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Higher dimensionality, polyhedral packing and transformation


Framing Cognitive Space for Higher Order Coherence (Part #5)

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Multidimensionality: This argument has further implications in the light of the assumptions too readily made regarding personal human experience of "globality" and "wholth" (Wholth as Sustaining Dynamic of Health and Wealth: cognitive dynamics sustaining the meta-pattern that connects, 2013).

To the extent that one assumes oneself to be "rounded", is this as a 2-sphere -- superficially, as with a bubble? Or is there an understanding of depth for which a 3-sphere would be more appropriate? More challenging -- if not inspiring -- is human identity better understood in terms of an N-sphere, with N being commensurate with the insights of physics? Such topological considerations could clarify the manner in which spherical experience of coherence could be transformed into toroidal experience -- with corresponding topological distinctions between 2-torus and N-torus.

Are human aspirations to "freedom" far more usefully recognized as being associated with such multidimensionality, as implied by arguments from various perspectives:

For some, speculation extends to the nature of "multidimensional humanity" and to "transdimensional humanity" (Alice Bryant and Linda Seebach, Multidimensional Potential of Human Beings).

Complexification through hexagram organization: There exist four traditional approaches to the organization of the 64 hexagrams of the I Ching as an 8x8 matrix, as presented and discussed separately (Classical Chinese Arrangements of 64 Hexagrams in Squares, 2008). The four patterns can be experimentally superimposed as indicative of alternation between frameworks, as presented separately (Fractal comprehension of coherence requiring an 8-fold uncertainty principle? 2019).

There are examples of a circular organization, as shown below left and discussed separately (Diagram of 384 Relationships between I Ching Hexagrams, 1983; Bagua and the sequence of 64 hexagrams, Shanghai Daily, 20 December 2015). One of these is named as the circle of Shao Yong (1011-1077) or the I Ching hexagram circle. It was an influential feature of the communication to Leibniz in 1701. A qualification to a prevailing conclusion by science (predictable as argued above) is offered by James A. Ryan:

Leibniz thought he had discovered evidence of a forgotten mathematical science in the Chinese past, and, in spite of his sinological knowledge, he never found evidence to the contrary. Thus, it has been left to contemporary scholars to explain the apparent correspondence. While the general trend in scholarship has rightly presumed that no forgotten mathematical science existed in ancient China, the conclusion that the Yijing / Binary System Episode was a mere coincidence has perhaps not satisfied scholars, in view of the intricacy of the binary geometrical progression and the temporal and spatial isolation of the Chinese Diagram from the European. (Leibniz' Binary System and Shao Yong's "Yijing", Philosophy East and West, 46, 1996, 1)

The pattern of 64 is relatively unique within the variety of polyhedra. However one interesting candidate for mapping purposes is the toroidal drilled truncated cube with 64 edges -- with which any set of 64 elements could be associated, as discussed separately (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). The issue is whether the manner in which they can be positioned on that framework constitutes a configuration which is meaningful in relation to particular cases, such as the hexagrams (or genetic codons). Furthermore, is it possible that known constraints in the patterning in such particular cases can together offer guidance in the attribution of the distinct elements -- of relevance to each case? The correspondence between hexagrams and genetic codons offers a provocative possibility that the hexagrams could be understood in terms of memetic codons, as notably argued by M. Pitkänen (Could one find a geometric realization for genetic and memetic codes? Semantic Scholar, 2013; Three new physics realizations of the genetic code and the role of dark matter in bio-systems, Semantic Scholar, 2018).

Preliminary experiments with the drilled truncated cube have been undertaken previously with respect to the hexagrams alone -- but only to get a sense of the possibility, as a "proof of concept" (Enabling Wisdom Dynamically within Intertwined Tori: Requisite resonance in global knowledge architecture, 2012).

Traditional circular configuration of hexagrams
(augmented with transformation pathways between them)		 Drilled truncated cube of 64 edges with random attribution of hexagram names
Selected faces transparent All faces transparent
Logo of Laetus in Praesens Drilled truncated cube of 64 edges with hexagram names Drilled truncated cube of 64 edges with hexagram names
Animated variant at Dynamic Exploration of Value Configurations (2008) Animations prepared using Stella Polyhedron Navigator

(Topology of a Renaissance "Stargate" of Higher Dimensionality: complementary ways of imagining engagement with otherness, 2018) ***

Nesting polyhedra and nested cubes: As noted above, the Fujitsu torus fusion memory organization involves a form of nesting, as shown in the image below left. Such nesting also features in the diamond cubic structure as shown in the animation below centre. The structure of the hypercube or tesseract (as discussed below) can be represented by nesting of cubes to a higher degree as shown in the animation below right -- where 7 cubes are nested within an eighth

Supercomputer memory organization
(detail from Tofu)		 Diamond cubic crystal structure
(animation)		 Nested cubes -- N-fold hypercube?
(rotation)
Detail of higher order sFujitsu supercomputer memory organization Animation of diamond cubic crystal structure Animation of 8-fold nested cubes as an N-fold hypercube
Detail from Fujitsu image above By MarinaVladivostok -- Own work Link  

Such a pattern invites reflection on yet another pattern for the periodic table of chemical elements of which there are some thousand indicated in the Internet Database of Periodic Tables. Of relevance however is the absence of nesting in such configurations with few exceptions (Tomás A. Carroll, Spherical and Russian Doll Formulations, 2008; Anthony Grainge, Elemental Periodicity formulation with concentric spheres intersecting orthogonal planes, 2019). Of particular relevance are recent arguments for a hypercube formulation (Ramon Carbó-Dorca and Tanmoy Chakraborty, Divagations about the periodic table: Boolean hypercube and quantum similarity connections, Journal of Computational Chemistry, 40, 2019, 30). This considers the possibility of a seven-dimensional Boolean hypercube.

The image below left suggests how the groups and periods might be related within a nested hypercube framework. It is however important to recall that such nesting is a representation of higher dimensionality. The image below centre shows another technique for the depiction of the four-dimensionality of a cube by projection into 3D. Another metaphor is offered by use of h polyhedra nested within one another, as discussed separately (Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra, 2015). Two cases are considered there Relative movement of nested Platonic polyhedra: pumping and rotation and Packing and unpacking of 12 semi-regular Archimedean polyhedra. The animation on the right is an illustration of the first.

24 cells
Nested Periodic Table of Chemical Elements
(Groups I to VIII in Periods 1 to 8)		 4D: Uniform polychoron from 3D vertex
24 cells, 96 faces, 96 edges, 24 vertices		 Nesting 5 Platonic polyhedra
"Pumping" motion (video mp4)
within Rhombic Triacontahedron (green)
Nested Periodic Table of Chemical Elements 4D: Uniform polychoron from 3D vertex Platonic polyhedra nested within Rhombic triacontahedron
Developed with X3D Edit and Stella Polyhedron Navigator

There are many references to nesting of cubes, including:

Of particular interest, given the emphasis here on comprehension of complexity, are references to the nested cube with respect to memory, as discussed in the Art of Memory Forum in relation to the technique of Gregor von Feinaigle (The New Art of Memory, 1812). There is also the sense in which links between the nested cubes can be recognized as directions of perception of inner or outer coherence. This is suggestive of the understanding associated with degrees of access, as cultivated with respect to classified or secret knowledge.

Inversion of cube: As discussed separately, so much of psychosocial organization is framed by the static architecture of the cube in 3D -- or through its compression into a square in 2D (Eliciting the dynamics of the cube: reframing discourse dynamics, 2018). This is the favoured modality for most explanatory tables. Through its 12-edges, the cube potentially offers clues to a relationship within any 12-fold pattern, but has not been extensively explored in that respect, although it is a feature of studies of oppositional logic, and a relationship to the 8-fold pattern valued in Chinese thinking (as discussed below). Missing are the paradoxical insights justifying reference to the Necker cube and the "4D" Klein bottle of relevance to this argument ((Steven M. Rosen, Topologies of the Flesh, 2006, Dreams, Death, Rebirth: a multimedia topological odyssey into alchemy's hidden dimensions, 2014).

The question is therefore whether the form that Paul Schatz extracted from the cube -- through the dynamics of its possible eversion -- offers indications of a way of transforming conventional preoccupation with its static form. The following images offer some indication of this.

Explorative 3D animations of the image on the left above are presented separately (Succinct mapping of multidimensional psychosocial dynamics? 2016). ****

In terms of the argument with respect to features hidden from the observer, this is especially evident in the case of the central image above. In that phase, the 24 sides are visible through the animation. But in the case of the static blue-green perspective or the static red-yellow perspective, only 12 sides are visible. Being hidden, the other 12 can only be inferred unless the structure was rendered transparent. In the reality of sociopolitical discourse opposing sides are never "transparent" to one another -- whatever the claims that are made. Cognitively each could be interpreted as a form of shadow for the other in the Jungian sense. The wireframe image on the right is indicative of the commercial product widely marketed as Hexyflex.

Schatz cube prior to inversion   Rotation of views of a phase
in inversion of cube 		Animation of selected phases
in inversion of cube
Schatz cube inversion Sergey Bederov of Cortona3D has produced an interactive vrml version of the complete cycle of the original, with formulae kindly provided by Charles Gunn.
Thanks to both.
See video of the complete cycle 		Rotation of views of a phase in inversion of cube Cube inversion animation
       

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