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Part I: The biology and the replication globes

Figure 1: “Pollency”: The ‘replicating power’ of biological entities and the ‘Chomsky hierarchy’
Pollency or reproductive power

I.1.1 “It has become almost a cliché to remark that nobody boasts of ignorance of literature, but it is socially acceptable to boast ignorance of science and proudly claim incompetence in mathematics”. Richard Dawkins.

This paper uses the DNA genetic code with its nucleotide codon syntax and semantics in combination with a four-dimensional hypersphere and the well-known invertible topological properties of a Möbius strip to thoroughly refute contemporary biology’s general consensus that universal laws and deductive logic are inapplicable, if not impossible, because biological phenomena are “too complex”. We instead show that topology’s “classification theorem” allows us to place all biological structures, entities, and populations into four equivalence classes based on the sphere, the torus, and the real projective plane. We thereby provide the four laws of biology, the four maxims of ecology, and the three constraints to which all biological entities and populations are subject.

I.1.2 Our model begins by perusing the DNA genetic code … but is not restricted to that bioinformatics approach. As in Figure 1, it reviews all biological-ecological events, assigning them to at least one of two distinct globes of bioactivity.

We provide one “biology globe”. This causes all the more generally biological behaviours. They are the “recurvature” interactions for a generation. We also provide a more specifically “replication globe”. It is responsible for all more explicitly reproductive recurvatures.

Each of our two globes or spheres of activity contributes to an overall biological generation by recurving events about its surface and around its interior and central point. Since each such globe has the potential to create a “circulation” about itself, then these various elements, taken together, allow us to declare that biological populations are “infinite cyclic groups”, with infinite capabilities. They come complete with their equally infinite cyclic subgroups. They can each generate infinitely many such distinct groupings and subgroupings, each over its potentially infinitely many generations. These are the manifold species.

I.1.3 Our four dimensions arise because our biology and our replication globes allow for four movements. Three are similar to the ordinary three dimensions of space. They cohere around the central point in each. The fourth is their movement towards and away from each other.

Just as a developing hurricane builds a “wind wall” all about itself, so also are all biological macromolecules, entities, and populations similar kinds of wind walls. They result from the recurvature movements about our two biology and replication globes. The recurvatures are then expressed in terms of their locations about each globe as latitudes; as longitudes; and as heights above each; as well as in terms of their overall circulation lengths and rates as regards those two globes moving towards and away from each other.

I.1.4 Our model imports the Chomsky(–Schützenberger) hierarchy’s power to conceptualize four main groups of biological actions. The infinite cyclic groups, subgroups, and wind wall recurvatures create the four main categories of biological artefacts:

• some are biological but fail to replicate;

• others are both biological and replicative;

• yet others not only replicate, but add a host of ancillary but non-replicative biological activities;

• a final group—including Homo sapiens—are entirely replicative, but give the impression of also being biological.

Our model shows that the four groupings listed above are inevitable results of DNA’s bioinformatic and genetic syntaxes and semantics. The biological Chomsky grammar our model establishes states the recurvature rules of selected sets of chemical components—moving as wind walls—about the two globes.

I.1.5 Our Chomsky grammar and its resulting hierarchy provide the backbone for the four-dimensional and topological framework in Figure 1. The four rows assign biological recurvatures to the two globes in accordance with their codings. Those produce the accelerative and decelerative behaviours that create the infinite cyclic groups and subgroups in terms of their distances about the globes for the four types.

I.1.6 Our model’s Chomsky grammar reflects the underlying DNA genetic language. A part of its power is the realization that it is always possible to find some cylinder of maximum volume, V, that can replace any sphere. The cylinder’s more rectilinear surfaces, S, give it the potential to turn and to extend infinitely outwards at each point. It can then roll as far in any direction as any sphere.

But if a generation is to complete, then there must always be recurvatures about the globes. Every biological interaction that emerges through some surface, S, originates in some volume, V, that also guarantees those continuing recurvatures. We can therefore describe a grammar for those V behaviours through those S presenting surfaces.

I.1.7 Our model thoroughly exploits the realization that a Chomsky grammar is both abstract and versatile. The latter’s dominion can be applied to any field desired. It can for example explain an urban sprawl. The relationship between our globes, our surfaces, and our volumes is then similar to that between a language, a grammar, its syntax, and its semantics.

Given that a Chomsky-style grammar describes an entire living language, L, it is the surprisingly concise:

[Σ, S, δ, α0, F],

where Σ is a finite input alphabet of discrete symbols that can represent any object desired; S is the set of possible alphabet combinations; δ is some “transition function” or set of “production rules” over the Cartesian product Σ × S; α0 is an initial state; and F is the set of permissible terminators in that language.

By convention, the intermediary nonterminals, constructors, and indicators that build a language are represented by upper case letters such as S and F; its completed sentences—or more strictly, its “terminals”—are represented by lower case letters; while Greek letters represent strings of both. So if it is our present desire to compile an English dictionary, then our input alphabet will (at its simplest) be the standard English one of ‘a’, ‘b’, ‘c’. A set of production rules similar to A a will then produce all possible English words for its terminals. But we can use the same structure and step up a hierarchy to let our input alphabet instead be the entire English lexicon. Our nonterminals will then be phrase structures. These can combine through a set of production or grammatical rules to create all possible well-formed English sentences. And with appropriate structures and rules, we can describe a town.

I.1.8 Biological processes and language are both highly complex. No given—and necessarily simple—example can convey the sophistication possible to a Chomsky grammar. But we can nevertheless consider a language, L, in which:

• our finite alphabet has only two symbols, and so that our input alphabet Σ = {a, b};

• our sole initiating nonterminal, α0, is A;

• we have a Rule 1 production or transformation of A ba;

• and a Rule 2 of A aAb.

We can now take up some A as an α0 to begin a production. We can next pick either of the two rules. If we pick Rule 1, we get the trivial production A ba. Since this is all terminals, we are done.

We can begin again. This time, our initiating α0 production is Rule 2. We now have → aAb. Since this is a mixture of terminals and nonterminals, we invoke Rule 2 again, to substitute for that A. We get aAb aaAbb. Invoking Rule 2 yet again gives aaaAbbb. And if we now pick Rule 1, we terminate with aaababbb.

This rudimentary grammar over L, with its only two rules and its two alphabet symbols, allows us to create all strings, of arbitrary length, of the general form αbaβ where α and β are equal-length strings of as and bs respectively. Its significance and meaning depend upon the field over which this is a language. If, for example, it is an architecture, then we are describing the different kinds of neighbourhoods in an urban sprawl.

I.1.9 Chomsky productions and grammatical rules are considerably more sophisticated but—similarly to strings of codons—they all produce strings of terminals and nonterminals. Their rules therefore fall into the four broad and well-known types of 3, 2, 1, and 0 that are the four grammars in the Chomsky hierarchy. Our model then assigns DNA’s biological–ecological productions to one or another of the four rows in Figure 1. They represent the transformation rules that distinguish between the different fertility spaces, fertility groupings, and fertility recurvatures that DNA uses to replicate; and also between the biological spaces, groupings, and recurvatures that the replicated entities then use to interact with the surroundings.

I.1.10 In the language of information science, DNA’s Chomsky style productions are physically and quantitatively measurable upon our globes. Both sets of the biological and the fertility materials are countable. They are each equipollent with 0, the set of countably infinite natural numbers (Weisstein 2015a).

I.1.11 The four groupings on the left in Figure 1 are the set of non-replication-based recurvatures about our biology globe. They are oriented “trivial cycles”. They bound definite regions on the globe’s surface, so creating the wind walls that surround their centres of activity.

The four groupings on the right differ by being reversible journeys about a Möbius strip. They do not create wind walls. They instead impose curvatures and accelerations. They create the recurvatures about our replication globe.

I.1.12 The genetic code requires both a transmitter and a receiver. There must be both a wind wall and a velocity upon one or both globes.

I.1.13 The successful transmission of hereditary information requires both (a) a syntax, and (b) a semantics. This is the propagation of discrete materials, and their circulating wind wall, about both of our globes. The receiver must therefore possess (a) a cipher to decode the necessary information, and (b) the structures that permit it to carry out whatever task is communicated … which is in this case to produce viable biological entities by replicating the winds and wind wall that begot it. They must therefore travel successfully about our two globes. We refer to the Chomsky production rules that achieve this recurvature, and on the right of Figure 1, as a “recursive function”, δ.

I.1.14 As again in Figure 1, the recursive functions in our model have their complementarities of “reversions” and “derivations”. They build our recurvatures about the globes, and from their initial symbols to their terminal ones for a generation. Every replication phenomenon derives from a recursive function journey about a Möbius strip. It is some hyperspherical “biovolume”, V, that can independently travel about our globes.

But that journey is not just a recurvature about the globes. It is also some associated biological presentation. That surface presentation is the “biosurface”, S. Its enclosed biovolumes as codons use their production rules to construct biological effects. The resulting biosurfaces, also codons, are their observed biological presentations. Since those surface presentations are, necessarily, terminals, we refer to them as “loops”.

I.1.15 Our model builds on the fact that DNA, acting as a genetic code, stores information in its codons of three nucleotides each. Genes are the functional segments, having four possible bases. The three nucleotides therefore give 43 = 64 different possibilities. Their combinations specify the 20 different amino acids used by all living organisms.

I.1.16 The use of a formal code, of this type, to accomplish such a purpose again requires the receiver of the code to understand the syntax and its rules, and to accord the correct meaning in order to accomplish either, or both, of the biological and/or fertility tasks implied by those symbols. The biovolumes, V, that DNA generates thus contain biological energies per some unit volume. As recursive functions, they are productions that are then incident upon their presenting biosurfaces as terminals. Their successful transfer thus begets some momentum per relevant unit area, expressed as a timed loop.

Figure 2: A biological generation
Figutr 2: A biological generation

I.1.17 It is unfortunately impossible to represent the four dimensions DNA uses to construct its biological hypersphere in the only two available to us on a piece of paper. Nevertheless, and as in Figure 2, our model will represent a biological generation as two globes. One can initially be regarded as nesting inside the other as they roll. The outer biology globe has the behaviours that supervise all biological interactions and recurvatures; while the inner replication one exclusively handles replications, originating all recursive functions.

The outer biology globe interacts directly with the surroundings to construct our DNA nucleotide codons using three dimensions equivalent to the left-right, front-back, and up-down of ordinary physical space. We can initially think of the fourth dimension—most generally known as ‘upsilon’–‘delta’—as a combination of a horizontal rolling over some absolute time period, T, and an internal pulsing, during that rolling, to and from each other of the inner replication and outer biology globes. They do so over some distance, τ. That combination of τ–T is a joint transition (a) to and from the replication point; and (b) forwards in time from the beginning to the end of a generation.

Our model gets its power by using topology’s classification theorem for compact surfaces to examine the conjoined surface these two globes create. All biological structures then fall into very specific equivalence classes based on the resulting normal forms they establish.

Our horizontal hyperglobe translation of time T, combined with our waxing and waning of distance τ, produce the complete set of DNA and biological-ecological interactions, λ, for a recurvature and a generation. Our hyperglobe achieves this by drawing in the needed resources to build its Möbius strip based Vn molecular reversions. Some of those are allocated to the biology globe, others to the replication one, and yet others to both.

All such “lifts” up into our linked globes, from the surroundings, are uniquely determined by their initiating values at any point. The productions are therefore some hyperspherical Vn. This is only sometimes replicative; but is always intrinsically biological. The constructed materials now extrude and loop rectilinearly outwards, from the globes, as derivations. They thus produce a set of observed Sn-1 biosurfaces and events. Biology therefore consists of this Sn-1Vn or else SnVn+1 coupling, with a replication point, ρ, acting as a pole.

I.1.18 The cells at bottom right in Figure 1 are V1 Möbius strip style biovolumes relative to the S0 molecules on their left that “tangle” to create their base pairs … with those same molecules manifesting the relevant biological behaviours in the surroundings as their presenting biosurfaces. But the manifest molecules have a parallel upwards recursion and complexity. They seem able to come together to build the observable S1 genomes above them. But those apparent productions are merely the biosurfaces belonging to the parallel reversion, on the right, of cells into the V2 biovolumes that are the enshrining entities for those same genomes. Those are built by our recursive functions.

The DNA codon grammar in our parallel structure is that only those biological constructs that pierce the inner replication globe and return, to interact with the surroundings, have the appropriate syntax and semantics. Only they are successfully fertile and replicative.

I.1.19 Our grammar stipulates that the period between the beginning of a generation and the pole or replication point is a “fibration”, θ; with that between the replication point and the end of the generation being the “cofibration”, ρ. The fibration takes in resources and energy from the surroundings; the cofibration then returns them to complete the generation.

Figure 3: Reproductive and pollency considerations
Reproductive and pollency considerations

I.1.20 As in Figure 3, the language that DNA speaks is that reproduction is a “mapping” of the form φ:X Y. DNA maps from a first group of elements in a “progenitor domain”, X. Those transform to a second set of elements in a “progeny codomain”, Y. All such φ:X Y and/or φ:Y X mappings are our recursive Chomsky production rules, δ.

Our mappings will generally occur via a “mapping cylinder”, Mλ. That mapping cylinder must then impose sufficient accelerations and velocities on both (a) the groups; and (b) the surroundings to create the needed recurvatures.

I.1.21 We also define the progenitor domain, X, as the “preimage”; and the progeny codomain, Y, as the “image”. A “homomorphism”—from the Greek homos for ‘same’, and morph for ‘form’—is then a structural mapping between the groups contained in these different globes and domains.

A biological homomorphism preserves a group’s essential characteristics. It does so throughout all the group’s possible translations and transformations, and quite irrespective of the biological and/or replicative spaces that group transitions through.

I.1.22 Our model very carefully separates structures from spaces. Since we have our two globes, then we can carefully analyse the realization that not all entities in any one group are obliged to recurve about our two globes at the same velocities and accelerations as they transition across a generation, and so from preimage to image and/or conversely. Entities can therefore map between domain and/or codomain without necessarily also keeping the spaces they each traverse—and so their recurvature values—invariant. They do not all recurve about our two centres in the same ways.

A “homeomorphism”, from the Greek homoios for ‘similar’, is a successful mapping between spaces, rather than between groups of elements. A biological homeomorphism therefore preserves the biological space’s characteristics, irrespective of the groups and/or transformations passing through it.

We shall only regard groups of entities as identical if they preserve both (a) their homomorphism, and so group structure; and (b) their homeomorphism, and so space-behaving characteristics. Any populations that successfully match a generation’s worth of biological activities with a generation’s worth of replicative ones have clearly preserved an appropriate structure across all spaces, and will then be both homomorphic and homeomorphic. This is “isomorphic” where iso comes from the Greek for “identical”.

I.1.23 The significance of so distinguishing between iso-, homo-, and homeomorphism is that, as in Wernicke’s aphasia, DNA syntax transmission over a generation is possible without semantic comprehension by the receiver; while as in Broca’s aphasia, it is possible to transmit the semantic comprehension capability that would make a generation possible, but then to fail to effect it by transmitting no—or the wrong—syntax for the decoding.

I.1.24 DNA’s interactive, complementary, and parallel language and grammar of homo- and homeomorphisms between (a) preimage and image, and (b) biology and replication globes, produces Figure 1’s four horizontal groupings. They are a relationship between our globes. Groups of entities use fibration and cofibration to transition back and forth between progenitor domain and progeny codomain, and so between syntax and semantics.

I.1.25 Each of our four topological style groupings—based as they are on the sphere, the torus, and the real projective plane—is characterized by its “pollency” or “fertility power”, taken from the Latin pollre ‘to be strong’. This reflects the relationship between the syntax and semantics that produce the different biology and fertility behaviours across our two globes. Each possible combination—being homomorphic and/or homeomorphic—is now “injective”, injective, and/or “surjective”, surjective, i.e. “one-to-one” and/or “onto”. The four possible groupings of biological artefacts that our model establishes across all biological Kingdoms are:

  • Nonpollent: non-injective and non-surjective. Replicatively open; biologically open. These are biological but not replicative. The replicative pervade the biological, but not conversely. These artefacts, entities, and/or populations correlate with Chomsky’s Type 3 or “regular grammars” and so with “finite state automata”. Linguistically, each of their distinct production rules can have only a single nonterminal, X, on the left, and may not have more than one terminal, x, on the right, as in X x. They may optionally have a nonterminal on the right, as in X Yx or X xY. These are Figure 3c. They are neither homomorphic nor homeomorphic. They neither create nor preserve their biological spaces or structures. They follow Figure 3a, but not 3b. They are replicatively open because all these entities, being only biological, must look elsewhere to be replicated. Their generation cycle is thus incomplete. They do not return to the replication globe. They are “intransitive”. Their biological activities are born from both the progenitor domain and the inner replication globe, via the intermediary mapping cylinder, then transitioning into both the progeny codomain and the outer biology globe. They always appear, in the surroundings, as if direct products from some preimage. They can therefore be analysed as biosurfaces, S, relative to some set of biovolumes, V. But they do not decelerate to undertake the reverse transition from progeny codomain back to progenitor codomain … and so do not replicate. Since no reverse process exists, they do not cross the replication point to reenter the replication globe, and again cannot replicate either their spaces or their groups. They are therefore non-injective, displaying the first of our two methods for falling in this category. And further since they do not return to the replication globe, they can only understand what they are told, by the biology globe, to that limited and nonreplicative extent. So irrespective of how many different production rules might be applied; and irrespective of any apparent biological complexity in their processes; they can form only the trivial biological cycles that restrict them to this biology globe. They have reversions (or “parents”), but no derivations (or “daughters”). They are produced of some Möbius strip, using the mapping cylinder, but do not themselves traverse one. And since these are only trivial cycles that do not traverse a Möbius strip, then they are biologically open. We designate them ‘many from none’ and/or ‘any to none’ for they are neither one-to-one nor onto. They are non-surjective because there are always more replicative elements in the progenitor domain as can produce them than there are in the progeny codomain for there are zero replicative elements in the latter … which is the first of our two methods for being non-surjective. Therefore: not all elements in any preimage or progenitor domain need map to ones in the image or progeny codomain; while not all elements in any image or progeny codomain are fully mapped to by elements in their partnering preimage or progenitor domain.
  • Unipollent: injective and surjective. Replicatively closed; biologically closed. The biological fully pervade the replicative; while, reciprocally, the replicative fully pervade the biological. These create and preserve both their biological and their replicative spaces. They correlate with Chomsky’s Type 2 or “context free grammars” and with “pushdown automata”. Linguistically, they may have arbitrarily many distinct linguistic production rules again of great complexity, but they can only ever have a single nonterminal symbol on the left side. They may, however, have any number of terminals and/or nonterminals upon the right. They thus have more possibilities than the nonpollent, but do have the limitation that irrespective of context, they behave the same way on every occasion. Like the two groups to follow them, these can only exist because they emulate Figure 3b by using the mapping cylinder to recurve, and so to successfully reenter the replication globe. Since they cross the replication point, they are transitive. In biological terms, however, the two globes do no more than pervade and sustain each other. The entities can therefore create a Möbius strip. That intermediary mapping cylinder imposes a recurvature for all necessary invertible, accelerative–decelerative and reproductive actions. They thus appear, in the surroundings, as direct injections first from preimage to image; and then from image back to preimage to complete the generation. But although these unipollent entities, of Figure 3d, can therefore replicate, they are both biologically and replicatively closed. They successfully traverse a Möbius strip to complete a cycle, but are more limited than the two groups to follow, for they support no additional biologically trivial cycles. They support no nonreplicative and/or extra-cellular materials in any domain or globe. They understand exactly what they are told by both the replication and biology globes; but also do not understand any more than they are told by either, the two again doing no more than pervade each other. Their Möbius strip operations are therefore somewhat restricted. All elements in the progenitor domain have exactly one image in the progeny codomain and so map to exactly one element; and all elements in the progeny codomain also have exactly one preimage in the progenitor domain, and so are mapped to by exactly one replicating element. These unipollent entities are thus both one to one and onto, which is a ‘one-to-one correspondence’. They recurve, but again permit no additional trivial cycles; and so no additional biological phenomena, in either of the biology or replication globes. They are both homomorphic and homeomorphic. They are “bijective”, which is both surjective and injective.
  • Pluripollent: injective and non-surjective. Replicatively closed; biologically open. These are again replicative, but also add additional biological activities. They can create all biological spaces required, but do not explicitly preserve them. They correlate, linguistically, with Chomsky’s Type 1 or “context sensitive grammars”, and so with “linear bound automata”. Their linguistic production rules allow for more than one nonterminal on the left side of any production rule … with the added proviso that any nonterminal production, X, may be surrounded by specified terminals and/or nonterminals as in αXβ; and so that when so surrounded, X may on those specific occasions be replaced by some third string to create αγβ. This nonterminal’s behaviour now depends entirely upon the context in which it is found. These materials, seen in Figure 3e, recurve; and so can again all be reproduced. They are injective. However, their reproductive elements can support additional trivial cycles in the biology globe. The entities can therefore produce a variety of additional, and biologically viable—but nonreplicative—products by invoking operations—as those trivial cycles—beyond each of their replicative ones. But since those added cycles and activities do not themselves replicate or traverse a Möbius strip, they are biologically open. Those additional trivial cycles emulate nonpollent behaviours by not being able to directly replicate themselves. These entities are therefore ‘many from one’ and/or ‘one to many’ (Latin plurimus, ‘very many’). They are, however, replicatively closed because the replicating elements can successfully traverse a Möbius strip. They are overall like the unipollent in being injective and circumscribing a Möbius strip; but differ by supporting those additional nonreplicative and/or extra-cellular materials. They are not homomorphic because the complete biological structures they build, with those supported trivial cycles over in the progeny codomain, are quite unlike those in the progenitor domain that produce them; and conversely. They are, however, homeomorphic because they can recreate their biological spaces in both globes. But when compared to the upcoming final group, then although the progeny can indeed use the biology globe to construct larger biological structures, and trivial cycles, from a more basic replicated unit, they are limited. They cannot reconstruct their replicative centres from yet smaller elements. They are, therefore, somewhat restricted in the replication globe capabilities, but are not restricted in the biology one. Their replicative elements, in the progenitor domain, can thus have multiple images in the progeny codomain; while all elements in the progeny codomain map to at most one element in the progenitor domain and replication globe, having exactly one preimage. They are one-to-one, but not onto, which is injective and non-surjective.
  • Totipollent: non-injective and surjective. Replicatively open; biologically closed. These can only be replicative because each replicative element can induce—but it cannot independently create—biological activity. They correlate with Chomsky’s Type 0 or “unrestricted grammars” and with “Turing machines”. They are recursively enumerable. The sole linguistic production rule is that the left side cannot be empty. These Figure 3f materials are ‘many from many’ and/or ‘many to many’ because, amongst other things, they can support trivial cycles and additional activities in the replication globe. Distinct reproductive centres must, however, link themselves to biological structures, and processes, so they can be viable and build their more complex reproductive structures. They are therefore replicatively open because zero entities in the progenitor domain can—unaided—replicate themselves. Distinct replicative elements must align themselves with others like themselves before they can create a locus that can then produce their joint progeny. They in other words construct replicating elements from smaller units. Those originating progenitor elements can nevertheless use their supporting trivial cycles, built in their progenitor domain, to create multiply sourced derivations, of great biological complexity, over in the progeny codomain. All elements and cycles in the image or progeny codomain are thus mapped to by more than one element from the preimage or progenitor domain. However, elements in the progeny codomain are also biologically closed because they must accelerate the replicative enclosures, and must force the biological structures they are embedded in to accelerate about the Möbius strip. All replicated elements in the progeny codomain can therefore map themselves back to multiple progenitor domain preimages, thereby having more than one generative element. But since elements in the progenitor domain must first ally themselves, with others, in that domain to create that extended progeny codomain, then they do not—individually—traverse a Möbius strip. They can only traverse one when biologically combined with others. Elements in the progenitor domain can thus have more than one image in the progeny codomain, mapping to multiple elements. But since none of the replicative elements are again individually responsible for any cycles created in the biology globe, then since those cycles can return, they are closed. Their ensuing trivial cycles are therefore both replicatively open and non-injective. And further since all elements in the progeny codomain must have at least one mapping from the progenitor domain, then these are onto but not one-to-one, which is non-injective and surjective.

I.1.26 We shall now prove that these various and parallel trivial cycle and Möbius strip interactions are DNA’s entire language and grammar of syntax and semantics. They invoke our two globes and our two domains to establish (a) the four maxims of ecology that in their turn establish the homeomorphic spaces that can support the wind walls that recurve about our two globes; (b) the four laws of biology that establish the homomorphic structures that can accelerate and decelerate between the biosurfaces and the biovolumes; and (c) the three constraints that link the homomorphic to the homeomorphic to create the circulations of the generations. We shall take Brassica rapa as a test case.