Relative movement of nested Platonic polyhedra: pumping and rotation

Psychosocial Implication in Polyhedral Animations in 3D (Part #2)

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The following two images were presented together in the earlier document as a means of drawing attention to the possibility of new insight into systemic relationships -- as proved so inspiring with the early image of Johannes Kepler, and continues to do so. The pattern indicated by him is now recognized to be fundamentally incorrect from a variety of perspectives. The point to be made, however, is that there are few patterns of that degree of integrity -- of relevance to current global psychosocial organization. Considered as a transitional exercise, the question is whether analogues to Kepler's model could be usefully explored at this time. Arguably the image that has recently offered some degree of equivalent symbolic inspiration is that of the Earth, articulated geographically, as photographed from the Moon.

Nesting 5 Platonic polyhedra: octahedron, icosahedron, dodecahedron, tetrahedron, cube
Rhombic Triacontahedron (green) as a nesting framework (virtual reality variants static: vrml or x3d; mutual rotation: vrml or x3d; "pumping": vrml or x3d; videos: "pumping" mp4; "rotation" mp4)	Polyhedral model of solar system of Johannes Kepler on Mysterium Cosmographicum (1596)
Developed with X3D Edit and Stella Polyhedron Navigator	Reproduced from Wikipedia entry

The significance of the distinct elements of the image on the left (above) were discussed as systemic holding patterns in the earlier document. Links are provided to facilities with which that pattern can be interactively explored in virtual reality as a potential source of inspiration.

Kepler found that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another, thereby producing six layers, corresponding to the six known planets known at that time -- Mercury, Venus, Earth, Mars, Jupiter, and Saturn. By ordering the solids correctly--octahedron, icosahedron, dodecahedron, tetrahedron, cube.The spheres could then be placed at intervals corresponding (within the accuracy limits of available astronomical observations) to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not precise enough.

This is recognized as a beautiful astronomical model. For example, explaining why there were only six planets: How could there be a seventh planet, when Euclid proved that there are only five Platonic solids! Of course, the model is completely false, the interplanetary distances it predicts are not sufficiently accurate, and Kepler was scientist enough to accept this eventually. But it is considered an excellent example of how truth and beauty are not always equivalent.

The potential significance can be further enhanced by introducing a dynamic into the image itself, moving beyond reliance on static configurations, as criticized in the earlier document -- whatever the degree to which dynamics may be implied (as in Kepler's model). As argued there, the dynamics which might be associated with the distinctive polyhedra -- especially relative to one another -- are then suggestive as templates of new levels of systemic insight.

Screen shots of a virtual reality rendering of an expansion/contraction ("pumping") dynamic Version as video animation (.mov); access to X3D variant

The earlier document was especially focused on a response to the current incoherence of global discourse with respect to governance, as has been remarkably demonstrated with respect to a variety of issues (climate change, resource overshoot, refugees, environmental degradation, nuclear power, and the like). As argued there, oversimplistic understandings of coherence and consensus may well call for more complex patterns, given the implications for governability (The Consensus Delusion: mysterious attractor undermining global civilization as currently imagined, 2011; Ungovernability of Sustainable Global Democracy? 2010).

Exploratory animations of tetrahedral morphing -- whether to be understood as distinctively 2-fold, 4-fold or 8-fold
Animations prepared using Stella Polyhedron Navigator

Further insights can be suggested by rotating the polyhedra around the common axis at different rates, rather than using that axis to guide the "pumping" motion indicated above

Relative rotation of nested Platonic polyhedra around a common axis

Version as video animation (.mov); version in virtual reality (.wrl)

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