Each base of the triplet can be connected by single transitions with the three other bases.

The result of this work was the structure that is isomorphic to the Boolean hypercube B⁶.

On this structure nine triplets originating from the ССС (3 C base substitution at each position of the triplet), and their relationships are shown in red.

It is L-structure of the Boolean hypercube. Triplets with substitutions in the first position are located at the left.

There is also a D-structure symmetric to it in which triplets with replacements in the first position are located on the right side of a hypercube.

The spatial structure of the genetic code containing transitions

C<-->G, U<-->A, C<-->U, G<-->A  and superimposed on it transitions C<-->A, U<-->G is a nine-dimensional simplex.

Fig. 9. The spatial structure of the triplet genetic code, which is isomorphic to the Boolean hypercube B⁶.

2.3.2.1. Properties of the structure of the triplet genetic code, inherited from the B⁶ hypercube

A. Longlines

Longlines

Triplets

The spatial structure of the triplet genetic code contains
7 longlines with different numbers of triplets:
1 triplet - in the I and VII longlines
6 triplets - in II and VI longlines
15 - in III-rd and V longlines and
20 triplets - in the IV longline

I

II

III

IV

V

VI

VII

1

6

15

20

15

6

1

Fig. 10. Longlined structure of the triplet genetic code.

B. Hierarchical Organization

а

 б

 в

Fig. 11. The hierarchical organization of the spatial structure of the genetic code.
a - two sets of 32 triplets: М₁ and М₂; b - four subsets of 16 triplets: SM₁, SM₂, SM₃, SM₄; c - eight octets: : О₁ - О₈.

2.3.2.2. The specific properties of the spatial structure of the genetic code

A. The single transitions

In Figure 9 possibility of formation of three types of transitions for each of three bases of triplets is shown:

C<-->G, U<-->A between complementary bases (solid lines);

C<-->A, G<-->U between non-complementary bases (rare dotted line);

C<-->U, G<-->A transitions pyrimidine - pyrimidine and purine - purine (frequent dotted lline).

In practice, in a general form on the presented model it is possible to consider only two types of transitions: C<-->G, U<-->A и C<-->U, G<-->A. The third type of transitions C<-->A, G<-->U which is an imposition on the initial structure, for technical reasons in a general view is difficult to portray, therefore they are presented in an enlarged view in separate drawing (see transitions C<-->A, G<-->U).

Single transitions C<-->G, U<-->A

In the first position of the triplet these transitions occur between two sets (M₁ и M₂) and are shown in Figure 12 a. In the second position of the triplet they are formed in octets (Fig. 12 b). In the third position of the triplet, these transitions connect the adjacent octets within subsets (Fig. 12 c). If necessary, they can be seen in more detail, having clicked the underlined words in each line in the caption under the picture 12 or pictures themselves.

 a

 b

c

Fig. 12. Single transitions of C<-->G, U<-->A (solid lines).
a - between the first triples bases (connect sets M₁ and and M₂);
b - between the second bases (bind triplets within octets from top to bottom);
c - between third bases (connect  triples within subsets).

Single transitions C<-->U, G<-->A

In the first position transitions of this type are observed within the octet of triplets down from right to left. They are shown in Figure 13 a. In the second position, on the contrary, they connect triplets in octets from top to bottom from left to right (Fig. 13 b). Finally, in the third position of triplet, these transitions connect subsets of triplets SM₁ - SM₂ and SM₄ - SM₃ (Fig. 13 c). If necessary, as for Figure 12, you can view them in more detail, having clicked the underlined words in each line in the caption under the picture 13 or pictures themselves.

 a

 b

c

Fig. 13. Single transitions of  C<-->U,т G<-->A  (frequent dashed line).
a - between the first bases (connect triplets within octets from top to bottom from right to left);
b - between the second bases (connect triplets within octets from top to bottom from left to right);
c - between third bases (connect triplets of SM₁ - SM₂ and SM₄ - SM₃).

It is known that in the Boolean hypercube В⁶ each vertex is connected to six neighbors. Shown above, six different single transition, actually, make up the structure of the triplet genetic code, which is isomorphic to the Boolean hypercube В⁶.

Single transitions C<-->A, G<-->U (see separate Figure 14).

B. An arrangement of the triplets coding for similar amino acids

On the spatial structure of the genetic code triplets coding for similar amino acids form either a quartets, or a pair of triplets occurring in quartets.

The quartets of triplets which encode the same amino acid, derived from «red doublets», are marked in red (Fig. 15) and are located in the top part of each set:  M₁¹ and M₂¹.

Quartet of triplets, which are usually encoded by only two pairs of amino acids derived from the "dark blue doublets" are marked in blue (Fig. 15) and are located at the bottom of each of the sets of the hypercube: M₁² and M₂².

To see them in more detail, click the underlined capital letters of the text.

Fig. 15. An arrangement of quartets of triplets coding for similar amino acids on spatial structure of a genetic code.

C. Rumer’s transformation

As seen in Figure 16, the triplets of the groups
 M₁¹ and M₂¹

are transformed in
triplets of groups M₁² and M₂²
by Rumer’s rule:

C <---> A, G <---> U,

occupying in the hypercube symmetric position
  (Group of symmetry C₂), for example:

ACC <---> CAA, GCC <---> UAA,

CCC <---> AAA, CCU <---> AAG etc.

Fig. 16. An arrangement of the triplets connected by Rumers’s transformation, on the spatial structure of the genetic code.