Changing Patterns using Transformation Pathways (Part #4)

The concern here is whether a laundry list of codons or hexagrams can be presented otherwise. In both cases there are example of presentation in tabular form as a kind of matrix offering a higher degree of order. Some degree of meaning can be associated with such a configuration. More interesting is the possibility that greater meaning can be associated with a configuration in 3 dimensions. Many insightful remarks in this respect are offered by Mark White (The G-ball, a New Icon for Codon Symmetry and the Genetic Code, 2007), notably:

The standard codon table is merely a data object, but it is conceptually a ***linear*** object that demonstrates an arbitrary arrangement subjectively chosen from a large number of logically equivalent structures. We might ***line up*** all codons in any old way that all equal the limited epistemic value of this arrangement. Therefore, the patterns observed in the data itself will always be largely subjective. The G-ball, on the other hand, is a data object chosen from only two possible objects of this form (it has a mirror twin). The data patterns seen here are un-weighted and therefore are the natural patterns of this form. The one chosen... reveals more aesthetically pleasing patterns in the data than does its mirror, so this degree of subjectivity should not be too disturbing. A dodecahedron is a real object, and it is shown here via Cayley**Os theorem that its symmetry elements can be used to create an isomorphic data object to completely represent the sequence symmetry of this specific set of nucleotides and codons. The G-ball wins out on comparisons of objectivity.

With respect to mutually informing the codon and hexagram cases, the initial assumptions made are as follows with respect to the codons:

• utility of the toroidal drilled truncated cube. Some other polyhedra could be considered, although the challenge of 64 is a constraint
  
• arbitrary attribution of C, T, A and G to the 4 traditional encoding combinations of broken and unbroken lines, as discussed separately (Archetypal otherness -- "DNA vs. I Ching", 2007):
  
  
  

  T	C	A	G

  
  Clearly other attributions could be made. It should be noted that the codons are conventionally understood as 3-fold combinations of any of those 4 -- as denoted by the letters above. A hexagram of 6 lines is then a combination of 3 pairs of 2 lines (whether broken or unbroken), but is traditionally represented as 2 groups of 3 lines (termed trigrams). It is unclear why the more fundamental binary code has not been used to make the 4-fold distinction, or considered as potentially offering further insight -- especially given its value in exploring patterns of distinctions (as discussed below). It would appear that the letters derive from Thymine, Cytosine, Adenine and Guanine.
  
• association of the distinction made between "whole" and "split" familiesof codons with distinctive configurations of lines on the drilled truncated cube (as shown below):
  • "whole" families were assumed to be triangular configurations of 3 codons with the fourth attached to one vertex. The singular nature of the latter suggested that it might be provisionally associated with a pair of unbroken lines.
  • "split" families were assumed to be 2 distinctive configurations of 4:
    • one in a star formation linked to a single vertex
    • one in a non-star configuration of 2 distinct forms
  • the configurations can be presented in contrasting orientations on the drilled truncated cube:
  • the amino acid specified by the codon combinations were associated as follows on the drilled truncated cube
    • the single amino acid specified by a "whole" family is indicated in relation to the (central) vertex of the triangular configuration
    • the two amino acids associated with each split family could be then associated with one inner vertex (or with both)
      
      
      

      Indication of contrasting combinations with which amino acid codes are associated (single vertex for each "whole" or split family highlighted)

      
      
• a tentative attribution of the family patterns to the structure was made as follows (with others to be considered)::
  • "whole" families on the outer "ends" of the form, in one particular orientation
  • "split" families as inner configurations of lines, in one particular orientation

Drilled truncated cube of 64 edges with tentative attribution of codon combinations(selected amino acids indicatively associated with central vertex of a combination)
Animation with faces non-transparent	Animation with faces transparent

Further guidance is suggested by the possibility that patterns oscillate, as suggested by animations below.