1. Introduction

Figure 1.1. The cycle of the metamorphosis of butterflies: butterfly-egg-caterpillar-chrysalis- butterfly.

Figure 1.1. The cycle of the metamorphosis of butterflies: butterfly-egg-caterpillar-chrysalis- butterfly.

2. Binary oppositions in genetic alphabets, Gray-type matrices, and Karnaugh maps

Figure 2.1. The arrangement of 4 nucleotides, 16 duplets and 64 triplets in matrices whose rows and columns are ordered by the corresponding Gray codes. The decimal equivalents of codewords or numbers of Gray codes are indicated in parentheses.

Figure 2.1. The arrangement of 4 nucleotides, 16 duplets and 64 triplets in matrices whose rows and columns are ordered by the corresponding Gray codes. The decimal equivalents of codewords or numbers of Gray codes are indicated in parentheses.

Figure 2.2. The location of 64 triplets and the amino acids and stop codons they encode in the Gray-type matrix (from Figure 2.1) in the case of the Vertebrate Mitochondrial Code. Common abbreviations for amino acids and stop codons are used. Semicircular brackets on the sides of the Gray-type matrix show the directions of the codewords order of the 6-bit Gray code, which numerates 64 matrix cells and their 64 triplets.

Figure 2.2. The location of 64 triplets and the amino acids and stop codons they encode in the Gray-type matrix (from Figure 2.1) in the case of the Vertebrate Mitochondrial Code. Common abbreviations for amino acids and stop codons are used. Semicircular brackets on the sides of the Gray-type matrix show the directions of the codewords order of the 6-bit Gray code, which numerates 64 matrix cells and their 64 triplets.

3. Genetic code degeneracy and genetic Gray-type matrices

Figure 3.1. Gray-type matrices of 16 duplets and 64 triplets from Figure 2.1 with black-and-white mosaics based on Rumer opposition. Strong duplets CC, CT, CG, AC, TC, GC, GT, GG and triplets with these strong roots are marked by black color.

Figure 3.1. Gray-type matrices of 16 duplets and 64 triplets from Figure 2.1 with black-and-white mosaics based on Rumer opposition. Strong duplets CC, CT, CG, AC, TC, GC, GT, GG and triplets with these strong roots are marked by black color.

Figure 3.2. Representations of mosaic Gray-type matrices of 16 duplets and 64 triplets from Figure 3.1 in forms of numeric matrices M₂ (at left) and M₃. The indices n = 2, 3 in the notation of these matrices correspond to the n-bit Gray codes used in numbering their rows and columns. Each black cell contains entry +1 and each white cell contains entry -1.

Figure 3.2. Representations of mosaic Gray-type matrices of 16 duplets and 64 triplets from Figure 3.1 in forms of numeric matrices M₂ (at left) and M₃. The indices n = 2, 3 in the notation of these matrices correspond to the n-bit Gray codes used in numbering their rows and columns. Each black cell contains entry +1 and each white cell contains entry -1.

3.1. Analysis of the genetic Gray-type matrix for 16 duplets

The dyadic-shift decomposition (described in Appendix B) of the matrix M₂, whose mosaic is based on Rumer opposition, shows that this Gray-type matrix is a sum of 4 sparse matrices e₀, e₁, e₂, and e₃ (Figure 3.3). Each of these sparse matrices is orthogonal since e₀e₀^T = E, e₁e₁^T = E, e₂e₂^T = E, e₃e₃^T = E, where T means the transposition and E is the identity matrix. The set of these sparse matrices is closed relative to multiplication and corresponds to a certain multiplication table in Figure 3.3, at bottom. This table matches to the multiplication table of the 4-dimensional algebra of Cockle split-quaternions [https://en.wikipedia.org/wiki/Split-quaternion], which is used in the Poincare conformal disk model of hyperbolic geometry [Karzel, Kist, 1985].

Figure 3.3. Dyadic-shift decomposition of the Gray-type matrix M₂ from Figure 3.2 into 4 sparse matrices e₀, e₁, e₂, and e₃. Their set corresponds to the multiplication table of Cockle split-quaternions (at bottom), used in the Poincare conformal disk model of hyperbolic geometry. An artistic symbol of this model is shown at bottom; it is taken from https://commons.wikimedia.org/wiki/Category:Poincar%C3%A9_disk_models , where the right to use this symbol for any purpose is granted.

Figure 3.3. Dyadic-shift decomposition of the Gray-type matrix M₂ from Figure 3.2 into 4 sparse matrices e₀, e₁, e₂, and e₃. Their set corresponds to the multiplication table of Cockle split-quaternions (at bottom), used in the Poincare conformal disk model of hyperbolic geometry. An artistic symbol of this model is shown at bottom; it is taken from https://commons.wikimedia.org/wiki/Category:Poincar%C3%A9_disk_models , where the right to use this symbol for any purpose is granted.

Concerning the topic of cyclic shifts, one can algorithmically transform the Gray-type matrix M₂ by a cyclic shift inside each of its 2-bit codewords, which numerate matrix rows and columns. Under this shift transformation, the previous set of numberings 00, 01, 11, 10 becomes a new set of numberings 00, 10, 11, 01, whose using is accompanied by a permutation of matrix rows and columns. In the result, a new Gray-type matrix M_2/perm arises with a completely different mosaic of arrangement of entries +1 and -1 (Figure 3.4).

Figure 3.4. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of position in 2-bit numberings of rows and columns generates a matrix with new mosaic (at left). Numeric representation M_2/perm of this new Gray-type matrix, when each black (white) cell contains entry +1 (-1), is shown at right.

Figure 3.4. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of position in 2-bit numberings of rows and columns generates a matrix with new mosaic (at left). Numeric representation M_2/perm of this new Gray-type matrix, when each black (white) cell contains entry +1 (-1), is shown at right.

The dyadic-shift decomposition of this new matrix M_2/perm shows that this Gray-type matrix is a sum of 4 sparse orthogonal matrices g₀, g₁, g₂, and g₃ (Figure 3.5). The set of these sparse matrices is again closed relative to multiplication and corresponds to the same multiplication table of the 4-dimensional algebra of Cockle split-quaternions, which was shown in Figure 3.3. In other words, Gray-type matrices M₂ and M_2/perm, which present structural features of the genetic code degeneracy, are two different matrix representations of 4-dimensional split-quaternions with unit coordinates.

Figure 3.5. Dyadic-shift decomposition of the Gray-type matrix M_2/perm from Figure 3.4 into 4 sparse matrices g₀, g₁, g₂, and g₃. Their set corresponds to the multiplication table of Cockle split-quaternions (at bottom), used in the Poincare conformal disk model of hyperbolic geometry. An artistic symbol of this model is shown at bottom.

Figure 3.5. Dyadic-shift decomposition of the Gray-type matrix M_2/perm from Figure 3.4 into 4 sparse matrices g₀, g₁, g₂, and g₃. Their set corresponds to the multiplication table of Cockle split-quaternions (at bottom), used in the Poincare conformal disk model of hyperbolic geometry. An artistic symbol of this model is shown at bottom.

In addition, cyclic shifts in the set of 2-bit numberings 00, 01, 11, 10 of rows and columns in the initial Gray-type matrix M₂ can be studied as well. More precisely, we mean the following tree possible orders of Gray-code members: 01, 11, 10, 00; 11, 10, 00, 01; 10, 00, 01, 11. Under using these three cyclic shift options for numberings of rows and columns of the genetic matrix of 16 duplets from Figure 3.1, matrix rows and columns are permutated and new Gray-type matrices arise with new arrangements of duplets and entries +1 and -1 (that is, with new black-and-white mosaics) (Figures 3.6–3.8). Numeric presentations of these three matrices are denoted as matrices P_2/01, P_2/11, P_2/10. Dyadic-shift decompositions of these three matrices into appropriate 8 sparse orthogonal matrices (Figures 3.6–3.8) reveal that each of them is a matrix representation of the Cockle split-quaternions with unit coordinates as well (by analogy with matrices in Figures 3.2–3.5).

Figure 3.6. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 01, 11, 10, 00 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/01 gives 8 sparse orthogonal matrices s₀, s₁, s₂, and s₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.6. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 01, 11, 10, 00 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/01 gives 8 sparse orthogonal matrices s₀, s₁, s₂, and s₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.7. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 11, 10, 00, 01 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/11 gives 8 sparse orthogonal matrices r₀, r₁, r₂, and r₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.7. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 11, 10, 00, 01 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/11 gives 8 sparse orthogonal matrices r₀, r₁, r₂, and r₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.8. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 10, 00, 01, 11 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/10 gives 8 sparse orthogonal matrices q₀, q₁, q₂, and q₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.8. The transformation of the Gray-type matrix with 16 duplets from Figure 3.1 under the cyclic shift of numberings of rows and columns to the option 10, 00, 01, 11 generates a matrix with a new mosaic and an arrangement of duplets (at top). The dyadic-shift decomposition of the corresponding numeric matrix P_2/10 gives 8 sparse orthogonal matrices q₀, q₁, q₂, and q₃ (shown at middle level), which defines a multiplication table of Cockle spit-quaternions used in the Poincare conformal disk model of hyperbolic geometry (shown at bottom jointly with the artistic symbol of this model).

Figure 3.9 allows comparing - in a comfortable form - of mosaics of these 4 Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10, in which cyclic changed sequences of Gray code members numerate rows and columns. Such cyclic changes of numberings transform matrices each into others.

Figure 3.9. Comparison of Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10 (from Figures 3.3, 3.6–3.8), in which cyclic changed sequences of Gray codewords numerate rows and columns (the corresponding sequence is shown next to each matrix). In matrices, each black (white) cell contains entry +1 (-1). All 4 matrices are different matrix representations of Cockle split-quaternions used in the Poincare conformal disk model of hyperbolic geometry, whose symbol is shown in the center.

Figure 3.9. Comparison of Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10 (from Figures 3.3, 3.6–3.8), in which cyclic changed sequences of Gray codewords numerate rows and columns (the corresponding sequence is shown next to each matrix). In matrices, each black (white) cell contains entry +1 (-1). All 4 matrices are different matrix representations of Cockle split-quaternions used in the Poincare conformal disk model of hyperbolic geometry, whose symbol is shown in the center.

Squaring each of these Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10 results in doubling it:

M₂² = 2M₂ , P_2/01² = 2P_2/01, P_2/11 = 2P_2/11, P_2/10² = P_2/10

(3.1)

Each of these Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10, when divided by two, is a projector, since its squaring results the same matrix, which is the criterion of projector operators:

(M₂/2)² = M₂/2 , (P_2/01/2)² = P_2/01/2, (P_2/11/2)² = P_2/11/2, (P_2/10 /2)² = P_2/10/2

(3.2)

The presented data (Figures 3.3–3.9) raise the question of why the genetic coding system is associated with the Poincare conformal disk model of Lobachevsky’s hyperbolic geometry? The author believes that this may be due to the theory of automorphic functions and the fundamental importance of cyclic organization for living bodies. The fact is that it was on the basis of the properties of Lobachevsky’s hyperbolic geometry that A. Poincaré developed the theory of automorphic functions, which are a generalization of periodic functions and are associated with the ideology of cycles. This algebraic theory uses actions of discrete groups of transformations on functions. Many inherited biological symmetries are connected with appropriate non-Euclidean symmetries of conform-geometrical types (cyclomeries, etc.), which expand similarity bio-symmetries by A.V. Shubnikov [Petoukhov, 1989].

All the genetic Gray-type matrices M₂, M_2/perm, P_2/01, P_2/11, and P_2/10 (Figures 3.3–3.9) have a general property associated with the operation of mutual replacement 0↔1 in the binary numbering of columns (this mutual replacement is called the complementarity operation): any two columns, whose numberings are complementary each to another relative to the replacement of 0↔1, have sets of entries, which differ only by their opposite signs (that is, these sets are also complementary each to another relative to the replacement operation of signs “+ ↔ -“ ). For example, in matrix M₂ (Figure 3.2), two complementary columns with complementarity numberings 00 and 11 contain correspondingly entries “1,1,-1,1” and “-1,-1,1,-1”, which differ only by their opposite signs. Two other columns with complementarity numberings 01 and 10 contain correspondingly entries “-1,1,-1,-1” and “1,-1,1,1”, which also differ only by their opposite signs. This general property of the considered Gray-type matrices can be conditionally called as the property of the double complementarity.

This double complementarity provides that the matrix operators M₂, P_2/01, P_2/11, and P_2/10 (Figures 3.3–3.9) are cyclic operators in the following sense. Let's consider an arbitrary 4-parameter system, the behavior of which in time is described by the vector Y=[f₀(t), f₁(t), f₂(t), f₃(t)], whose coordinates are voluntary periodic functions f_i(t) independent of each other. The action of any of the named genetic Gray-type matrices on this vector generates a vector whose coordinates are not only mutually dependent, but also consist of pairs equal in magnitude and opposite in signs. The expression (3.3) shows an example of the action of the Gray-type matrix M₂ (Figure 3.2) on such vector:

$$\left[{\mathrm{f}}_0\right(\mathrm{t}),{\mathrm{f}}_1(\mathrm{t}),{\mathrm{f}}_2(\mathrm{t}),{\mathrm{f}}_3(\mathrm{t}\left)\right]\ast {\mathrm{M}}_2=[{\mathrm{f}}_0+{\mathrm{f}}_1-{\mathrm{f}}_2+{\mathrm{f}}_3,-{\mathrm{f}}_0+{\mathrm{f}}_1-{\mathrm{f}}_2-{\mathrm{f}}_3,-({\mathrm{f}}_0+{\mathrm{f}}_1-{\mathrm{f}}_2+{\mathrm{f}}_3),-(-{\mathrm{f}}_0+{\mathrm{f}}_1-{\mathrm{f}}_2-{\mathrm{f}}_3\left)\right]=\overline{\mathrm{X}}$$

(3.3)

In the generated vector $\overline{\mathrm{X}}$ (3.3), the cyclic behavior of coordinates X₀(t) and X₂(t), denoted by yellow, is synchronized: at each value of t, these coordinates are equal in amplitude and opposite in sign (the complementarity relative to signs), that is, their periodic behavior occurs in antiphase. The same holds for relations between the coordinates X₁(t) and X₃(t) denoted by green. It means that the action of genetic Gray-type matrices M₂, P_2/01, P_2/11, and P_2/10 on arbitrary 4-dimensional vectors with periodic coordinates introduces a cyclic ordering into the coordinates of the generated vector. Figuratively speaking, cycles beget cycles.

Similar properties exist in the case of the action of the Gray-type matrix M₃ (Figure 3.2) on arbitrary 8-dimensional vectors with periodic coordinates. In this direction, an algebraic theory of code hypercycles based on cyclic Gray codes is developed by the author in some distant analogy with the biochemical theory of hypercycles of Nobel laureate M. Eigen modeling life origin via the scheme of combining self-replicating macromolecules into closed autocatalytic chemical cycles [Eigen, Schuster, 1977].

One more kind of interconnections of genetic Gray-type matrices M₂, M_2/perm, P_2/01, P_2/11, and P_2/10 (Figures 3.3–3.9) with the complementarity operation 0↔1 is revealed due to a special decomposition of each of these matrices into the sum of two sparse matrices, whose example in the case of the matrix M₂ is shown in Figure 3.10. Each of these two sparse matrices contains only those non-zero columns whose 2-bit numberings are complementarity: more precisely, these columns numberings are 00 and 11, and also 01 and 10 (one can remind that these numberings are based on the molecular binary opposition “purines-vs-pyrimidines”).

Figure 3.10. The decomposition of the genetic Gray-type matrix M₂ from Figure 3.2 into the sum of two sparse matrices, having only those two non-zero columns, whose 2-bit numberings are complementarity each to another.

Figure 3.10. The decomposition of the genetic Gray-type matrix M₂ from Figure 3.2 into the sum of two sparse matrices, having only those two non-zero columns, whose 2-bit numberings are complementarity each to another.

Each of these two sparse matrices in Figure 3.10 occurs to be a matrix representation of 2-dimensional hyperbolic numbers with unit coordinates. It is revealed by a decomposition of each of these sparse matrices into the sum of two other sparse matrices shown in Figure 3.11. The set of these two new sparse matrices is closed relative to multiplication and defines the multiplication table of the algebra of 2-dimensional split-complex or hyperbolic numbers [https://en.wikipedia.org/wiki/Split-complex_number].

Figure 3.11. Decompositions of the two sparse matrices from Figure 3.10 into sums of other two sparse matrices v₀, v₁ and v₂, v₃. The set of the matrices v₀ and v₁ is closed relative to multiplication and defines the shown multiplication table of the algebra of 2-dimensional split-complex numbers (at top). The same holds for the set of the matrices v₂ and v₃ (at bottom).

Figure 3.11. Decompositions of the two sparse matrices from Figure 3.10 into sums of other two sparse matrices v₀, v₁ and v₂, v₃. The set of the matrices v₀ and v₁ is closed relative to multiplication and defines the shown multiplication table of the algebra of 2-dimensional split-complex numbers (at top). The same holds for the set of the matrices v₂ and v₃ (at bottom).

3.2. Analysis of the genetic Gray-type matrix for 64 triplets

Now let us turn to a similar analysis of the genetic Gray-type matrix M₃ for 64 triplets, which reflects the genetic code degeneracy for the alphabet of 64 triplets by means of the black-and-white mosaic based on Rumer opposition (Figures 3.1 and 3.2). The dyadic-shift decomposition of the matrix M₃ shows that this matrix is a sum of 8 sparse orthogonal matrices p₀, p₁, p₂, p₃, p₄, p₅, p₆, p₇ (Figure 3.12). The set of these 8 sparse matrices is closed relative to multiplication and defines a multiplication table of 8-dimensional algebra of hypercomplex numbers (Figure 3.12, at bottom), which can be considered as associated with the multiplication table of 4-dimensional algebra of Cockle split-quaternions in Figures 3.3, 3.5–3.9.

Figure 3.12. The dyadic-shift decomposition of the genetic Gray-type matrix M₃ of 64 triplets (from Figure 3.2) into the sum of 8 sparse orthogonal matrices p₀, p₁, p₂, p₃, p₄, p₅, p₆, p₇, whose set is closed relative to multiplication and defines the shown multiplication table of 8-dimensional algebra of hypercomplex numbers (at bottom).

Figure 3.12. The dyadic-shift decomposition of the genetic Gray-type matrix M₃ of 64 triplets (from Figure 3.2) into the sum of 8 sparse orthogonal matrices p₀, p₁, p₂, p₃, p₄, p₅, p₆, p₇, whose set is closed relative to multiplication and defines the shown multiplication table of 8-dimensional algebra of hypercomplex numbers (at bottom).

Concerning the topic of cyclic shifts, one can algorithmically transform the Gray-type matrix M₃ by a cyclic shift inside each of its 3-bit codewords, which numerate matrix rows and columns. Under this shift transformation, the previous set of numberings 000, 001, 011, 010, 110, 111, 101, 100 becomes two net set of numberings. Under the cyclic shift for 1 position, the following set arises: 000, 010, 110, 100, 101, 111, 011, 001. Under the cyclic shift for 2 positions, the following set arises: 000, 100, 101, 001, 011, 111, 110, 010. These new sets of numberings of rows and columns are accompanied with a permutation of matrix rows and columns. In the result, new Gray-type matrices M_3/perm1 and M_3/perm2 arise with a completely different mosaic of arrangements of entries +1 and -1 (Figure 3.13).

Figure 3.13. Transformations of the genetic Gray-type matrix M₃ from Figure 3.2 under the cyclic shifts - on 1 and 2 positions - in 3-bit numberings of its rows and columns generate corresponding matrices M_3/perm1 (at left) and M_3/perm2 (at right) with new mosaic of arrangements of +1 and -1.

Figure 3.13. Transformations of the genetic Gray-type matrix M₃ from Figure 3.2 under the cyclic shifts - on 1 and 2 positions - in 3-bit numberings of its rows and columns generate corresponding matrices M_3/perm1 (at left) and M_3/perm2 (at right) with new mosaic of arrangements of +1 and -1.

The same dyadic-shift decomposition of each of these Gray-type matrices M_3/perm1 and M_3/perm2 represents it as a sum of 8 sparse orthogonal matrices, whose set is closed relative to multiplication and corresponds to the same multiplication table of 8-dimensional algebra of hypercomplex numbers shown in Figure 3.12 (with accuracy up to reindexing of these 8 sparse matrices).

Possible cyclic shifts of members in the set of Gray 3-bit codewords 000, 001, 011, 010, 110, 111, 101, 100 produce 7 new sequences of Gray codewords (3.4):

001, 011, 010, 110, 111, 101, 100, 000;
 011, 010, 110, 111, 101, 100, 000, 001;
 010, 110, 111, 101, 100, 000, 001, 011;
 110, 111, 101, 100, 000, 001, 011, 010;
 111, 101, 100, 000, 001, 011, 010, 110;
 101, 100, 000, 001, 011, 010, 110, 111;
 100, 000, 001, 011, 010, 110, 111, 101

(3.4)

Under using these seven cyclic shift options for numberings of rows and columns of the Gray-type matrix of 64 triplets from Figure 3.1, matrix rows and columns are permutated and new seven Gray-type matrices arise with new arrangements of triplets and entries +1 and -1 (that is, with new black-and-white mosaics) (Figure 3.14).

Figure 3.14. Seven additional Gray-type matrices, which arise under cyclic shifts in sets of Gray codewords numberings of rows and columns (3.4) in the initial Gray-type matrix from Figure 3.1. This family of 8 Gray-type matrices represents the genetic code degeneracy.

Figure 3.14. Seven additional Gray-type matrices, which arise under cyclic shifts in sets of Gray codewords numberings of rows and columns (3.4) in the initial Gray-type matrix from Figure 3.1. This family of 8 Gray-type matrices represents the genetic code degeneracy.

Figure 3.15 shows all these 8 Gray-type matrices of 64 triplets from Figures 3.1 and 3.14 in their numeric forms, where each black (white) cell contains +1 (-1). These matrices are transformed one into another by appropriate cyclic shifts of numberings of its rows and columns.

Figure 3.15. A comparison of black-and-white mosaic of 8 Gray-type matrices G_i (i = 0, 1, 2, …, 7) of 64 triplets from Figures 3.1, 3.2 and 3.14. Each black (white) cell in matrices contains +1 (-1).

Figure 3.15. A comparison of black-and-white mosaic of 8 Gray-type matrices G_i (i = 0, 1, 2, …, 7) of 64 triplets from Figures 3.1, 3.2 and 3.14. Each black (white) cell in matrices contains +1 (-1).

All 8 Gray-type matrices G_i of 64 triplets in Figure 3.15, which represent the genetic code degeneracy in connection with cyclic changes in the sets of 3-bit numberings of their rows and columns, possess general algebraic properties: raising each matrix to the fourth power generates its quadrupling; each matrix, when divided by four, is a projector operator (3.5):

G_i² = 4G_i ; (G_i/4)² = G_i

(3.5)

All 8 genetic Gray-type matrices in Figure 3.15 have a general property associated with the complementarity operation of the replacement 0↔1 in the 3-bit numberings of their columns: any two columns, whose numberings are complementary each to another relative to the replacement of 0↔1, have sets of entries, which differ only by their opposite signs (that is, these sets are also complementary each to another relative to the replacement operation of signs “+ ↔ -“). For example, in the matrix M₃ from Figure 3.2, the column with numbering 000 contains the set of entries +1, +1, +1, +1, -1, -1, +1, +1, and the complementary column with numbering 111 contains the same set of entries but with opposite signs: -1, -1, -1, -1, +1, +1, -1, -1. This general property of the considered Gray-type matrices of triplets can be also conditionally called as the property of the double complementarity.

This double complementarity provides that the Gray-type matrix operators G_i (Figure 3.15) are cyclic operators in the following sense. Let's consider an arbitrary 8-parameter system, the behavior of which in time is described by the vector $\mathrm{Y}¯$ =[f₀(t), f₁(t), f₂(t), f₃(t), f₄(t), f₅(t), f₆(t), f₇(t)], whose coordinates are voluntary periodic functions f_i(t) independent of each other. The action of any of the named genetic Gray-type matrices G_i on this vector generates a vector $\overline{\mathrm{X}}$ whose coordinates become mutually dependent and synchronized in cyclic behaviors of its separate coordinates (3.6). In other words, instead of a random set of oscillations f_i(t) in a multi-parameter system, order arises in the form of strictly ordered relationships of cyclic processes occurring in different subspaces of the system configuration space, due to which this system acquires the features of a whole entity. For example, the action of the Gray-code matrix M₃ (from Figure 3.2) on the vector $\overline{\mathrm{Y}}$ =[f₀(t), f₁(t), f₂(t), f₃(t), f₄(t), f₅(t), f₆(t), f₇(t)] generates the following vector $\overline{\mathrm{X}}$ = [X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇], where cyclic coordinates with identical amplitudes at any fixed t are marked by identical colors:

$$\begin{array}{ll}\overline{\mathrm{X}}={\overline{\mathrm{Y}}}^{*}{\mathrm{M}}_3=& [{\mathrm{f}}_0+{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4+{\mathrm{f}}_5+{\mathrm{f}}_6+{\mathrm{f}}_7,\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{f}}_0+{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5+{\mathrm{f}}_6+{\mathrm{f}}_7\\ & -{\mathrm{f}}_0-{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5-{\mathrm{f}}_6-{\mathrm{f}}_7,\hspace{1em}\hspace{1em}\hspace{1em}-{\mathrm{f}}_0-{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5-{\mathrm{f}}_6-{\mathrm{f}}_7\\ & -({\mathrm{f}}_0+{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5+{\mathrm{f}}_6+{\mathrm{f}}_7),\hspace{1em}-({\mathrm{f}}_0+{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5+{\mathrm{f}}_6+{\mathrm{f}}_7)\\ & -(-{\mathrm{f}}_0-{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5-{\mathrm{f}}_6-{\mathrm{f}}_7),\hspace{1em}-(-{\mathrm{f}}_0-{\mathrm{f}}_1+{\mathrm{f}}_2+{\mathrm{f}}_3-{\mathrm{f}}_4-{\mathrm{f}}_5-{\mathrm{f}}_6-{\mathrm{f}}_7)]\end{array}$$

(3.6)

According to (3.6), X₀(t) =X₁(t) = -X₄(t) = -X₅(t) and X₂(t) = X₃(t) = -X₆(t) = -X₇(t). It means that 4 yellow coordinates X₀, X₁, X₄, and X₅ are identical in their amplitudes at any t, but coordinates X₄ and X₅ are cyclically changed in opposite phases in comparison with coordinates X₀ and X₁. Analogically, 4 green coordinates X₂, X₃, X₆, and X₇ are identical in their amplitudes at any t, but coordinates X₆ and X₇ are cyclically changes in opposite phases in comparison with coordinates X₂ and X₃.

4. Interrelation between the Hilbert curve and Gray codes for using in genetic studies

Figure 4.1. Examples of the Hibert curve: a – Hilbert curve, the first order; b – Hilbert curve, first and second orders; c – Hilbert curve, first to third orders; d – Hilbert curve of a higher order; e - 3-D Hilbert curve with color showing progression (taken from https://en.wikipedia.org/wiki/Hilbert_curve , permission is granted to copy, distribute and adapt these images under the Creative Commons Attribution-Share Alike 3.0 Unported, 2.5 Generic, 2.0 Generic, 1.0 Generic license and also under the Creative Commons Attribution-Share Alike 4.0 International license).

Figure 4.1. Examples of the Hibert curve: a – Hilbert curve, the first order; b – Hilbert curve, first and second orders; c – Hilbert curve, first to third orders; d – Hilbert curve of a higher order; e - 3-D Hilbert curve with color showing progression (taken from https://en.wikipedia.org/wiki/Hilbert_curve , permission is granted to copy, distribute and adapt these images under the Creative Commons Attribution-Share Alike 3.0 Unported, 2.5 Generic, 2.0 Generic, 1.0 Generic license and also under the Creative Commons Attribution-Share Alike 4.0 International license).

Figure 4.2. Recursive construction of the fractal Hilbert curve of the first, second, and third orders to fill space (taken from https://en.wikipedia.org/wiki/Moore_curve, permission is granted to copy, distribute and adapt the work under the Creative Commons Attribution-Share Alike 4.0 International license).

Figure 4.2. Recursive construction of the fractal Hilbert curve of the first, second, and third orders to fill space (taken from https://en.wikipedia.org/wiki/Moore_curve, permission is granted to copy, distribute and adapt the work under the Creative Commons Attribution-Share Alike 4.0 International license).

5. Gray codes for modeling genetic structures and inherited cyclic phenomena. Cyclic biocomputing

Figure 5.1. Example of a transformation of one 3-bit Gray code into another 3-bit Gray code under the complementarity operation 0↔1 in its set of binary codewords.

Figure 5.1. Example of a transformation of one 3-bit Gray code into another 3-bit Gray code under the complementarity operation 0↔1 in its set of binary codewords.

Figure 5.2. An example of a cyclic system with 8 subsystems, each of which can be in one of 4 states, denoted by codewords 00, 01, 11, 10 of the 2-bit Gray code. The top row shows a color representation of each of these codewords and states. Figures 5.2a-e illustrate the relay race-like process of cyclic change in the states of all 8 subsystems (see text for details).

Figure 5.2. An example of a cyclic system with 8 subsystems, each of which can be in one of 4 states, denoted by codewords 00, 01, 11, 10 of the 2-bit Gray code. The top row shows a color representation of each of these codewords and states. Figures 5.2a-e illustrate the relay race-like process of cyclic change in the states of all 8 subsystems (see text for details).

Figure 5.3. An example of cyclic kinematics of a system of 5 rods, consisting of 4 mutually connected cyclic subsystems of angular types. Each of these subsystems can be in 4 angular states, characterized by the angle between adjacent rods and denoted by the code words 00, 01, 11, 10 (for clarity, these angular states are assigned color labels, as shown at the top). The initial angular states in the sequence of 4 subsystems correspond to the sequence of code words 00, 01, 11, 10. The kinematics of the rod system is shown during a relay race-like process of cyclic shift of the angular states of the subsystems (for details, see the text).

Figure 5.3. An example of cyclic kinematics of a system of 5 rods, consisting of 4 mutually connected cyclic subsystems of angular types. Each of these subsystems can be in 4 angular states, characterized by the angle between adjacent rods and denoted by the code words 00, 01, 11, 10 (for clarity, these angular states are assigned color labels, as shown at the top). The initial angular states in the sequence of 4 subsystems correspond to the sequence of code words 00, 01, 11, 10. The kinematics of the rod system is shown during a relay race-like process of cyclic shift of the angular states of the subsystems (for details, see the text).

Figure 5.4. An example of algorithmic fractal-like growth of a cyclic system, initially consisting of 4 mutually connected cyclic subsystems. In the presented system, each of the 4 subsystems can be in 4 states. These states cyclically replace each other in a relay race manner and are designated by Gray codewords 00,01,11,10, which are additionally marked with 4 colors for clarity. As a result of the algorithm, according to which each subsystem is transformed into 4 new cyclic subsystems, an enlarged cyclic system with 16 cyclic subsystems is formed (see text for details).

Figure 5.4. An example of algorithmic fractal-like growth of a cyclic system, initially consisting of 4 mutually connected cyclic subsystems. In the presented system, each of the 4 subsystems can be in 4 states. These states cyclically replace each other in a relay race manner and are designated by Gray codewords 00,01,11,10, which are additionally marked with 4 colors for clarity. As a result of the algorithm, according to which each subsystem is transformed into 4 new cyclic subsystems, an enlarged cyclic system with 16 cyclic subsystems is formed (see text for details).

6. Genetic Hadamard matrices, Walsh functions, and Gray codes

Figure 6.1. Schematic images of nucleotides T, A, C, and G. Big circles denote amino groups NH₂ in nucleotides A, C, and G. Small black circles represent carbon atoms and small white circles represent hydrogen atoms. The letter N stands for nitrogen atom.

Figure 6.1. Schematic images of nucleotides T, A, C, and G. Big circles denote amino groups NH₂ in nucleotides A, C, and G. Small black circles represent carbon atoms and small white circles represent hydrogen atoms. The letter N stands for nitrogen atom.

Figure 6.2. Matrices H₂, H₄, and H_8, which are numerical representations of symbolic genetic matrices of nucleotides, duplets, and triplets from Figure 2.1. Cells with entries +1 are shown in black, and cells with entries -1 are shown in white. Rows and columns of matrices are ordered according to appropriate sequences of codewords of cyclic Gray codes, similar to Karnaugh maps (see explanation in the text).

Figure 6.2. Matrices H₂, H₄, and H_8, which are numerical representations of symbolic genetic matrices of nucleotides, duplets, and triplets from Figure 2.1. Cells with entries +1 are shown in black, and cells with entries -1 are shown in white. Rows and columns of matrices are ordered according to appropriate sequences of codewords of cyclic Gray codes, similar to Karnaugh maps (see explanation in the text).

Figure 6.3. Dyadic-shift decomposition of the Hadamard genetic matrix H₄ (from Figure 6.2) represents it as the sum of 4 sparse matrices, the set of which is closed under multiplication and defines the symmetric multiplication table shown. The location of the + and – signs in this multiplication table is similar to their location in the shown Hadamard matrix V.

Figure 6.3. Dyadic-shift decomposition of the Hadamard genetic matrix H₄ (from Figure 6.2) represents it as the sum of 4 sparse matrices, the set of which is closed under multiplication and defines the symmetric multiplication table shown. The location of the + and – signs in this multiplication table is similar to their location in the shown Hadamard matrix V.

Figure 6.4. The family of Hadamard matrices, which arise from matrix H₄ (Figure 6.2) under a simultaneous cyclic shift in its number sequence of rows and columns by 1 position, by 2 positions, and by 3 positions.

Figure 6.4. The family of Hadamard matrices, which arise from matrix H₄ (Figure 6.2) under a simultaneous cyclic shift in its number sequence of rows and columns by 1 position, by 2 positions, and by 3 positions.

Figure 6.5. A complete orthonormal system of 4 Walsh functions, whose set of sequency 0, 1, 2, 3 are in one-to-one correspondence with the 4 codewords of the 2-bit Gray code. Here these 4 Walsh functions are ordered in the ascending sequency order.

Figure 6.5. A complete orthonormal system of 4 Walsh functions, whose set of sequency 0, 1, 2, 3 are in one-to-one correspondence with the 4 codewords of the 2-bit Gray code. Here these 4 Walsh functions are ordered in the ascending sequency order.

7. Gray codes, dyadic clock, and the sequency theory by Harmuth

8. Universal stochastic rules of genomic DNAs, binary-genomic numbers, and complementarity operation

Table 8.1. A unified vector representation of families of probabilities (percentages) of n-bit numbers characterizing the statistical structure of binary sequences of single-stranded DNAs (that is, their binary-genomic numbers), which are analyzed using the HBS method described in the text. The percentages of n-plets starting with 0 are shown in red and those starting with 1 are shown in blue.

Table 8.1. A unified vector representation of families of probabilities (percentages) of n-bit numbers characterizing the statistical structure of binary sequences of single-stranded DNAs (that is, their binary-genomic numbers), which are analyzed using the HBS method described in the text. The percentages of n-plets starting with 0 are shown in red and those starting with 1 are shown in blue.

Table 8.2. Phenomenological percent values of each of the n-plets of purines (symbol 0) and pyrimidines (symbol 1) in binary-genomic number corresponding to the single-stranded DNA of human chromosome № 1 (n = 1, 2, 3, 4, 5). This binary-genomic number contains about 250 million purines and pyrimidines. Initial data on this chromosome were taken from the GenBank: https://www.ncbi.nlm.nih.gov/nuccore/NC_000001.11. Percentages are given in fractions of one and are shown as the coordinates of the vectors $\overline{D}$_n from Table 8.1.

Table 8.2. Phenomenological percent values of each of the n-plets of purines (symbol 0) and pyrimidines (symbol 1) in binary-genomic number corresponding to the single-stranded DNA of human chromosome № 1 (n = 1, 2, 3, 4, 5). This binary-genomic number contains about 250 million purines and pyrimidines. Initial data on this chromosome were taken from the GenBank: https://www.ncbi.nlm.nih.gov/nuccore/NC_000001.11. Percentages are given in fractions of one and are shown as the coordinates of the vectors $\overline{D}$_n from Table 8.1.

Figure 8.1. Diagrams of the percentages of binary n-plets of purines and pyrimidines in the 4-duplet, 8-triplet, 16-tetraplet, and 32-pentaplet families taken from Table 8.2. The order of the columns corresponds to the order of the coordinates in the corresponding vectors $\overline{D}$_n of Table 8.1. The numbers of these coordinates $\overline{D}$_n are plotted along the abscissa axis, and the values of these coordinates are plotted along the ordinate axis, that is, the percentages of binary n-plets of DNA of the first human chromosome in its binary representation as a sequence of purines and pyrimidines (in the form of its BG-number for the sequence of purines and pyrimidines).

Figure 8.1. Diagrams of the percentages of binary n-plets of purines and pyrimidines in the 4-duplet, 8-triplet, 16-tetraplet, and 32-pentaplet families taken from Table 8.2. The order of the columns corresponds to the order of the coordinates in the corresponding vectors $\overline{D}$_n of Table 8.1. The numbers of these coordinates $\overline{D}$_n are plotted along the abscissa axis, and the values of these coordinates are plotted along the ordinate axis, that is, the percentages of binary n-plets of DNA of the first human chromosome in its binary representation as a sequence of purines and pyrimidines (in the form of its BG-number for the sequence of purines and pyrimidines).

Figure 8.2. Diagram comparing the percentages of binary n-plets of purines and pyrimidines from Table 8.2 at different values of n and also the corresponding fractal tree of probability dichotomies (right). At each level of the diagram, the lengths of the intervals are proportional to the percentage values of the corresponding n-plets from Table 8.2. The relationships of dichotomy (or twinning) between the lengths of the corresponding intervals at adjacent levels are visible.

Figure 8.2. Diagram comparing the percentages of binary n-plets of purines and pyrimidines from Table 8.2 at different values of n and also the corresponding fractal tree of probability dichotomies (right). At each level of the diagram, the lengths of the intervals are proportional to the percentage values of the corresponding n-plets from Table 8.2. The relationships of dichotomy (or twinning) between the lengths of the corresponding intervals at adjacent levels are visible.

Table 8.3. Comparison of the percentage X of each n-plet from Table 8.2 with the sum ∑ percent of two (n+1)-plets, differing from it by the presence of suffixes 0 and 1. Numbers are rounded to the fourth decimal place. The right column shows virtually no ∑-X difference between these quantities.

Figure 8.3. Hadamard matrices H₂, H₄, H₈, H₁₆, whose Walsh functions w(k,n) are ordered in ascending order of their sequency parameters. To the right of the matrices, indicated by bold boundaries, the Walsh functions corresponding to their rows are shown, indicating the sequency parameter k and the order n.

Figure 8.3. Hadamard matrices H₂, H₄, H₈, H₁₆, whose Walsh functions w(k,n) are ordered in ascending order of their sequency parameters. To the right of the matrices, indicated by bold boundaries, the Walsh functions corresponding to their rows are shown, indicating the sequency parameter k and the order n.

Some concluding remarks.

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