Changing Patterns using Transformation Pathways (Part #3)

Rather than engage immediately in two-culture controversy regarding any relation between 64 codons and 64 hexagrams, the concern here is theidentification of a framework which could hold this variety in a manner which offers a sense of coherence and comprehensibility -- beyond conventional presentations of such patterns. The focus is therefore on patterns and how they may be fruitfully configured.

Previous exercises have considered the use of spherically symmetrical polyhedra for such mapping purposes (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008). ). The switch to 3-dimensional mapping, in contrast to any 2D ("flat Earth") framework, is potentially consistent with requirements for coherence and comprehensibility. Experiments to that end have been undertaken with respect to traditional symbol systems (Representation of Creative Processes through Dynamics in Three Dimensions, 2014).

The polyhedral approach to representation of the genetic code has notably been explored by Chi Ming Yang (The naturally designed spherical symmetry in the genetic code, 2003; On the 28-gon symmetry inherent in the genetic code intertwined with aminoacyl-tRNA synthetases -- the Lucas series, Bulletin of Mathematical Biology, 2004). He makes use of a quasi-28-sided polyhedron (an icosikaioctagon). A relation to the Kazakh approach is considered by Tidjani N**gadi (A Connection between Shcherbak**Os arithmetical and Yang**Os 28-gon polyhedral ***views*** of the genetic code, Internet Electronic Journal of Molecular Design, 2003). Use is made of the complementarity of the much simpler icosahedron and dodecahedron by Mark White (The G-ball, a New Icon for Codon Symmetry and the Genetic Code, 2007). Ironically, in the light of the argument above, although Chi Ming Yang is based in China he makes no reference to the I Ching pattern of hexagrams.

The pattern of 64 is nearly unique within that polyhedral context. However one interesting candidate is the toroidal drilled truncated cube with 64 edges -- with which any set of 64 elements could be associated. The issue is whether the manner in which they can be positioned on that framework constitutes a configuration which is meaningful in relation to particular cases, such as the codons or the hexagrams. Furthermore, is it possible that known constraints in the patterning in such particular cases can together offer guidance in the attribution of the distinct elements -- of relevance to each case?

Preliminary experiments with this polyhedron have been undertaken previously with respect to the hexagrams alone -- but only to get a sense of the possibility, as a "proof of concept" (Enabling Wisdom Dynamically within Intertwined Tori: Requisite resonance in global knowledge architecture, 2012).

Drilled truncated cube of 64 edges with random attribution of hexagram names (reproduced from Enabling Wisdom Dynamically within Intertwined Torir: requisite resonance in global knowledge architecture, 2012; all images and animations below were prepared using Stella Polyhedron Navigator)
Selected faces transparent	All faces transparent

Possible use of this form as a means of interrelating codons is illustrated by the following.

Drilled truncated cube of 64 edges with random attribution of codon combinations
Animation with faces non-transparent	Animation with faces transparent

The juxtaposition above immediately raises the questions:

• how best to assign the elements into significant patterns in both cases
• whether mapping assignments in one case offer guidance and constraints for the other
• which of the preliminary assumptions made below should be called into question
• whether assignments could be better considered as dynamic rather than static (as suggested by resonance, and discussed below)

The above images are reminiscent of Rubik's Cube with all the widely appreciated challenges it has represented. Is it possible that a virtual form would enable the "movement" of the 64 edges of the drilled truncated cube (by recolouring them). It is appropriate to note that one variant of Rubik's original (3x3x3), known as Rubik's Revenge, has 64 coloured faces (4x4x4). Such a process could be reminiscent of the symbolic value associated with manipulation of circlets of beads (Designing Cultural Rosaries and Meaning Malas to Sustain Associations within the Pattern that Connects, 2000). A step in this direction is a game inspired by the genetic code, called Mutation, invented by Mark White and played on the surface of a sphere.