Association of the Szilassi polyhedron with cube inversion

Time for Provocative Mnemonic Aids to Systemic Connectivity? (Part #7)

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Szilassi polyhedron: The strange form of the Szilassi polyhedron offers another approach to 12-foldness. As shown below left, it has the highly unusual property that each face shares an edge with every other face. Its dual is the Császár polyhedron which has no diagonals, every pair of vertices being connected by an edge. Both polyhedra, having a central hole, bear a strange relationship to a torus. They invite consideration of their potential as mapping surfaces. The image on the right derives from mapping question pairs (Mapping of WH-questions with question-pairs onto the Szilassi polyhedron, 2014; Potential insights into the Szilassi configuration of WH-questions from 4D, 2014).

Szilassi polyhedron of 7 faces each in contact with the other
Rotation (reproduced from Wikipedia)	12 types of the 21 edges (highlighted by colour)	7 Pairs of vertices with indication of question-pairs (coloured by vertex pair, as with 12 edge types; edge lengths not to scale in this rotated perspective of the polyhedron, with faces transparent)
Reproduced from Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity (2017)

Animating a ring configuration of Szilassi polyhedra: Of particular interest in relation to the Schatz cube, is that the Szilassi polyhedron bears an intriguing similarity to the 6 polyhedral elements of which that cube is composed. 6 Szilassi polyhedra can be similarly linked by the extreme edges at right angles to each other (those above and below in the lleft hand image above). This configuration is show below with screen shots of an animation of the same complexity as that of the Schatz cube.

Screen shots of animation of 6 Szilassi polyhedra in circular Schatz linkage (wireframe rendering on right)
Sergey Bederov of Cortona3D substituted 6 Szilassi polyhedra for the 6 double terahedral elements in the Schatz cube, as originally enabled above with the formulae kindly provided by Charles Gunn. The animations of the Szilassi cycle alone include: interactive 3D variants (vrml; x3d); videos (solid mp4; wireframe mp4). The relation between the Schatz and Szilassi forms is further clarified by combining them into a single animation, an elegant extension by Sergey Bederov of that prepared for the Schatz cube alone (above, with the formulae and features of Charles Gunn), including: interactive 3D variants (vrml; x3d); videos (solid mp4; wireframe mp4) The VRML version of the Szilassi polyhedron was derived from Stella Polyhedron Navigator.
Technical note (Sergey Bederov): The animations with or without the Schatz cube are internally different from the early variant. That implementation of the Schatz cube closely followed the code provided by Charles Gunn, where the script was directly manipulating individual vertices of an IndexedFaceSet. This was natural for a low-level Java application, but turns out to be quite awkward in a powerful high-level environment such as VRML or X3D, because the triangles and colors are described in one place and vertices in another, it complicates the code and makes visual editing of shapes almost impossible. When envisaging the addition of the Szilassi polyhedron, this inconvenience became apparent with the realization that, in fact, each tetrahedron in the Schatz cube does not undergo any distortion, it only moves and rotates, and therefore it would be more natural to place static geometry inside a Transform and let the script merely calculate the Transform's translation and rotation. So the script has been modified to output translation and rotation instead of vertices' coordinates. Several different shapes were also added with a Switch and an IntegerSequencer to change them. Now it's much simpler to add new shapes because then can be added into the Switch, changing the IntegerSequencer animation. The Szilassi polyhedron had indeed to be a bit distorted. It required some rotation and non-uniform scaling. In fact, the Szilassi polyhedron is not a strictly defined geometric shape, it's rather a large family of polyhedra having the required topological qualities. Therefore, it can be scaled by different factors along different axes, and it will remain a Szilassi polyhedron. Moreover, it is possible to move individual faces and vertices as long as faces remain flat and keep the desired topological properties, and it will remain a Szilassi polyhedron. So it is indeed possible to fit the Szilassi polyhedron into the shape of the "smaller" Schatz cube tetrahedron, of which the initial, cube-shaped Schatz cube is composed.

Given the unique chracteristic of the Szilassi polyhedron, with its 7 faces touching each other, the characteristics of the unusual flexible linkage of 6 such forms calls for further study. It has 42 faces, 120 edges (given that 6 are common), 72 vertices (given that 12 are common). Of potential significance, in contrast with the double tetrahedra of which the Schatz cube is composed, the "hole" in the Szilassi polyhedron is an indication of what is required to enable all "faces" (in a discourse) to be in contact with one another.

Of related interest are the mapping possibilities offered by usse of the toroidal drilled truncated cube (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015; Relating configurative mappings of 64 I Ching conditions and 48 koans, 2012). With respect to the latter form, possibilities of interest are suggested by the following animation.

Drilled truncated cube of 64 edges
Animation with faces non-transparent	Screen shot of cyclic movement of parallels
Codons tentatively attributed to the structure for illustrative purposes.	Slower variant as video animation (.mov); access to X3D variant Faster variation as video animation (.mov); access to X3D variant
Animations prepared with the aid of Stella Polyhedron Navigator

Other variants of the animation on the right are accessible and discussed separately (Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization?, 2015)

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