## Tuesday, April 10, 2012

### Exploring the Dodecahedral Model

Per the work introduced at http://www.codefun.com (authored by Mark White, MD et al.), the current author has expanded and refined ideas relating to the Book of Changes.  Briefly, White invites his readers to reappraise the genetic code (and related constructs) as language based on shape, rather than through the conventional reductive viewpoint of nucleotides and amino acids.
Here, we adapt the dodecahedral model to the system of Change in an attempt to discern what insights or refinements to extant notions we might derive.  Hand-held models are dear to the author, particularly for the practical value they provide in exploring the systems thus represented.  This exploration proves no exception.  The model displayed below employs color to characterize the faces.  Each face is blazoned with McNeil notation (q.v.) to assist in diagramming the model.  The blazon allows for quick identification the attached vertices, face, and edges.

FACES
Faces stand for symbols (xiang).  Our symbol set is described by the letters [E W F A], standing for the four elements recognized by the ancients (earth, water, fire, and air). In the "alphabet" of Change, these would be represented by xiang di-grams [⚏ ⚎ ⚍ ⚌].  Dodecahedral modeling provides three faces per symbol, possibly to provide for each position in a sequence; that possibly being a defining characteristic of a symbol -- where it falls within a sequence.
We should like to investigate the possible significance of the pentagon as it relates to the symbols.  "Why five sides?"  Five does not often appear in the lexicon of Change.  The Chinese did have a theory of Five Elements (the four mentioned plus 'Metal').  We further observe that the maximal path length connecting any two vertices on the model is five.

VERTICES
the dodecahedral model of Change places hexagrams at the vertices, where faces (symbol triplets) converge.  Since the dodecahedron has twenty vertices, each vertex identifiable by a McNeil symbol triplet, with each of the twenty triplets having six possible symbol sequences, the model appears to allow for 120 possible hexagrams, providing for some redundancy.  In the discussion that follows,

Isochromous (same color) triplets (e.g. AAA) have only one distinct sequence; four such triplet sequences exist:
EEE    WWW    FFF    AAA

Heterochromous (all different colors) triplets (e.g. AFW) have six valid sequences; twenty-four such triplet sequences exist:
EWF    EFW    EWA    EAW    EFA    EAF
WEF    WFE    WEA    WAE    WFA    WAF
FEW    FWE    FWA    FAW    FEA    FAE
AEW    AWE    AEF    AFE    AWF    AFW

Dichromous (two colors) triplets (e.g. AAE) have only three valid sequences, one for each position of the odd symbol; thirty-six such triplet sequences exist:
EEW    EWE    WEE    EEF    EFE    FEE    EEA    EAE    AEE
WWE    WEW    EWW    WWF    WFW    FWW    WWA    WAW    AWW
FFE    FEF    EFF    FFW    FWF    WFF    FFA    FAF    AFF
AAE    AEA    EAA    AAW    AWA    WAA    AAF    AFA    FAA

Author Mark White, MD proposes that each nucleotide in a triplet is inherently distinct, and that all of the the six possible isochromous sequences, AeAfAw (subscripted for clarity) for example, are meaningfully different from one another.  Likewise treating heterochromous and dichromous triplets, we discover that 56 (120 - 64 = 56) "isotopic" sequences exist that are generally indistinguishable (unless subscripted) from one of the basic 64 triplet sequences.  We should note, however, that this inability to distinguish isotopic sequences derives from our abstraction of the sequence from the physical model and its orientation in space.

EDGES
Edges define faces, each edge held in common by two adjoining faces.  Edges also connect vertices.  Thirty distinct edges exist on the dodecahedron. Since vertices stand for hexagrams, edges act as paths connecting vertices.  This role suggests that an edge defines a change from one hexagram to another, further implying that the paths are directed, hence numbering sixty in total.  We have elsewhere noted that the hexagrams may be understood as forming thirty-two pairs, under at least two schema.  Efforts to integrate edge-as-pairing would have to account for the missing paths/edges.

ANGLES
It is arguable that the pentagon, with its vertex angle of 108° and central angle of 72°, defines the dodecahedron, as its adjoining faces meet at the same 108°angle.  A pentagram results when the vertices of a pentagon are connected.  The diagram at left shows that 36°angle is fundamental to the pentagon (and pentagram), thus to the dodecahedron.  In fact, the pentagonal net is fractal of a single 72-36-72 isosceles triangle.
If a side of the pentagon (purple) is taken as unity (1), a diagonal (blue) measures phi.  The red segment equals (phi -1), and the yellow segment equals (2 - phi).

The pentagon's relation to phi is worthy of mention in this section since the author had long-sought a relation between Change and phi, now obtained, albeit indirectly.