Main menu
You are here
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] [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  This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License For further updates on this site, subscribe here |