Polyhedral catalysts of global imagination

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #6)

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The following images display the truncated icosahedron from the animation above. The 5-pointed star and 6-pointed star are however displayed in white backgrounds with an alternative colouring scheme. The remaining images show various geometrical transformations of the form using the features of the Stella Polyhedron Navigator. The purpose is to encourage imaginative reflection through alternative windows on the relationship between the 5-fold and the 6-fold -- as it may apply in particular to the Middle East. As with the various map projections of the globe, the images are included to provoke the question as to what they might variously offer, if anything, as a new way of seeing the relationships there.

Truncated icosahedron (folded)	Truncated icosahedron (partially unfolded)

The polyhedral dual of the truncated icosahedron is the pentakis dodecahedron into which it may be transformed by various morphing processes illustrated by the following images. Especially intriguing is the distinctive emergence of the triangle. If considered representative of Christianity, as the pentagon and hexagon are associated here with Islam and Judaism respectively, these highlight the potential of a polyhedral pattern language to explore the cognitive challenges of Jerusalem and the holiness for wqhich it is esteemed -- irrespective of the political implications. However the dynamics of the transformation through morphing -- constrained by the geometry -- suggest that the "language" needs to be understood dynamically, rather than with respect to any particular static extreme. This argument relates to that with respect to resonance, as explored below.

Morphing to dual by sizing	Morphing to dual by truncation

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Morphing to dual by expansion (stage 1)	Morphing to dual by expansion (stage 1)

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Morphing of transparent variant (example 1)	Morphing of transparent variant (example 2)

The implications of any morphing process can also be explored from the perspective that the truncated icosahedron is a form of "compromise" between an embedded dodecahedron with 20 vertices (touching the 20 hexagonal faces) and an embedded icosahedron with 12 vertices (touching the 12 pentagonal faces). With one or other emerging dynamically to a greater degree through morphing, this recalls the importance attached by R. Buckminster Fuller to a related form, the cuboctahedron, which he described as being a vector equilibirum because of the "pumping" process  through which it could be transformed (Vector Equilibrium and its Transformation Pathways, 1980).

Selected faces of the truncated icosahedron can also be "augmented" as illustrated by the following pair of images.

Augmentation of pentagonal faces with pyramid	Augmentation of hexagonal faces with prism

The truncated icosahedron may be transformed into a variety of geodesic spheres -- according to the frequency, namely the number of subdivisions along each edge of that original polyhedron.

Transformation into geodesic sphere (frequency 3)	Transformation into geodesic sphere (frequency 5)

The original truncated icosahedron may also be transformed into a zonohedron according to various rules, as illustrated by the following.

Zonohedrification (method 1)	Zonohedrification (method 2)

The original truncated icosahedron may be subjection to a stellation process, namely producing a new polyhedron with faces that lie in the same planes as the faces of that original. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. There are 1117 stellations identified by Stella Polyhedron Navigator of which particular examples are given below.

Stellation (over 12 pentagons)	Stellation (over 20 hexagons)

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Stellation (all cells above 3-fold axis)	Stellation (all cells above 5-fold axis)

In addition to the final product of the stellation process (shown below), various transformations of the original truncated icosahedron are possible within Stella Polyhedron Navigator in order to give a "glimpse" of their structure projected in four dimensions. One of these is presented below.

Final valid stellation (of 1117)	Elaboration of four-dimensional prism

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