Nested Numeric/Geometric/Arithmetic Properties of shCherbak’s Prime Quantum 037 as a Base of (Biological) Coding/Computing

Numerous arithmetical regularities of nucleon numbers of canonical amino acids for quite different systematizations of the genetic code, which are dominantly based on decimal number 037, indicate the hidden existence of a more universal ordering principle. Mathematical analysis of number 037 reveals that it is a unique decimal number from which an infinite set of self-similar numbers can be derived with the nested numerical, geometrical, and arithmetical properties, thus enabling the nested coding and computing in the (bio)systems by geometry and resonance. The omnipresent fractal structural and dynamical organization, as well as the intertwining of quantum and classical realm in the physical and biological systems could be just the consequence of such coding and computing.

Thus, the crucial challenge biological organisms meet is the generation, transmission, reception, and storage of information with high fidelity due to continuous noise in the system, which according to the information theory requires involving the error-correcting codes.The simplest realization of error-correcting codes is achieved by a layered structure referred to as the nested codes, a special type of the concatenated error-correcting codes where the result of a previous encoding process is combined with new information and then encoded again, so that the deepest nested information is also the best protected one and does not demand very efficient individual codes, which in terms of biocodes means "...the older and more fundamental it is, the better it is protected" (Battail, 2007).Such biocodes nesting, i.e. their coexistence and overlapping, was first discovered by Trifonov (1980;1989) for the coexisting triplet code (a sequence of instructions for protein synthesis) and chromatin code (a sequence of instructions for nucleosome positioning).The logic of nesting or fractality reduces the problem to the first generative set/mapping -the fractal generator, which means, in terms of biological coding/computing, the reduction to the first biocode/biocomputation -genetic code and translation, since Woese (1965) shows in his early work that the translation process was highly developed at the bottom of the universal phylogenetic tree, even in comparison to the simpler process of transcription, while replication still did not exist at that level.Since the genetic code, as the first biocode, represents not only the origin of life, but also the link between physical and biological coding/computing, the understanding of mathematical logic of the genetic code is of special interest, and was suddenly made possible by shCherbak's revealing arithmetic inside the universal genetic code (Shcherbak's, 1994).

ShCherbak's arithmetic inside the universal genetic code
Almost two decades ago, shCherbak (1994) made astonishing discovery of the arithmetic regularities of nucleons inside the genetic code.The history of this discovery (shCherbak, 2003) began soon after the code decrypting, when it was recognized that the correlation between amino acid mass and codon distribution existed in a sense that the smaller amino acid size requires the greater number of codons for its translating and vice versa (Schutzenberger et al., 1969).This antisymmetrical correlation was confirmed by introducing an integer-valued parameter -a nucleon number (a sum of protons and neutrons in atomic nucleus) (Hasegawa and Miyata, 1980), which motivated very extensive shCherbak's researches of arithmetical regularities in the genetic code (Shcherbak, 1994;2003;2008).The initial shCherbak's key result is revealing the determination of symmetrical architecture of genetic code by decimal number 037 through arithmetical regularities of nucleon numbers for the free molecules of canonical amino acids and nucleotide bases, with remark that the number 037 is unique in decimal system in the sense that its three digit multiples remain multiples modulo 9 by cyclic permutations (Tab. 1) and that similar numbers also exist in some other numeral systems ([13] 4 , [25] 7 , [49] 13 ) (Shcherbak, 1994).The number 037 was called a Prime Quantum -PQ by shCherbak (Shcherbak, 1994;2003) or later just a Prime Number -PN (shCherbak, 2008).
ShCherbak pointed out a variety of different nucleon arithmetic regularities, including those for the free form amino acids and peptide bonded amino acids (the standard block residues and the ionized and protonated side chains); for the compressed, life-size, and split representation of genetic code; for Rumer's and Gamow's division of genetic code, and many other regularities (shCherbak, 2003;2008).As an explanation for the found arithmetical regularities based on PQ 037, shCherbak (2003) suggested that "the divisibility by PQ as a validation criterion, if any, simplifies molecular machinery and facilitates the computational procedure of hypothetical organelles working as biocomputers", since to very simple divisibility rule of 37 for base 10 by the checking divisibility for the sum of three digit block of number, which "…requires only the three digit register".Generally, this divisibility "…criterion is valid for the is applied and the n-digit reading frame is used" (shCherbak, 2003).
ShCherbak's results motivated other researchers to further reveal arithmetical regularities and their deeper mathematical and physical principles (Verkhovod 1994;Downes and Richardson, 2002;Rakočević, 1998;2004;Négadi, 2009;2011;Mišić, 2004;2010), but the purpose of such evident correlation between the genetic coding and the quantized nucleon packing of its constituents through the 037 nucleon packing quantum has remained unclear.So, the question arises -why 037?

Self-similar numbers
A good way of understanding the properties of a number is its generalization.Let us call Self-similar Number (S) every integer which has the property of decimal number 037 in an arbitrary numeral system, i.e. the analogue multiplicative table of number 037 (Tab. 1) (Mišić, 2010).This property of S has been defined as a special case of cyclic equivariability or, more precisely, an equidistant cycling digit property2 .It can be proved that the definition of S is valid both for the condition of multipliers equidistance and the condition of digits equidistance.Definition 1A (Mišić, 2010) Self-similar number ( )  p q = S S of a given numeral system, radix ∈ N \ {1} q , is the smallest nontrivial p-digit number, ∈N \ {1} p , whose successive cyclic permutations are equal to its own equidistant multiples, except for the permutation which results in S.
, with cyclic digit property whose digits are equidistant.
The trivial forms of numbers with equidistant cycling digit property are represented by the repdigits, q p aa aa , where ∈ − {0,1,2, , 1} a q is q-nary digit.
Both definitions indicate that general solution, i.e. the solution for each p, exists only for right-shift cyclic permutations, while in the case of left-shift it does not exist, i.e. the solution exists only in the case of doublets (biplets) when it is equalized with that of the right-shift.This general solution for S is determined by the following equation (Mišić, 2010):

S
(1) where − is a q-nary digit on the ith position.
The solution could be expressed in a simple form (Mišić, 2010;cf. shCherbak, 2003): where ( ) p R q is pth repunit in the numeral system of radix q, given as 1 2 ( ) 11 11 1 Eq. ( 1) gives the necessary and sufficient condition for the existence of S in numeral system of arbitrary radix q, which is minimally satisfied for = −1 p q , and thus in each numeral system there exists at least one S.
The graphic representation of Eq. (1) (Fig. 1) shows interdependence between the equidistant multiplying and equidistant digit distribution of numbers with the cycling digit property.In Fig. 1, it can be observed that each S has the analogue numbers for other radix q and multipletness p, so called S analogues ( A S ) (Mišić, 2010).Two groups of A S can be distinguished -those with the same number of digits (vertically arranged) and those with similar digits [horizontally arranged, Eq. ( 6)], which is leading to the definition of "vertical" and "horizontal" S analogues (Fig. 1).
Definition 2 Self-similar numbers ( ) p q = S S for constant p and variable q are the vertical analogues of class p or p-plets.

The successive vertical A
S have the radix difference p and the digit difference 0,1,2,..., 2, 1 p p − − , respectively.Definition 3 Self-similar numbers ( ) p q = S S for variable q and p with the constant ( 1) q p − are the horizontal analogues of order ( 1) q p − .The successive horizontal A S , for instance of order 3, enable the next transformation: The fact that some S have vertical and horizontal analogues in the same numeral system, enables the defining of the third kind of analogues (Fig. 1).
Definition 4 Self-similar numbers ( ) p q = S S for variable q and p so that = + 2 1 q p are the diagonal analogues.
Figure 1.The regular digits distribution of S in dependence of numeral system radix q and digit multiplicity p.The colored arrows denote three types of S analogues.Vertical and horizontal analogues are shown for the case of number 037, while the diagonal analogues are unique and independent of the particular case (Mišić, 2010).
The main property of S results from Eqs. ( 2) and (3), which shows that S for (prime) multiplicity p are related to (irreducible) cyclotomic polynomials and thus to the pth roots of unity in complex domain, (Mišić, 2010).This relationship of self-similar numbers with cyclotomic polynomials, which describe regular polygons, is in the correlation with their definition as the numbers which are equidistantly multiplied by regular cycling of their digits.
Cyclotomic polynomial, Eq. ( 7), can be regarded as a complementary form of generalized Golden polynomial whose largest root on the open interval (1, 2) is the generalized Golden Mean (Miles, 1960), wherefrom the relation of S to generalized Golden Means follows (cf.Tab. 6) (Mišić, 2010).
In the case of the basic form of Golden Mean, , and its golden polynomials, complementary cyclotomic polynomials are obtained, 2 3 ( ) 1, q q q Φ = + + (11) whose roots are , give the vertices of regular hexagon (this complementarity for more general case is given in Tab. 6).This is in consistence with the fact that triplets S represent the centered hexagonal numbers (Mišić, 2004;2010).

Fractal properties of decimal varieties of number 037
In the previous Section it has been shown that S has analogues in other numeral systems, so it is questionable whether it has its varieties in the same numeral system.The extension of S for the given numeral system can be done by modifying repunit in Eq. (2).
Since S is fundamentally related to equidistantness (Defs.1A and 1B), then the equidistant extension of repunit, Eq. ( 3), with preserving divisibility in Eq. ( 2), can be done by the two operations -equidistant insertion ( ) , that are respectively described by ( ) According to Eqs. ( 2), ( 13) and ( 14), it is possible to define two types of S varieties Definition 5 The nth vertical variety of S,

S S
, for the given numeral system q and digit multiplicity p is , ( ) Definition 6 The nth horizontal variety of S,

S S
, for the given numeral system q and digit multiplicity p is From Eqs. ( 13) and ( 14) it follows that V S can be extended only in the direction of increasing values of q and p ( → n q q and → × p p n), while A S can be extended in both directions ( − ← → + q p q q p and − ← → + 1 1 p p p ), which is the main difference in their notation (Tab.2).
Table 2.The notations of ( )  p q S modifications.
q -base radix, p -number of digits, m -order of analogue, n -order of variety.
The V S are defined so that the first S variety, for = 1 n , reduces to S and Eqs. ( 15) and ( 16) to Eq. ( 2).For ≥ 2 n , Eq. ( 15) reduces to which means that the initial extension of pplet S to p×n-plet in same radix q is equivalent to the scaling of p-plet S in radix n q (for instance,

S S
), and which is actually vertical A S for radix n q and thus this type of varieties is also named vertical (Tab.3A).
In contrast, the extension of p-plet S according Eq. ( 16) results in has not p-plet S counterpart for the scaled radix n q , but leads to p×n-plet in the same radix q, which is comparable to horizontal A S (variable p-plets, ↔ S ) and hence the name horizontal V S , → S .The relation between vertical and horizontal V S for the number 037 is given in Tab.3A.

S
show very interesting property of numerical scaling, we will focus on them, especially on the triplets S which are the centered hexagonal numbers, in order to examine their potential geometrical and arithmetical scaling.It can be proved that all ↓ S for the triplets ( 3 ( ), for n q n ↓ ∀ ∈N

S
) also represent the centered hexagonal numbers, which are given specifically for the vertical decimal varieties of number 037 (Tab.3B).
The erasing of digits in V S (Tab. 3)which result from inserted and concatenated positions by Eqs. ( 13) and ( 14) (normal formatted digits in V S and in the indexes) reduces expressions to the original ones (the first row in Tab. 3).T n -nth trigonal (triangular) number, c H n -nth centered hexagonal number.
Each ↓ S reflects the same kind of vertical and horizontal analogy as the original S (Defs. 2 and 3; Fig. 1), since ↓ S is reduced to S for powered radix, i.e. n q .Consequently, the vertical successive analogues of ↓ S have the radix difference p and the digit differences, for instance in the case − , respectively, while the horizontal successive analogues of ↓ S , for instance of order 3, enable the following analogue transformation to Eq. ( 6): Further scaled or nested properties of ↓ S are manifested in their multiplicative table (Tabs.4 and 5), because according to Eqs. ( 17) and (18) all these numbers satisfy and thus are related to the p×nth roots of unity, , and represent the numbers with cyclic digit property.Concretely, for

S
are both centered hexagonal numbers (Tab.3), it is interesting to examine whether their multiples also have some geometrical meaning.For the first three multiples of 3 (10) it is shown that 3 (10)   S are related to polygonal numbers (Fig. 2A-C) and according to nested principle it is also valid for

S
, which is consequently valid for 037 and its varieties.13) and ( 15)].Grey digits are the result of the inserted positions introduced by Eq. 13, and their erasing reduces all numbers to the original Tab. 1 (for bold and normal formatted numbers, respectively).Normal numbers are obtained by the right-shifting of leftmost digit and, together with the first congruence class, they consist of a set of most regular (almost perfectly equidistant) multipliers whose successive differences are 10 (grey shaded fields).700033366 733400066 766766766 800133466 833500166 866866866 900233566 933600266 966966966 The explanation of Tab. 5 is the same as in Tab. 4, except the fact that the multiplicand is 000333667 and the scaling is done by 3 [n=3 in Eqs. ( 13) and ( 15)], as well that the most equidistant multipliers successively differ by 100 (grey shaded fields).Nested properties of decimal varieties of 037 can be also deduced from the fact that 037 almost perfectly divides all its varieties, for instance: 3367=37×91, 333667=37×9018+1, 33336667=37×900991, 3333366667=37×90090991, 333333666667=37×9009018018+1, and so on (cf.Tab.6B).
The indicated arithmetical regularities of ↓ S (Tabs. 4and 5) are actually the consequence of a deeper regularity in the numeral system.Namely, it follows from Eqs. ( 2) and ( 4) that the biggest S for the given numeral system is for 1 p q = − , when a number is obtained in the form (the second raw in Fig. 1): 1 ( ) 0123 ( 3)( 1) The importance of 1 ( ) q q − S is in the fact that it represents the period of q-nary numeration of natural numbers, so called fundamental period of q-nary numeral system (Fig. 4) (Mišić, 2004).Fundamental period 1 ( ) q q − S and its length − 1 q are related to the basic arithmetical periodicity (modularity) in q-nary system, and thus to the divisibility rule which follows from modular arithmetic.Decimal numeration of natural numbers results in the periodic number with period 012345679, while its triple value results in the periodic number with period 037 (Mišić, 2004).

S
for p prime and − 1 p q represent the elementary periods of q-nary numeral system where p is their period length.
is in its resulting from the product of ( ) p q

S
and its vertical variety, so that for = − ( 1) n q p and from Eqs. ( 17), (18), and (23) it follows q p q p q p p p p q q q q q S S S S S (24) For diagonal A S (Def.4), when − = 2 1 q p , Eqs. ( 23) and ( 24) are reduced to

S
. ( 28) From Fig. 3 and Eq. ( 27), it also follows that the fundamental period of the decimal system is the product of two polygonal numbers, and thus it can be considered as a composite polygonal number (Fig. 5).
Definition 7 Composite polygonal number is a positive integer that can be entirely factorized into two or more other polygonal numbers.A particular property of composite polygonal numbers is that they have multivariate geometrical form, as in the case of composite number 259, which is 7 th multiple of 037 (Fig. 5).
The last particular property of S to be mentioned in this paper is their relation to the generalized Golden Mean, Eqs. ( 8), also valid for ↓ S according to Eq. ( 17), thus transforming Eqs. ( 8)-( 12) as it is shown in Tab. 6.These two types of complementary polynomials that differ only in the signs, give two geometrically complementary solutions.The first solution relates to the Golden Mean and the corresponding numbers 89, 109 with their scaled values, but also to the Fibonacci numbers (Tab.6A).The second solution relates to the nth roots of unity and cyclotomic values 111, 91 with their scaled values, but also to 037 and its decimal varieties (Tab.6B), and thus to the centered polygonal numbers (Fig. 3).Table 6.Comparation between Golden Mean polynomials and cyclotomic polynomials for q = 10.

S S S
and

S S S
and which means that 037 analogue precursors are the analogue successors of the previous diagonal analogue 03 5 , Eq. ( 30), and vice versa for next diagonal analogue 048D 17 , Eq. (31) (Fig. 1).Thus, Eqs. ( 30) and ( 31) enable general concatenation of diagonal analogues, which are also the only

S
with a fundamental period of q-nary system in the form of the product of ( ) p q

S
and its scaled value in the powered radix, Eq. ( 26).The uniqueness of 037 and thus the decimal system follows from Eq. ( 29) which has the general form for and shows that only 10 n -nary systems are completely determined by the centered hexagonal numbers, and thus correlated with the hexagonal lattice or equilateral triangular lattice.This lattice belongs to the Bravais lattice, an infinite regular array of points in which each lattice point has exactly the same environment in 2 R (and in 3 R ).
The condition for this regular point arrangement is ∈ 2cos( 2) N π Z, which implies that = 1,2,3,4,6 N , for which N is said to be crystallographic number and lattice Nfolded Bravais lattice.Using the cyclotomic ring of order N in the plane, i.e. by the Zmodule, it can be obtained:  2), ( 3) and ( 7), and thus really correlated with hexagonal lattice, Eqs. ( 33).Similarly, it can be proved that the doublets, 2 ( ) n q ↓ S , are correlated with square lattice and with centered square numbers, which will be better explained in the next paper.
All the analyses carried out in this Section indicate that, from the aspect of selfsimilarity, the number 037 and decimal system are unique integer and numeral systems in the whole realm of numbers and systems, and thus the number 037 has a central place in the whole set of A S (Fig. 1).

Discussion
The most general correspondence between the 037 based coding/computing and the genetic code and its evolution in Woese's sense (Woese, 2002;Vetsigian et al. 2006) can be derived from the existence of horizontal and vertical 037 analogues and its central place among them.According to Woese's dynamical theory of genetic code evolution, such process represents the first stage in cellular evolution at the root of the universal phylogenetic tree and it is the result of communal evolution of the early life by the horizontal gene transfer, so that the universal genetic code is actually a generic consequence of such process and the precondition for the later individual evolution by the vertical gene transfer.
Similarly, the horizontal 037 analogues are geometrically very distinctive, while the vertical are self-similar.This mathematical logic is also comparable to Barbieri's mechanism of macroevolution (Barbieri, 1998;2008) based on two distinct processes, coding and copying, the first of which creates absolute novelties and involves a collective set of rules (for instance, translation), while the second creates relative novelties and operates on individual molecules (transcription).Because of the need for simultaneous action of both mechanisms, the existence of a universal mechanism which can act in both directions is demanded, and that is the characteristic of 037 based coding/computing by its diagonal analogues.
The reflection of 037 based nested coding/computing must be embodied in other biocodes, like the genomic code.It means that the genomic code must be generally established on the sequence length of 1000 nucleotide as the basic computing frame of number 037 (Mišić, 2010), consistent with the detrended fluctuation analysis that "…clearly supports the difference between coding and noncoding sequences, showing that the coding sequences are less correlated than noncoding sequences for the length scales less than 1000, which is close to characteristic patch size in the coding regions" (Havlin et al., 1995).Moreover, the frequencies of the 64 codons in the whole human genome scale are a self-similar fractal expansion of the universal genetic code and strongly linked to the Golden Mean, indicating that the universal genetic code table predetermines global codon proportions and populations, thus governing both micro and macro behavior of the genome (Perez, 2010), which is also the reflection of 037 based nested coding/computing, since we have shown the complementarity of the number 037 with the Golden Mean (Tab.6).Rakočević (1998) pointed out that the universal genetic code table is in itself determined by Golden Mean.
The next correspondence with 037 based nested coding/computing will be examined on the nucleic level, since shCherbak (1994)  , where 125 = 3 5 is 5 th cubic number), the same pattern which appears in Eq. ( 42).Although the nucleon sums of canonical DNA base pairs and RNA base U are the multiples of 037, they can be also expressed in the form of the composite polygonal numbers (Fig. 5 and 2C) and S arithmetic, (5) ( ).
In spite of the "imperfect" divisibility of nucleon number for C G ≡ pair, it can be a perfect computational feature due to the modular arithmetic, since which is the universal periodical pattern (GCU) n in mRNA and appears to be a fossil of a very ancient organization of codons (Trifonov and Bettecken, 1997), and the reason for that can be the fact that the repetitive sequence of this triplet also enables counting.
The last correspondence is on the level of shCherbak's arithmetic inside the genetic code.Starting from the first shCherbak's result of the arithmetical regularities of the genetic code compressed representation for division according the amino acids degeneracy (Shcherbak, 1994) [also Fig. 9 in (shCherbak, 2008)], the relation of nucleon numbers for blocks+chains=whole molecules can be expressed in the form of figurate numbers and/or S arithmetic for the four-codon amino acids as and where 4 n Py is the nth square pyramid [037 is also the number of points in a square lattice covered by a disc centered at (0,0) in the form of an octagon (Sloane and Teo, 1984)].
According to the Gamow's division of genetic code [Fig. 7 in: (shCherbak, 2008)], the sum of the side chains for the one half of the set is 3 4 packing quantum which describes a collective mass of particles and they are also in correlation with the regular geometrical arrangement and space measurement); 5) the involving of periodic, crystal-like lattice dipole structure with long range order (the doublet and triplet S represent the packing quantums for square and hexagonal lattice).Penrose and Hameroff (2003) proposed cytoskeletal microtubules as a biological structure particularly suitable for quantum computation and for whose coherence sustaining, among others, an important role belongs to the coherently ordered water trough the dynamical coupling to the protein surface.It is important that microtubules are the cylinders whose walls are hexagonal lattices of subunit proteins known as tubulin, while the ordered water next to hydrophilic surfaces such as the tubulin, according to research by Pollack and his collaborators, behaves like a liquid crystalline (Zeng et al., 2006) with the ice-like structure in the form of hexagonal layers whose oxygens are not linked by proton bonds like in the ice, but the layers are stacked by interaction of opposite charges (Pollack, 2012).Since mathematical properties of number 037 are also fundamentally related to hexagonal lattice, then biological coding/computing might be actually fundamentally based on hexagonal symmetry and packing.Therefore, water might be the perfect biological coding/computing medium, as the DNA sequence reconstitution from the treated water indicated (Montagnier et al., 2010), and could have had a crucial primordial role (Pollack et al., 2009) both in the selection of life building block, such as canonical nucleic and amino acids, and in principally predeterminate evolution of genetic code with small degree of freedom.
Generally, the presented mathematical properties of number 037 and its realization in the genetic code and to a lesser presented extent in genomic code, indicate that the biological coding/computing is essentially the process both geometrical in nature and determined by the self-similar symmetry, giving the base for the biological large-scale coherence systems and biological quantumclassical intertwining.

Conclusions
ShCherbak's arithmetic inside the genetic code has a firm mathematical foundation in the sense that it is related to the number 037 -a unique decimal number from which an infinite set of self-similar numbers can be derived, with the nested numerical, geometrical, and arithmetical properties.Their correlation with self-similar symmetry, but also with the cyclotomic polynomials and thus the crystallographic lattices, can explain the numerous consistent arithmetical regularities of nucleon numbers of canonical amino acids for quite different systematizations of the genetic code.Biological coding/computing based on the self-similar numbers enables the realization of the nested organic codes, not only as the simplest error-correcting codes, but also as the biological systems with the holistic fractal structural and dynamical organization and thus the large-scale coherence systems, which is one of the main properties of biological organisms.Since such coding/computing is based both on the geometry and optimal space quantization, and on resonance and long-range interactions, the biological organisms reflect the principles of coding/computing in the physical world and deeply interact with it, which also justifies the fact that they are understood as information determined systems.The suggested coding/computing is also correlated with liquid crystalline water, emphasizing its crucial role in life origin and evolution.Mathematical possibility of the infinite nested coding/computing with selfsimilar numbers enables a limitless physical domain transition, which can potentially explain biological quantum-classical intertwining and general quantum-classical duality.

Acknowledgment
Author thanks to Professors Tidjani Négadi, Aleksandar Tomić, and Miloš Milovanović for their valuable comments.This research has been partially funded by the Ministry of Science and Technological Development of the Republic of Serbia, through Projects TR-32040 and TR-35023.

Dedication
This article I dedicate to my scientific "teacher", Professor Miloje M. Rakočević, who bridged my scholarly knowledge to the original scientific work.
much more obvious if the extension of digit notation for higher radix digits is based on decimal numbers, i.e . (20)].The partial multiplicative table of number 003367, given in Tab. 4, represents its most regular multiplier distribution whose successive differences for (a)-rows are 1, and for (b)-rows are 10.The second and third (a)-rows in Tab. 4, for numbers made up of different digits, are obtained by cyclic permutation of two digit blocks of the first (a)-row according to the same rule in Tab. 1, and generally, (a)-rows (bold numbers) represent original multiplicative table of 037 scaled by 2 [ = 2 n in Eqs.(13) and (15)].The (b)-rows, obtained by right-shift cyclic permutation of the leftmost digit of congruent numbers in (a)-rows, together with the first congruence class form the set of most regular (almost perfectly equidistant) multipliers whose differences are 10 (grey shaded fields in Tab.4).Similarly, the partial multiplicative table of number 000333667, given in Tab. 5, represents its most regular multiplier distribution whose successive differences for (a)-rows are 1, for (b)-rows are 10, and for (c)-rows are 100.The second and third (a)rows in Tab. 5, for the numbers made up of different digits, are obtained by cyclic permutation of three digit blocks of the first (a)-row according to the same rule in Tab. 1, and generally (a)-rows (bold numbers) represent the original multiplicative table of 037 scaled by 3 [ = 3 n in Eqs.(13) and (15)].Erasing every third digit in the numbers of (b)-rows in Tab. 5, reduces them to (b)-rows in Tab. 4 (for instance, a multiple ), while erasing the grey digits in the numbers of any row results in Tab. 1, indicating that in Tab. 5 are nested both Tab. 1 and Tab.

Figure 2 .
Figure 2. Polygonal numbers related to the first three multiples of number 037.A) The first multiple of 037 exactly corresponds to the 4 th centered hexagonal number, c H 4 = 37; B) The second multiple of 037, with a unit difference, corresponds to the 4 th star number, S 4 = 73; C) The third multiple of 037 exactly corresponds to the 2 nd composite triangular number, T 2 × c H 4 = 111; D) The third multiple of 037, with a unit difference, corresponds to the sum of the 4 th centered hexagonal and star number, S 4 + c H 4 = 110.

Figure 3 .
Figure 3. Nested polygonal numbers related to the first three multiples of number 037 decimal vertical varieties.The first multiples of 037 varieties exactly successively correspond to the …3334th centered hexagonal numbers, while the second multiples are bigger for 1 than successive …3334 th star numbers.The third multiples of 037 varieties again exactly correspond to the 2 nd composite triangular numbers.

Figure 4 .
Figure 4. Decimal numeration of natural numbers results in the periodic number with period 012345679, while its triple value results in the periodic number with period 037(Mišić, 2004).

Figure 5 .
Figure 5. Noncommutative multiplication of the composite polygonal numbers, shown in number 259, which is the product of two centered hexagonal numbers and also triplet S, gives two different geometric forms [cf.Eqs.(34), (35), and (37)].
lattice.Comparing this with S reveals that the triplets, 3 ( ) the centered hexagonal numbers (Figs. 2 and 3), are also the reduced cyclotomic values for 3 p N = = , Eqs. ( be geometrically interpreted as in Fig.2D.For the bases which do not individually correspond to nucleon number 037divisibility, i.e. bases T, A, and G, their nucleon number differences satisfy the squares of the first three Pythagorean numbers (

(
the nucleotide sequence into the nucleon binary string, so that the pair C G ≡ would have the meaning of the unit element and counting, while T=A would have the meaning of the neutral element for additive operation.For the triplet, the similar modular unit mass has the combination of U, C, and G bases

Table 4 .
Partial multiplicative table of number 003367.the multipliers of 003367, while big numbers are the proper multiples of 003367.Bold numbers result from the original multiplicative table of 037 (Tab. 1) scaled by 2 [n=2 in Eqs. (

Table 5 .
Partial multiplicative table of number 000333667.