Polyhedral configuration of questions

Now as the Ultimate Cognitive Strange Attractor (Part #7)

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The 2-dimensional configuration above naturally raises the question as to how the pattern might be configured in a more integrative manner in a 3-dimensional representation. For it to be "more integrative", the relations between the question-pairs need to be held in ways which offer memorable insight into mutually reinforcing links -- readily lost in 2 dimensions. The concern would then be whether the 3-dimensional configuration constitutes a more appropriate container for the sense of "now" in the process of "feeling alive". The emphasis is on the identification of more appropriately facilitative mnemonic aids, as separately discussed (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007).

There is a certain irony to the arguments in the main paper regarding emergence of a cognitive "container culture" -- potentially well-framed by Max Weber's above-mentioned "iron cage" of rationalization -- given the typically polyhedral rectangular form of conventional containers and the global dependence on them. In terms of this argument, are their 6 sides, 12 edges and 8 vertices indicative of an overriding preference for a particular mode of organization and mapping -- with unexamined cognitive implications. The issue here might be framed in terms of how to design more appropriate containers with more fruitful cognitive implications. Also of relevance is the question as to why this is not considered?

As is evident by inspection, the configuration above can be understood as embedded in what is known as the Heawood graph (portrayed below). This is an undirected graph with 14 vertices (of 7 types) and 21 edges (of 12 types). It is a toroidal graph, namely it can be embedded without crossings onto a torus. As shown in the animation below, one embedding of this type places its vertices and edges into three-dimensional Euclidean space as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus: the Szilassi polyhedron (one of the Stewart toroids). [Thanks are due to Heiner Benking for triggering research into this possibility and its relation to previous discussion of orbifolds (see below)]

Rather than associating each WH-question with a vertex (as in the 2-dimensional configuration above), each of the 7 WH-questions could then be associated with one of the 7 hexagonal faces -- of which there are 4 types (all of irregular shape). The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face -- of significance to graph colouring and map colouring. Its relatively comprehensible complexity, and the toroidal form, then offer a richer and more integrative framing of "now" in the light of the above argument. The dual to the Szilassi polyhedron is the Császár polyhedron, which has no diagonals; every pair of vertices is connected by an edge.

Although seemingly obscure, the Szilassi polyhedron figures as one of the works of monumental art -- with the Mobius Strip -- in Reconciliation Place (Canberra). This is located between the National Library and High Court of Australia, as a tribute to the Indigenous people of Australia.

Heawood graph (reproduced from Wikipedia)	Rotation of Szilassi polyhedron (reproduced from Wikipedia)

A readily accessible interactive version of the Szilassi polyhedron is available (Lajos Szilassi and Sándor Kabai, The Parametrized Szilassi Polyhedron, Wolfram Demonstrations Project, 2008).

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