Current relevance of the simplest torus?

Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (Part #2)

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From the perspective of logical geometry, as noted previously, the 16 Boolean connectives are reduced to 14 -- enabling them to be configured on the vertices of a rhombic dodecahedron. Given the apparently fundamental significance of 16-fold patterns, whether in that respect, or in the standard model of particle physics, there is a case for exploring what polyhedra might be suitable (if only for mnemonic purposes) to display such a pattern -- rather than as a checklist, a matrix or in tabular format. Potentially this is of relevance to the 16 (+1) Sustainable Development Goals by which global governance is purportedly framed at this time. NB: For convenience, this introductory section is reproduced from the conclusion of the more extensive previous discussion.

It is therefore of interest to note what has been termed the "simplest torus", together with its dual -- as shown in the following animations. The dual appears not to have been named, although it bears a degree of relationship to what has been named as a star torus (itself distinct from the preoccupation of astronomers with the possibility of a "toroidal star" and toroidal magnetic fields). The dual, as explored through its variants below, could well be more appropriately named as a star torus.

Of particular interest is the less than obvious notion of "face" in each case, since some faces have two seemingly separate components when passing from inside the model to outside, or being contiguous (but not separate as appears). This is most evident from the colouring of the faces in both models. Each "double" face exhibiting such characteristics is coloured the same -- although parallel faces across the model may use the same colour. This confusing characteristic is clearest in the case of the simplest torus, rather than its dual.

Rotation of simplest torus (faces visible and transparent) (16 vertices and 12 faces: 4 hexagonal, 8 square)		Rotation of dual of simplest torus -- a "star torus" (faces visible and transparent) (12 vertices and 16 faces: 8 triangular, 8 square)
Animations prepared with Stella Polyhedron Navigator

These unusual forms then raise the question as to how they may be used to map 16-fold patterns, rather than the 14-fold discussed above. Most obviously, in the absence of any mapping worthy of the challenge to governance in the case of the 16 Sustainable Development Goals, what valuable counter-intuitive meaning is derived from the use of mapping surfaces which have an "inside" and an "outside" -- with such faces being continuous within the model (despite appearances). An initial attempt to apply the standard thumbnails for each of the 16 Goals to the relevant surfaces in the dual model proved problematic with the application package. Appropriately perhaps, the image was reversed when slid across the face from "outside" to "inside", for example.

A further clarification, using the thumbnail images of the 16 Goals, suggested the mapping in the following animations. Here it is to be noted that 2 opposing clusters of points (of the 4) are each used for 8 such mappings. The 2 intermediary clusters of points have faces which are extensions of those on the other 2 (as indicated by the similar colouring of the faces). As before, parallel faces are of the same colour. The thumbnails images are arbitrarily oriented. The inner surfaces of the torus do not seemingly have any images on them -- since those faces are an extension of those visible on the outside. These various subtleties invite recognition of the subtler relationships between the Goals and alternative mapping conventions -- perhaps with the images shifting between positions of the surface.

Animations showing experimental mapping of 16 UN Sustainable Development Goals onto 16 faces of dual of "Simplest Torus" (with 12 vertices)
"Vertical" rotation	"Horizontal" rotation
Animations prepared with Stella Polyhedron Navigator

Note: In exploring variants of the "star torus" in what follows, there is a distinction to be made from a variant form of self-crossing toroidal polyhedra, namely what are more widely recognized as crown polyhedra (or butterfly polyhedra). Also termed a stephanoid, this is a toroidal polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting and topologically self-dual. The construction of crown polyhedra is possible if prisms or antiprisms are used as a base -- the vertices of which are connected by the edges of crossed quadrilaterals, as discussed further below. Examples are given by Ulrich Mikloweit (Facetings of uniform polyhedra with crossed quadrilaterals, Polyedergarten). The models shown there were created by the same software used in the above animation.

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