can be associated with the non-orientability of the Möbius strip, or still of the Klein Bottle, where both chiralities are fused.

A new paradigm of chemistry, and in particular of organic chemistry, has been developed in which non-orientable



topologies are the very basis of the subject (Sokolov, 1973).

We may conclude: both enantiomer-rism and Newton’s Third Law of action and reaction have a topological protoform in which enantiomers and opposed vector forces are united not as twins but as dynamical single objects through a vortical motion along the Möbius strip.

Furthermore, from the existence of the human body as a whole through the integrated coexistence of both right and left hand chiralities, it follows from the previous observations that the toroidal configuration of its vertical symmetry, which is proper of metazoans-encom-passing from cnidarians to vertebrates and particularly humans (Jockusch and Dress 2003; Isaeva, 2014, Maresin and Presnov, 1985; Rapoport, 2011c, 2014a), has a built-in Möbius strip or Klein Bottle architecture that integrates both chiralities.

We recall that the elastic deformation of morphomechanics produces this archi-tecture, as discussed in Part I.

Indeed, both surfaces are immersed in the 2-torus with the 2:1 harmonics (Rapoport, 2013) and the topographic maps of the visual and somatosensory modes proves that this Klein Bottle architecture is indeed the case (Schwartz, 1977a, 1977b; Werner, 1970; Werner and Whitsel, 1968). It is quite remarkable that this observation seems not to have been made before.

Kant himself discussed the status of chirality vis-à-vis that of space as absolute or relational (van Cleve and Frede-rick,1991), but the notion of a non-orien-table human body-plan seems to have been left unidentified.

It is no less remarkable that the figure-8 or ∞ corresponding to the lemniscate, the one-dimensional project-tion of a Möbius strip on a plane, is the signature of flying, natatory and earthly motion of animal gait (Bell Pettigrew, 1873, 1908), as a projection of the metazoan bilateral symmetry organized as a 2-torus as the double covering surface of both Möbius strip and Klein Bottle.

Particular manifestations of the non-orientable bodyplan are the X-cross association of right (left) hand with the left (right) brain hemisphere, the X-cross

morphology of the optic nerves with respect to the eyes, the lemniscal cardiovascular loop in birds, in amphibians upon becoming terrestrial and in humans (Furst, 2014), etc.

As already argued in §1.4, they appear as the solution of non-linear elasticity problem of development. 4. Introducing the Klein Bottle Logophysics: Cognitive Semantics, and some remarkable biological examples of eversion and the non-orientable topologies of embryos, the human body, genomes, light waves and supernovas.

4.1 Fracturing Nature into a

Dualistic Image-Schema, Language-Games, Mathematical Modelling and Whither Reality is Located

Cognition is framed in terms of image-schemas which play a crucial role in structuring our language and our conceptions; in this organization of cognition, metaphors play a crucial constructive role (Johnson, 1987; Lakoff and Johnson, 2003).

This role of image-schemas is usually unacknowledged, yet implicitly used as the fundamental element for knowledge and modellisation (Goertzel, 2013).

In this section, we shall introduce a particular image-schema to the effect of presenting the non-dual Klein Bottle logophysics which is pervasive to physics, cosmology, geophysics, perception, chemistry and biomechanics (Rapoport, 2013), and biology (Rapoport, 2011b, 2011c, 2012, 2014), and particularly to genomics, to be presented below.

We shall illustrate this with several examples. Indeed, one of the most pervasive image-schemas is CONTAIN, which organizes spatially cognition in terms of a boundary or a distinction acting as such.

This distinction can be either material or imaginal (say, an ideological ascription of any kind), dividing the world into an Outside and an Inside.

The philosopher and epistemologist Gaston Bachelard argued about the perva-sive character of this image-schema and its seemingly inevitable character in framing



our representations of our experience of the world, and its role in a phenomenology of imagination (Bachelard, 1964).

Indeed, the discourse on objectivity and subjectivity is framed in terms of the Exterior/Interior divide. It is said that consciousness consists of ideas, emotions, memories and the likes, but etymologically this comes from the Latin, ‘to stand together’, ‘to stand firm`, ‘to exist’, which later in Medieval times it became to ‘be part of’, or still to ‘be in’. Presently it is somewhat ascribed to consciousness the function of containment, which appears to follow the latest etymology: to be Inside (or its single alternative, Outside), is only one way of being.

In the setting of a study of the unity of action and perception, G. Vesey has called the Inside/Outside dual categorization a “baleful … philosophical myth” alike the “ghost in the machine” or “the myth of the world as external” (Vesey, 1991).

For Goertzel instead it is a “justified system of belief… Nieztsche and Whorf… shared the following radical view: external and internal reality are belief systems.

Further, they both maintained that one of the main roles of consciousness and language is to maintain these belief systems. Beings without consciousness and language, according to this perspective, do not perceive a split between external and inner reality” (Goertzel, 2003).

However, Goertzel kept the dualistic divide as an implicit ontology in his stu-dies of cognition and self-reference (Goer-tzel, 2003).

Hao Tang, upon discussing Wittgen-stein’s (of language-games reknown) take on the Inside/Outside divide and its dualistic ontopoietic role, qualified it as “…an intellectual disease… an intellect-tualist conception of language.” (Tang, 2014).

It was Wittgenstein (Glock, 1996; Hark, 1990) who confronted this duality and identified its source “…in the very process of learning a sensation-language…

what I call the incision of language, i.e., the infliction of cuts upon certain natural and primitive unities between the inner and the outer” (Tang, 2014).

As for the import of such incisions, we can quote Wittgenstein’s take on what their effect come to be:

“A picture held us captive. And we could not get outside it, for it lay in our language, and language seemed to repeat it to us inexorably” (Wittgenstein, Philoso-phical Investigations §115).

Wittgenstein call is for us to experience language beyond its descriptive operation to convey thoughts, which is the case of enhanced or altered states of consciousness which elicit a non-dual logophysics (Shanon, 2003).

Remarkably, language operates upon the violation of dual logic, as a cognitive entanglement, since it “it enables us to understand but at the same time also tends to make us misunderstand” (Tang, 2014), a double-bind, as it were, a cursed-blessing, which sets us on the path of learning otherwise.

It is attributed to Nagarjuna, that the four-state logic that he proposed was meant precisely with this intention (Westerhoff, 2009).

According to the CONTAIN image-schema, the world of objects resides in our Exterior, while subjectively is a thoroughly Interior issue.

An interesting testimony of this dual ontology, the failure to surmount it while studiously refurbishing it and further presenting its critique is the case of (Atmanspacher and Dalenoort, 1994).

This established paradigm was contested by Lehar, by literally proposing an alternative turning Inside-Out of this model (Lehar, 2003). Merleau-Ponty also proposed a similar Inside-Out turning of the ‘flesh’, as operating perception and beings (Mazis, 1988).

In other words, Lehar proposed that the world which we conceive as Exterior actually is Inside our physical skull.

Although it generated interesting debates (Velmans, 2007), since there is no absolute frame of reference but the illusory one as setup in the dual Inside/Outside divide, it seems impossible to establish whether the usual understanding of the real world as placed Outside or Inside -relative to us- is indeed the case.

Quantum holography (Marcer and Schempp, 1997; Mitchell 1998, Schempp,



2001), in which phase conjugation which is crucial to the Nilpotence Universal Rewrite System (Rowlands, 2007) -and is tantamount to the rotational action of the TIME operator in Matrix Logic (see note. no. 11), was proposed as the physical process that explains the cognition of the physical world as if Exterior (Marcer and Schempp, 1997), and indicated as operating the process of brain hemispheric integration supporting binocular vision and cognition (Rapoport, 2010b).

Of course, as in coupled oscillators, the generic phenomena of resonance does not abide to the Interior/Exterior divide. In particular, the fundamental oscillation of the brain known as the 40hz signal, as the outcome of the coupling of many oscillations, already indicates that this dual divide is not the case of the fundamental embodied oscillation which is usually considered to be the signature of consciousness (Freeman, 2000; 2007b); the topology of the cortical neural networks that produce it is suggested to be the Möbius strip (Wright, 2014).

Most remarkably, there exists an homologous surmountal of the categorical Interior/Exterior divide, which occurs at the very foundations of mathematics as formalized in naïve set theory.

The latter is associated to CONTAIN conceived as a belonging to. Set theory is widely considered to be the basis of contemporary mathematics, though an alternative formalization which still uses sets as fundamental has appeared.

Upon revising the problems elicited by the usage of CONTAIN –though wthout framing it in its terms- it has been proposed to go beyond dual logic for the construction of mathematics (Lin Forrest, 2013).

Still, as a a kind of vengeance to the failure of providing a semantics-free axiomatization of mathematics, the alternative founded on the theories of topos and categories brings interpretation and non-classical (intuitionistic) logic to the core of this proposed alternative.

However sets still remain in this setting as a basic example but they are subjected to interpretation in terms of an Exterior/Interior categorization (Gold-blatt, 1984).

The intuitionistic school initiated by Brouwer emplaced finite constructivism as the basic methodology for mathematics: a mathematical entity must be constructed in a finite numbers of steps.

An appeal to a potential infinity as if an actual infinity is forbidden.

Indeed, intuitionism avoids working with actual infinities; only potential infinities are permissible as mathematical entities.

Furthermore, Tertium non-datur is only valid for finite sets, and should not be used when dealing with infinite sets.

The concept of infinity is keenly associated with the violation of Tertium-non-datur, since actual infinities and potential infinities can be and cannot be identified (Lin Forrest, 2013) while mathematical induction implicitly assumes their identification, thus Brouwer’s proposals. Mathematical reasoning assu-mes classical logic by default.

Galileo already observed that mathe-matics appeared to be the ‘language’ of Nature, which can be further precised as the language which we use and discover/invent to study Nature.

However mathematical cognition appears to be common among animals, though very limited.

Wigner coined the famous term of the ‘unreasonable effectiveness’ of mathema-tics.

Yet mathematics springs from the imaginal domain further elaborated intel-lectually, and image-schemas (Lakoff and Johnson, 2007) which subsume funda-mental conceptual elements of mathema-tics no less than of our colloquial language.

Mathematics appears to be sourced in the embodied mind (Lakoff and Nuñez, 2000). However, the imaginal domain cannot be accounted by dual logic (Corbin, 1969, 1983; Durand, 1969), but instead by a four ontological loci logic (Günther, 1962), the Klein Bottle logic (Rapoport, 2014b).

Thus, whether mathematics describes an Exterior world or an Interior domain is a query that follows its framing in terms of dual logic.

A rather striking example seems appropriate.



We recall that in the setting of mathematics in terms of defining properties of a collection of objects by reference to its members as contained in a set, we may rather proceed by referencing to its External relationship with respect to other collections.

The former approach is the current basis for the construal of mathematics and is associated to Boolean logic.

However, in adopting the latter approach which is the case of the theory of topoi and categories, this categorical divide is surmounted by replacing membership or containment by the symbol of an arrow, as an abstraction of function-nal relationship or of linkage.

Actually, the theory of topoi and of category was developed to do away with CONTAIN by replacing the property of belonging with the ‘arrow’.

Thus a category is an algebraic structure that comprises ‘objects’ that are linked by ‘arrows’.

A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.

A simple example is the category of sets, Set, whose objects are sets and whose arrows are functions.

A topos is a category that behaves alike Set.

The operations of union,intersection and complement of sets are associated to Boolean logic. However, the logic of Set as a topos is not Boolean; actually it is a Heyting algebra for which the law of the excluded middle is not valid (Goldblatt, 1984).

The theory of topos not only contex-tualizes CONTAIN, but also gives it a particular relevance vis à-vis establishing whether Boolean logic is the case or not of a topos; in this setting a topos may have an Interior and Exterior logics which differ in kind, one which can be interpreted from the Outside as being non-classic while upon being interpreted from the Inside, the logic is Boolean and thus so is the topos (Goldblatt, 1984). So departing from surmounting CONTAIN as a basis for mathematics in terms of set theory we observed that category theory externalizes it in terms of ‘arrows’ as representations of

linkages, which places us in the ontological grounding of non-classical logic for mathematics in terms of topoi.

The cognitive consequences of this are most remarkable with regards to whether reality is to be placed as Inside or Outside.

Indeed, in the particular case that the logic of topos is Boolean, this generates an algebra which is isomorphic to the algebra of quaternions (Trifonov, 2007).

This algebra is pervasive to physics, and crucial to represent rotations. However, this algebra has an independent representation in Matrix Logic, derived from the torsion in cognitive space introduced by the Klein Bottle (Rapoport, 2011a; Stern, 2000).

However, as an algebra with a unit it can be extended to include the invertible elements, and thus produce a Lie group, which has a relevant class of metrics, the Minkowski metrics. Through a simple coordinate transformation, they are transformed to the closed Friedmann-Lemaitre-Robertson-Walker (FLRW) metrics of cosmology.

These are solutions of Einstein equa-tions in GR, and are important in cosmology, since in particular they are appropriate to describe spiral galaxies (Carmeli, 1999); would we add to it torsion produced by a spin density we obtain that the visible Universe is the resultant of a turning Inside/Outside of a collapsing black hole (Poplawski, 2010).

However, disconsidering the latter more general case, this identification of basic elements of theoretical physics including the four-dimensionality of spacetime, the Minkowski metric and the FLRW metrics of cosmology decurs from elementary considerations on foundational issues in mathematics and the single basic hypothesis of the Boolean logic of the observer (Trifonov, 2008).

Actually, this derivation does not require actually solving Einstein’s equa-tions for the metrics, nor any equations at all, for that matter.

However, upon restricting the non-classical logic of topoi to Boolean logic, we observed that the algebra of non-zero quaternions which is both present as an algebra of physical spacetime and in Matrix Logic, is associated to the

https://en.wikipedia.org/wiki/Algebraic_structure

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Minkowski metric of physics and a metric of spiral rotating galaxies, indistinctly associated to the physics of spacetime as a modellization of an Exterior physical space, or to the logophysics associated to the Klein Bottle.

This is a common model for physical spacetime as if Exterior, or for cognitive space or two-state quantum systems as Matrix Logic associated to the Klein Bottle (Stern, 2001; Rapoport, 2009, 2011a), in which indistinctly we obtain the rotational logophysics of spiral galaxies as if Exterior or as if Interior.

Returning to Lehar’s contention that the world which we experience as if Exterior actually is Interior relative to ourselves, applies as well to the modellization of the world in terms of surmounting the dualistic image-schema CONTAIN.

To resume: Upon interpreting Set in this non-dual ontology in terms of classical logic, we can not assert anymore whether these abstract yet simple models refer to Exterior or Interior relative to ourselves.

A Klein Bottle ontology appears to be natural. CONTAIN can serve no better. 4.2 The Dual Image-schema CONTAIN, its Non-dual Surmountal: the Distributed Self, Biological Development, Genomics and Evolution The surmountal of the CONTAIN image-schema by a non-dual logic, actually the Klein Bottle Logic, is crucial to creation in terms of self-reference as embodying a non-dual ontology -an ontopoiesis, or still its hetero-referential extension as embodied by the HyperKlein Bottle. CONTAIN as an image-schema is further associated to the boundary as a logophysical dual divide, then a two-state (true and false) logic appears.

This is the classical dualism, in which usually true is associated with Inside, false with Outside; or an inversion, as we shall see.

The organization of cognition in terms of image-schemas is operated transparen-tly, just alike the impediment to conceive the Self as a model to which we attach

ourselves as if real (Metzinger, 2004) as argued in the note no. 12.

As a belief system, CONTAIN is extre-mely pervasive due to its bearing as a sense of reality, as already discussed, so it appears to be impossible to challenge its validity due to its transparency and so far unique ruling.

The virtual world of the so-called information technologies appears to ins-truct us that whatever the environment for the enactment of our experience may be, this image-schema literally takes hold of our sense of reality, as Wittgenstein warned us about language.

Indeed, as already discussed a sense of reality is largely enacted in terms of this image-schema, giving us the impression of immersion as Inside a world, be that virtual or of our more usual experience which increasingly is notoriously fabricated.

As argued by Marcer and Schempp (1997) this experience appears to be related to holography, which Pribram (1999) ascribed to the workings of the brain.

However, in the case of the infor-mation representation on the visual cortex, the receptive fields appear to be describa-ble by the Gabor functions of holography which self-organize, rather than the previous categorical divide being the case.

The resultant topology for the cortical mapping of optimal orientation and phase of the receptive fields appears to be the Klein Bottle, which integrates Inside and Outside as a non-dual logophysics (Tanaka, 1996).

So, underlying the Boolean logic (actually the single-state projection of it to the hegemonic true value) of the experience of a sense of reality lies a non-dual ontology, already operating at the level of neural networks, but not restricted to them since this ontology operates the topographical cortical representations of the sensorium and their motor response as the unity of perception and action.

However this is not restricted to a single sensorial mode. Indeed, due to synesthesics, this topology which is also embodiedas the topographic represen-tation ofthe somatosensory mode (Werner, 1969) and the visual mode (Schwartz,



1977), extends to all the sensorial modes involving the Self as a whole.

So the issue of co-ordination of modes is crucial, and as already discussed, the heterarchies and the HyperKlein Bottle operate co-ordinations, these in particular.

The ontology for the dualistic take of CONTAIN stems from the reflective character of the single negation operator of the dual logic, i.e. Aristotelian-Boolean logic (ABL) as a monocontexture, the Objective Exterior world (Rapoport, 2014b).

Early in the 1960s Günther introduced the notion of selfhood as being distributed between the I-subjectivity and the Thou-subjectivity (Günther, 1979), not an Inside/Outside divide. Günther associated this distribution to the multivalued logics ontology of the heterarchical polycontextu-res already alluded.

In contrast with Günther’s distribu-tion of identity and of rationality (Günther, 1979 ; Kaehr, 2007) the dualistic CON-TAIN places Inside –or pushes Outside to the Eye-of-God of Objectivity stance, with meaning as emplaced in an unmediated Outside, to be downloaded, as it were (Johnson, 1987; Lakoff and Johnson, 2003).

This distribution appears to introduce the Hyper Klein Bottle as the metaform of selfhood, since Thou is unseparable of I, actually entwined, interpenetrating beings. Hence, co-ordination also operates at a social level which is crucial to the construal to the self, as elicited by the mirror-neuron phenomenology. Remarkably, Günther anticipated neuro-scientists and philosophers of conscious-ness for which action- particularly imitation (Frith and Wolpert, 2003)- is ontologically more fundamental than passive perception would the latter be the case at all (Freeman, 2000, 2007a, 2007b; Merleau Ponty, 1965), so that the motor representations underlie “the conscious experience of being a self in the act of imitating” (Metzinger, 2004).

Dual logic was first proposed by Aristotle, and algebraically by George Boole and it is very much the basis for informatics and digital computers, and in particular to bioinformatics (Petoukhov and He, 2010, He and Petoukhov, 2012).

The importance of this to biology cannot be overstated: Embryology is fra-med in terms of the foldings of the ecto-derm and endoderm (as Exterior and Interior tissues) of zygotes.

Yet, rather than this upholding a two-state logic, due to the existence of the mesoderm appearing with gastrulation and altogether forming the three germ layers, points to a non-dual logophysics as operating in biological development and differentiation (Rapoport, 2011b, 2011c, 2014a, 2014b).

Also the biology of the cell is concei-ved in terms of the membrane, acting as a dual operator-barrier, disconsidering thus the fact that the cytoskeleton is continued through the membrane by the extracellular matrix, which is crucial to the organism’s integration.

The logophysics that arises by consi-dering that the membrane is actually a non-orientable surface, has been conside-red in Rapoport (2011c, 2014).

CONTAIN is ubiquitous and in most cases it might be associated to a non-dual logic of metamorphosis which creates a multiple Inside and Outside which are process-wise connected.

Indeed, eukaryogenesis -after Margu-lis, is conceived as a process of endosym-biosis of a proteobacterium with an ancestral archaeon, with the endomem-brane system and particularly the nucleus evolving as defense against intron invasion (Koonin, 2012).

This can be assimilated to a cybernetic process creating a HyperKlein Bottle (fig.3. II B,D,E & F), the eukaryote cell. CONTAIN is crucial to the theory of evolution based on partially nested developmental systems (Oyama, 2000; Jablonka and Lamb, 2005), which do not ascribe to dual logic.

Topologically, they are HyperKlein Bottles with multiple Inside and Outside distinctions which cannot be conceived as a hierarchy, but a mutually and self-penetrating heterarchy conformed as a HyperKlein Bottle.

These may include cultural distinc-tions as evolutionary elements (Distin, 2011).

In particular, the reduction to a dualistic logophysics is the root of the



nature-nurture divide (Oyama, 2000), among the many dualisms such as the mind-body divide, biological system-environment, etc.

As a notice of a cognitive turning Inside-Out, the developmental system approach to evolution proposes as a methodological recourse to examine in addition to which are the genes in a certain genome, the environment in which the genome bearing organism is placed in (Oyama, 2000).

Furthermore, this partial nesting is suggested to be linked with the surmountal of the Central Dogma of biology in the framework of System Biology (Oyama, 2000).

Indeed, the Central Dogma conceives genes in the dualistic setting of CONTAIN, as independent of the environment, and omnipotents agents of control in a first-order cybernetics.

An apologist of this take, Richard Dawkins puts it as:

“Now they [genes] swarm in huge colonies, safe inside gigantic lumbering robots, sealed off from the outside world, communicating with it by tortuous indirect routes, manipulating it by remote con-trol. They are in you and me; they created us, body and mind; and their preservation is the ultimate rationale for our existence.” (Dawkins, 1976). As we shall see, already genes both in

their generation and in the nucleus where they interact with histones, embody a non-orientability, which belies this purported isolation, as an external all-commanding deity. CONTAIN as a dualistic image-schema is also basic to embryological develop-ment. Indeed, in the standard paradigm of developmental biology, tissue or cell-type differentiation is conceived as a “process that goes on in a particular direction unless diverted by an outside stimulus” (Dawkins, 1976). Instead of this paradigm, we have proposed (Rapoport, 2011b, 2014) that tissue differentiation appears as the outcome of a non-dual Klein Bottle logophysics associated to the TIME operator, acting through contraction and

expansion waves which are initiated by a biodevice.

This biodevice, called the cell state splitter, was conceptually introduced by Gordon and Björklund-Gordon, having observed a wave dynamics associated to tissue differentiation.

This biodevice purportedly enacts the mechanism which is setup by the cell –say, by microtubules among other structures- to produce the contraction and expansion waves which operate alternatively as a dual-state morphogenetical timing mecha-nism on the cell as a tensegrity structure. These waves have been observed in Axolotl (Gordon, 1999).

It has been proposed that a non-dual logophysics associated to vortical fields is the case rather than linear geometry and dual logic (Rapoport 2011c, 2014).

Contraction waves play a crucial role in initiating embryological development of Volvox (Kirk 1998), to which we shall re-turn below.

This dual image-schema appears in the theory of evolution, as a kind of hege-monics (Rapoport,2 014b).

Indeed, as argued in the setting of a non-adaptive evolution critique of Darwi-nian selection, the latter is based on the identification of evolution as a kind of Exterior selection ‘force’. In this setting mutation appears as an Interior phe-nomenon “…creating variation but never controlling the ultimate direction of evolutionary change” while ignoring the capability of “mutation to overcome the force of selection” (Lynch, 2007a).

Yet studies which were crucial to the later formulation of the Central Dogma of genomics, elicited that bacteriophages po-pulations mutate in anticipation of the environmental cue, so that the Inte-rior/Exterior divide is not the case of mutations in phages and the causal rela-tion appears not to follow it neither (Summers, 2006).

As we already mentioned, the tem-poral logic of biological organisms as who-les is non-dual, as is evident in the case of conchoids (Illert and Santilli, 1995).

However, current studies in compara-tive genomics have suggested the role of the sizes of populations in promoting whe-



ther a rejection or an assimilation of alien elements by a species is the case.

This is a matter of a non-dual hetero-penetrating HyperKlein Bottle logophy-sics, indicating that “genome composition is governed by biases in mutation and gene conversion, some of which (e.g., mobile-element proliferation) operate via internal [my emphasis] drive-like mecha-nisms” (Lynch, 2007a; 2007b).

As further observed in the setting of comparative genomics of populations, Darwinian evolution has been surpassed, literally turned Inside-Out, since: “Inter-nal mutation pressure plays a central role in genomic evolution, often completely overwhelming the external forces of natural selection and driving change in a directional manner” (Lynch, 2007b).

Later on we shall relate these “selective pressure”, “internal mecha-nisms” and the mobile elements with transposons.

We shall see that these operate as the material organization which carry out topological transformations related to the topological-algorithmic Klein Bottle gene-ration of genomes which assimilates the diversified environment, and operates in material terms through these mobile ele-ments.

In other worlds, evolution is closely related to the problem of the Self as distri-buted with the Thou, whether the latter is the environment or other agents.

This stands in agreement with the notion of the non-dual temporal logic of biological systems and particularly of metamorphosis and symbiosis, as we shall elaborate below.

In fact, this distribution is traced down to evolution as arising from cellular interactions which are thought to have intervened in the major transition from the unicellular to the multicellular level. Lynch: “the roots of the cellular inter-actions that constitute development must reside in the resolution of adaptive conflicts between the advantages of cell-cell cooperation and the individual bene-fits of going it alone” (Lynch, 2007b).

So again,what has transpired is the conflict between assimilating other-refe-rence or keeping the idealized singular

entity, the latter being very much the ideo-logical scaffold for (social) Darwinism.

The latter operates by trivially projecting subjectivity to the world of ma-terial objects (Günther, 1962) assimilated to a transphysical Natural Selection, which stands breaking the real phenomenology of the distribution of selfhood vis-à-vis otherhood, which is a non-dual heterar-chical phenomenology.

The survival of the ‘fittest’ versus the integration of self and Thou, which is experienced even in conflicts, which may also result in the demise of any of them, or both, in the unending cycle of destruction and regeneration. Image-schema CONTAIN is no less crucial to genomics. Indeed, whether the Double Helix model (DH) or alternatives such as the Side-by-Side Model (SBS) is the case of genomes, crucially depends on the existence of a well defined Inside and Outside. For the proponents of the latter model, the Double Helix does not have a well defined side, though non-orientability is unconsidered (Rodley, 1995), while the SBS model claims the existence of a double strand with well defined sides.

The latter structure is not a double helix, but consists of a pair of poly-nucleotide strands lying side by side and held together by Watson-Crick base pairing (Sasisekharan, 1978).

For the DH, the Interior is given by the pairs of basis holding the Outside, where the two phosphate chains are loca-ted (Crick, 1979).

The DH came in the wake of the observation of the X-cross shape of the x-ray photograph of B-DNA (the famous photograph 51), taken by Rosalind Franklin (Franklin and Gosling, 1953) and of the Pauling-Corey model, which some-what turns Inside-Out the disposition of the DH, placing the sugar-phosphate basis Inside and the basis pairs Outside (Pau-ling, 1953).

It was the X-cross shape which was interpreted as a double helix; see note no. 18. Yet, the X-cross shape is pervasive to anatomy and physiology (Rapoport, 2013; Werner and Whitsel, 1968; Werner 1968), as the signature of the Klein Bottle one-

http://en.wikipedia.org/wiki/Photo_51



sided non-orientable surface of the sensorium, as we shall discuss further.

This shape is no less ubiquitous to Nature, say in geophysics (Rapoport, 2013), to the point of being the meta-pattern that arises from the pattern recognition of arbitrary landscapes, on ca-rrying the statistical analysis of the pixels of their digital photographs (Carlsson, 2008; 2009).

In fact, the universality of this meta-pattern is elicited from studying the sphe-rical harmonics of the pattern that arises from a sinuosoidal wave impinging on an arbitrary boundary: the first two terms (and dominant at that) of this harmonics identifies the X-cross topo-logy of the Klein Bottle,or still, of the Möbius strip (Rapoport, 2013).

It is also elicited as the topology of three-dimensional light waves with a non-uniform phase, as we shall discuss below.

Yet, these non-orientable surfaces which do not have a global Inside nor Outside but only local ones, and in terms of which we have obtained a coding for the genome, are not exclusive to genomics nor chemistry.

Particularly interesting is the fact that they appear in developmental evolution. Indeed, the algae Volvox (the “fierce roller” for Linneus) which is considered to be the case study of morphogenesis, namely that of epithelial bending, and the “Rosetta stone for deciphering the origins of cytodifferentiation” including its genetic basis, remarkably undergoes a change of orientability as a gastrulation-like morp-hogenetic process (Kirk, 1998).

Indeed, Volvox turns Inside-Out its original spherical configuration with its nascent gonidia on the External surface and the flagella of its somatic cells facing Inside the organism.

The cause for eversion is attributed to change of shapes of cells, i.e. a topological deformation, and their movements.

In order to achieve its final adult form, each embryonic Volvox must effectively turn itself Inside-Out, a process known by biologists as “inversion”.

It is known by mathematicians as the eversion of a sphere, in which Outside and Inside are intertransformed through a singularity free transformation which

requires intermediary non-orientable surfaces, a most striking topological theorem due to Smale, known as the Smale Paradox (Levy, 1995); see note no. 19.

Eversion occurs by a change of the monolayer cells that conform Volvox, curling outwardly (Hohn and Hallman, 2011; Hohn, 2015).

This eversion of a sphere, is functionally crucial to Volvox, since “in the fully cleaved Volvox embryo… the poten-tial flagellar ends point inward… if that arrangement would persist in the adult, it would make swimming rather difficult” (Kirk, 1998).

With the completion of eversion, the embryological phase of Volvox ends.

So the turning Inside-Out of the Volvox as a hollow sphere appears to be Nature’s solution to a similar inversion of the vertebrate retinal cells in which the light sensitive retina at the back of the eye is turned Outside-Inside pointing to the optic nerve, with the photoreceptors pointing away from the light instead of towards the light.

Rather than swimming as Volvox does, we are seeing.

This has prompted the notion of a “bad design” would a design exist at all (Dawkins, 1996).

It has been pointed out that the complex logarithmic topographic map of the visual sensorial mode, elucidated by Schwartz (1977a, 1977b), achieves mathe-matically the said inversion (Rapoport, 2013), with the V1 neurocortical image of the data impinging the eyes has the Klein Bottle topology (Schwartz, 1977a, 1977b; Swindale 1996; Tanaka 1991, 1997).

This exchange of Outside and Inside which in the Klein Bottle operates in both ways, is the logophysical principle which we claim that operates as Nature, rather than being an “unreasonable” oddity (Dawkins, 1986).

Indeed, for the dualistic CONTAIN image-schema, such an operation is out of its logical –actually, ontological- scope. Thus the “paradox” tagged to Smale’s theorem on the eversion of the sphere.

This movement develops through an intermediate state whereby the initially orientable spherical embryo becomes a likewise orientable sphere, yet through a



non-orientable intermediary state; geome-trical mechanical models of the eversion of Volvox have been elaborated as an inter-disciplinary effort (Goldstein 2015, Hohn and Hallman 2011; Hohn, 2015).

The process of eversion is associated to a contraction wave causing a dent which initiates the eversion.

It has been discovered that a kinesin is essential for this process, and that a transposon may arrest the rotational mo-vement of cells that produce the eversion (Boehm, 2012).

Remarkably, this suggests that transpo-sons, which we shall later associa-te with the non-orientable topologies of genomes, may be related to the eversion of Volvox in undergoing development, by also changing the orientability of its surface.

In distinction with monolayered Vol-vox, coelenterate Hydra also everts, throughex changing ectoderm and endo-derm. Both Hydra and Volvox do so by migrating cells.

Yet, Hydra’s cellular motions are such that the eversion is reversible, it will turn Outside the “rightside”-out again as the cells migrate across the mesogloea, the translucent, non-living, jelly-like substan-ce found between the two epithelial cell layers in the bodies of coelenterates and sponges, to reach the “correct” side.

On completing the eversion, Hydra starts to create new polyps and thus se-cures, in principle, its lack of senescence: Hydra is, in principle, biological immortal (Lenhoff, 1961).

Furthermore, it has remarkable rege-nerative properties.

As Goethe observed (Lewes, 1855), this as well as the turning Inside-Out are possible because all parts are indistin-guishable among them and so is the whole with respect to them.

Nowadays this is identified as a characterizing property of holographs, and associated to the generative order of Bohm, to be discussed below in the context of genomics.

So, the case of Hydra signifies a turning-Inside-Out unending process, Out-side reverting Inside further turned Inside-Out, which in each stage produces

the novel organization necessary for its self-preservation.

We shall later see that the material elements of biological systems as they arise from supernova explosions of demising stars, are another manifestation of an eternal cycle, now revealed as the life cycles of stars, rather than of immortal Hydra.

4.3 The Klein Bottle Logophysics and the Topology of the Action/Perception Cycle

Yet, what is most remarkable is that the Klein Bottle has a crucial role in physiology.

This is related to the existence of maps of the sensorial periphery as projected into two-dimensional sections of the neurocor-tex.

Indeed, the topographic maps of both the visual mode and the somatosensory modes appear to embody a Klein Bottle topology.

This is the topology of the cartogram-phic projections on the neurocortex that represent the three dimensional body’s periphery, say the data impinging on the eyes and the skin (Rapoport, 2013; Schwartz 1977a, 1977b; Swindale, 1996; Tanaka, 1995, 1997; Werner, 1970; Werner and Whitsel, 1968); see note no. 20.

Again, data appear to have a shape, actually the Klein Bottle as a metapattern, rather than merely being reducible to quantification, as the current notion of information in genomics and biology at large take for as the exclusive case (Godfrey-Smith and Sterelny, 2008).

The original work on the somato-sensory mapping, developed on macaque monkeys by Werner and Whitsel, was later extended to cats (Dyke and Rues, 1986). These topological mappings appear to be based on an analytical mapping , namely the complex logarithmic mapping, as argued in Schwartz for the visual mode (Schwartz, 1977a, 1977b) and by Werner for the somatosensory system (Werner and Whitsel, 1968; Werner, 1970).

Yet, this logarithmic mapping, due to its periodicity, has helicoidal structures associated to it as is well known in the theory of complex functions in mathe-matics (Cohn, 2010).



In the case of the somatosensory system, these helicoidal representations appear to be dermatoidal trajectories on the limbs.

Along these helicoidal trajectories the somatosensory mapping does not distinguish between the Inside, as muscle and joint receptors, and the Outside, as skin receptors.

A most crucial characteristic of these topographic mappings is their usually partial plasticity.

Namely if the body is locally severed, the topology of the representation does not change, despite images of the severed regions may be shifted to overlap neigh-bouring areas on the cortex, as is the case of phantom limbs phenomenology (Rama-chandran and Blakeslee 1998); this suggests that non-orientability is related to preservation through topological deforma-tion which conserves the metapattern, to be further discussed in the context of genomics.

Yet, while this cortical mapping of the periphery requires the cortical hyperco-lumns, in some examples these appear to fail to provide the elementary units for the representation of the sensorium.

In this cases, by considering instead small-world neuronal networks which evolve early in development, they appear to have a multitwisted Möbius strip archi-tecture (Wright, 2014) as if reflecting the multitwisted shape of Möbius light waves, to be discussed below.

It appears that the notion of data having a shape at all scales may be worth examining; more of this below.

4.4 The Turning Inside-Out of Star Cycles and the Non-Orientable Topology of Light

Materials of most systems are born from the explosion of demising stars, following the formation of the higher atomic number atoms and their isotopes, at their core.

In particular, the atomic constituents of all biological systems arise from these explosions. Literally, we are made of the stuff in the Interior of demising stars that explode. Until recently, it was believed that demising stars undergoing gravitational

collapse, explode forming a spherical shock wave front.

Cassiopeia A (Cas A) is a supernova remnant in the constellation Cassiopeia and the brightest extrasolar radio source in the sky.

The original star, about 15 to 20 times more massive than our sun, “died” in a cataclysmic supernova explosion relatively recently in our own Milky Way galaxy.

A new x-ray study of the remains of Cas A reveals that the higher atomic number elements, rather than being found at the centre of a spherical shock wave produced by the supernova explosion, they have literally turned Inside-Out, being present at the Exterior while the lighter elements appear to be closer to the centre (Hwang and Laming, 2012).

Yet, this turning Inside-Out of the demising star as it collapsed and was recreated as a new star through the supernova explosion, is compatible with the Klein Bottle topology of the Mendeleev table of elements.

The overall structure of atoms and their stable isotopes, is such that the higher order elements appear to be layed on a local side of a Klein Bottle, as identified with the lower order elements lying on the other local side of this surface.

In fact, this topology is related to the Aufbau rule for orbitals, in terms of which the orbitals of the higher atomic numbers turn Inside-Out; due to the high pressure at the core of the demising star (Boeyens 2005, 2010).

Thus, the turning Inside-Out of a supernova remnant which initiate a novel cycle of life of a star, appears to be related to the non-orientable topology of the overall structure of atoms and their stable isotopes, and still of their orbitals, which can be traced to the shapes on neutrons in the ultrahigh pressure states of supernova and neutron stars.

Mutatis mutandis, replacing codons for atoms and their isotopes, the same principle appears to be the case of genomes, as we shall discuss in the present work.

Yet, this raises the issue of a fractal logophysics, operating at all scales and at all levels of organization (physical, chemical, biological, cognitive, semiotic,

http://en.wikipedia.org/wiki/Supernova_remnant

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etc.), that is essentially non-dual and non-linear, and which sustains a kind of memory all along the evolution of the structures and processes of the cosmos.

Most remarkable for its basic role in physics and the enactment of phenol-menology, is that light waves can literally be turned Inside-Out and Outside-In, alike Hydra.

The phase of light waves with a cir-cular or elliptic polarization may develop singularities; such a light wave presents an helicoidal shape as it rotates on its pola-rization plane.

These are the so-called optical vorti-ces; very much alike the hypercolumnar vortices as the basis for the Klein Bottle shape of the sensorium, the principle of development of non-orientable singulari-ties appears to be the case for both phenomenae, as is also the case of sin-gularities of liquid crystals (Bouligand, 1978, 1999).

We further note that the mathematical background and its physics appears to be the same for all three cases.

Indeed, through the superposition of light waves, vortical light waves may deve-lop a non-planar three-dimensional dyna-mics which is no longer a rotation on the polarization plane as its wave front advan-ces.

These singularities may further produ-ce Möbius strips for the shape of light waves, with any number of odd twists! (Freund, 2010). Recently, using liquid crystals, Möbius light waves have been produced (Bauer, 2015).

This can in principle be extended to sound waves (Ruane, 2015).

The importance of this is that DNA strands, which physically speaking are li-quid crystals, have been shown to emit and absorb both light and sound waves (Gariaev, 2001).

Furthermore, this non-orientable nature of light waves shows a most remar-kable complementarity of form and function of the visual system, that com-prises both the architecture of the body and of the physical stimuli.

Indeed being the case that the eyes in mathematical terms turn Inside-Out the images of the world on the retina (Rapoport, 2013), for a start, the light

waves have an homologous behaviour which is further carried out to the Klein Bottle turning Inside-Out and Outside-Inside topology of the cortical retinotopic map (Schwartz, 1971a, 1971b; Swindale, 1996; Tanaka, 1995, 1997).

Thus, the complementarity of physio-logy and structure of the visual system goes down to the light waves themselves, and as already discussed, we may suggest that it comprises the level of the small-world architecture of neural networks.

We shall later propose that this complementarity of form and function is also the case of the genome as a coherent system operating through quantum holography (Gariaev, 1994, Berezin, 1996).

In the following Part III we shall introduce the Klein Bottle logophysics in the context of genomics, to later discuss its bearing to chemistry and holography, as the physical setup of intragenome commu-nication, to further introduce a theory of evolution related to it.

Conflicts of Interest The author declares no conflict of

interest.

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Notes (we continue with the numbering of

Part I) 15. A simpler and literally hands-on experience

with palindromes is to take both hand and

superpose them, either palmwise or the opposite.

The latter corresponds to the Möbius strip

palindrome as in the previous Crab Canon

representation, while the former is a collapsed

torus-like compactification of the palindrome.Thus

linear palindromes can be topologically represented

either as a collapsed two-torus or a Möbius strip,

the former not being the topology of metazoans

unlike the latter upgraded in its self-referentiality

Klein Bottle form. It is the latter case which we

shall consider in regards to the DH and non-

orientable topologies of genomics. Yet, as

discussed in (Rapoport, 2013), both the Mobius

strip and the Klein Bottle are embodied in the 2-

torus, which is their double-covering, in

mathematical terms.

16. These authors posit an implicit first-order

cybernetics, the controller of the system being

detached of the system, rather than participating in

it. The latter is the case of second-order cybernetics

whose metaform is the Klein Bottle (Rapoport

2011a, 2011b, 2011c). They further chose as the

metaform of this cybernetics the yo-yo, as a double

funnel with an entrance at one end, and an output

on the opposite emerging end, which has no self-

reentrance. Thus, linear causality is superposed

with non self-reentrant vortical dynamics. Also,

according to Morin, “the nature of nature” is that of

self-reentrant vortices, rather than linear causality

(Morin,1992). In Morin’s epistemology, complexity

is associated to these structures, but orientability is

assumed by default.We shall later see, upon

discussing the topology of liquid crystals, genomics

and proteomics, that complexity –as folding, is

keenly associated to non-orientability.

17. Yet, this contextuality does not embody the a-

priori Boolean character of a context assumed in

(Primas, 1981), by “…dropping incompatible

features that a context precisely specifies our

deliberate lack of interest” (Primas, 1981), but a

non-dual ontology. Yet, though the outcome of

measurement puts up the Boolean logic,

measurements are carried upon contextuality as a

plurality of conditions, and so is the case of the

very notion of probability (Khrennikov, 2010).

Rephrasing Primas’ contention, once we assume the

ontology given by the classical logic of Aristotle

and Boole, even a context may be impoverished –

erasing the incompabilities which Boolean logic

cannot embody- to yield an understanding which is

purported to be curtailed from the very outset: As

Primas non-naively choose to select a definition for

a ‘context’, one which is exhausted by Boolean

logic (based upon an a-apriori intention to do so).

This was indicated by a critically minded scientist-

philosopher, who understood that quantum

mechanics cannot be set in terms of this ontology,

despite the Boolean character of measurement. In

short: we deliberately delete all exceptions

incompatible with a-prioris, and determine this to

be a context, which is exhaustively described by

classical logic, a closed context, indeed. At best, we

ponder that incompatibilities -which already

transpire in quantum systems- might also be the

case of biological systems; for these systems, due to

their wholeness, the dual logophysics, i.e. Boolean

logophysics, is certainly not the case, as Primas

himself noticed (Primas, 1981).Yet, as already

argued this is not exclusively the case of biological

systems, but more generally is the case of re-entrant

processes of any kind, particularly climatological -

say, Milankovitch cycles (Rapoport, 2013),

cosmological cycles, etc.

18. Remarkably, the X-shape in photograph 51

together with the Chargaff law were the elements

that led to the setting of the DH; see page 6 (Suzuki

and Griffiths, 1976). Yet, not only the X-cross

form is that of the Klein Bottle as a metapattern,

but also the Chargaff law will appear in the sequel

to be derived from the Klein Bottle topology as the

single element of a recursive algorithm to generate

a genome.

19. In this regard the eversion of the sphere as an

ideally elastic surface is carried without breaking

its continuity, actually smoothly. On the other hand,

the eversion of Volvox requires as in gastrulation

the production of a discontinuity.So, in the actual

material world, it appears that singularities, actually

contextual ones in distinction to the purported “big

bang”, are required for this imaginal transformation

to be materially eventuated. See note no. 27; see

also (Isaeva, 2014).



20. In account of this and the fact that additional

parameters on V1 for the visual mode, and on S1

for the somatosensory and haptic modes are

reduced to only two dimensions (as the Klein

Bottle is a 2d surface), then the (human) phenol-

menological dimension of space is two. Yet, most

remarkably, this identification of a phenome-

nological dimension is independent of the field of

the real numbers over which is experienced (say a

2d space over the complex numbers is four

dimensional over the real numbers); this appears to

be left open. This certainly does not coincide with

the higher-dimensional space advocated by

physics,for which space is still considered to be an

absolute given, whether the geometry is Euclidean,

Minkowski or provided with a more general non-

Riemannian metric. As for time, it is associated to

the dynamical depth value which puts forward

multistable perception as a gestalt, which enacts the

non-dual logophysics and shows the invalidity of

the principle of non-contradiction, in the phenome

nological domain (Rapoport, 2011a; Rosen 2008).

We recall the importance of these non-orientable

topologies with regards to biocomputation for the

visual mode (Schwartz, 1993). Yet, most remar-

kable is that the higher than two dimensions, and

particularly the higher dimensions of string theory

which are deemed to be unreifiable appear most

naturally in Matrix Logic (Stern, 2001) which is

based on the Klein Bottle and its 2d character.

Stern’s extraordinary work is -intentionally in some

cases- ignored even by logicians working in multi-

state logics, and cybernetists alike. Dimension two

appears to be singled out in the phenomenology of

dimensional reduction that follows from both

empirical findings and the modellization with

neural networks using self-organizing maps, as in

the theory of perceptrons (Ritter,1992). For

example, olfaction appears to indicate a highly

dimensional phenomenology, not less than 32

dimensions. However, the application of these maps

allows a reduction to two dimensions, while

keeping invariant the topology of the olfactory

network (Mamlouk & Martinetz, 2004). Other

models have also shown that a reduction of the

multidimensional olfactory sensory space to

dimension 2, one of them related to eigenvalues of

molecules’ connectivity matrix, while the other is

correlated with measures of molecules’ polarity,

appears to represent accurately the relevant

dimensions of olfaction (Koulakov, 2009).

Applying the same modellization yet for the visual

mode departing from the Gabor functions of

holography theory, it turns out that a dimensional

reduction is achieved, and the visual stimuli space

has the topology of the Klein Bottle, consistently

with empirical findings (Swindale and Bauer,

1998; Tanaka, 1990). We already discussed the

same dimensional reduction to 2d for the Klein

Bottle of the somatosensory mode. As very briefly

indicated by von Foerster, the topographic maps of

the sensorium should play the major role in our

body computations (von Foerster, 2003). He further

indicated the genome as the main biocomputational

operator. As a late elaboration of his discovery of

the Hadamard matrices (which represent the Klein

Bottle), representation of the 64 codon/anticodon

genomic matrix which we shall derive from the

Klein Bottle logic further below, Petoukhov

suggested that the algebra associated to them with

their underlying system of signals, provide a

system of harmonics as the in-formational agency

for biocomputation, and still for the metric

description of morphomechanics (Petoukhov,

2016). As we shall see below, the genome as

generated by the Klein Bottle Logic does indeed

produce a system of harmonics, which several

genomes appear to embody (Perez, 2009, 2013,

2015).

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