Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization?

Psychosocial Implication in Polyhedral Animations in 3D (Part #3)

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]

The earlier document focused extensively on the value of the drilled truncated cube as a mapping surface, given its relatively unique characteristic amongst regular polyhedra of having 64 edges. As a mapping template, these could then be associated with the 64 conditions of change of the I Ching -- encoded in its 64 hexagrams and rendered memorable both by the notation and by distinctive metaphors. as separately discussed.

As a trigger to further reflection, the challenge presented was that of rendering memorable the pattern of 384 transformations between those conditions, as described separately (Transformation Metaphors derived experimentally from the Chinese Book of Changes (I Ching) -- for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997).

Such a visual rendering can at least be partially achieved by allowing the edges of the polyhedron -- understood as encoding conditions of potential change -- to move across the polyhedral template to other positions. This would then be indicative of one condition transforming into another -- as encoded in the pattern of that classical "Book of Changes". In the process, the integrity of the polyhedral pattern as a template is both decomposed and recomposed -- as indicated by the animations below in virtual reality.

The diagram on the left (below) was the basis for the distinctive colouring of the edges of the polyhedron in diagonally opposed clusters (as on the right). This gave rise to the following virtual reality representation

Drilled truncated cube -- a Stewart toroid with 64 edges (prepared using Stella Polyhedron Navigator)
Virtual reality variant	Virtual reality variant

Especially interesting is the manner in which the appropriateness (or viability) of transformations can be distinguished in terms of the parallelism between the source edge and the destination edge of a given movement. The parallelism is especially relevant to perception of the set of movements and their memorability.

Transformations distinguished in terms of parallelism in a cubic context
Inner cube movements (#1/2)	Outer cube movements (#3/4)	Framed movements
Access X3D variant	Access X3D variant	Access X3D variant
Framed inner cube movements (#1/2) Access X3D variant	Framed outer cube movements (#3/4) Access X3D variant	Access X3D variant

Perspectival parallelism was provisionally used to limit transformations to patterns of lines between which this obtained -- recognizing the manner in which the transformations then followed a distinctive cycle according to the type of transformation within the polyhedron. The following gives some indication of the range of lines moving to parallel positions. The exercise focused on those which do not move via the centre, or with respect to the implied diagonals of the inner cube. Whether or not they should be considered, the concerns are:

• how many movements can a given line make to parallel positions. Does each line have six parallels to which it can move on the array?
• how many kinds of moves are then to be recognized

How might these be detected systematically by appropriate maths? How do these relate to the 9 types of lines distinguished in the profile sheet of Stella Polyhedron Navigator from which the model was exported?

Drilled truncated cube coloured by edge type (numbered 0-8, but excluding reflections; generated by Stella Polyhedron Navigator)

Drilled truncated cube coloured by parallels (with indication of edge type, numbered 0-8; generated by Stella Polyhedron Navigator)

The following table identifies parallels of various edge types between which two-way movements could be assumed to occur.

The above table focuses on bidirectional movement between two parallels of the same or dissimilar types. Of potentially much greater interest, notably in terms of memorability, is the movement between a greater number of particular parallels -- thereby taking the form of a cycle of parallels. Selected cycles are presented below as they may effectively define the polyhedral form -- usefully understood as loops.

Of further relevance is that the direction of movement in each cycle may also be reversed. The table is selective because it raises the question of the possibility of a more systematic analysis of parallels to enable a complete set of cycles to be identified -- given the significance that might then be attributed to cycles of different types, and greater complexity, notably with respect to issues relating to chirality.

The following is one example combining cycles from the above table.

Alternative views of selected cycles of movement of parallels along edges of the drilled truncated cube Video version (.mp4); virtual reality (.x3d; .wrl)

Arguably the drilled truncated cube structure is the most compact (succinct) representation of a comprehensive set of patterns of change. Through the focus on parallelism as a symmetry effect, such effects serve to highlight comprehensibility and memorability, and to suggest correspondences of systemic significance (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). Given the objective of using the structure as a mapping device for the 64 conditions of change of the I Ching, to what extent can their attribution to the structure be rendered consistent with the transformations they respectively encode -- from one line position to another (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015; Relating configurative mappings of 64 I Ching conditions and 48 koans, 2012 ).

The issue is then to use visual triggers, most notably colour, to render such dynamic patterns comprehensible. The earlier document referred to previous analysis of the hexagram pattern in the light of the unit cube (below left).

Association of Ba Gua trigrams with unit cube and drilled truncated cube
Association of Ba Gua trigrams with unit cube (reproduced from Z. D. Sung, Symbols of Yi King, 1934)	Adaptation to drilled truncated cube of unit cube encoding (on left)

The patterns indicated are a step in the investigation of how each line in the drilled truncated cube -- with which a hexagram can be associated -- might be recognized as transforming into 6 other conditions indicated by lines parallel to it. How might the structure then be understood as encoding 6x64 transformations, namely 384 (or 2 ⁶, or 3 x 2 ⁷) ?

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]