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27 August 2018 | Draft

Time for Provocative Mnemonic Aids to Systemic Connectivity?

Possibilities of reconciling the "headless hearts" to the "heartless heads"

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Roman dodecahedron, Chinese puzzle balls and Rubik's Cube?
Interweaving disparate insights?
Inversion of the cube and related forms: configuring discourse otherwise?
Projective geometry of discourse: points, lines, frames and "hidden" perspectives
Eliciting the dynamics of the cube: reframing discourse dynamics
Association of the Szilassi polyhedron with cube inversion
Dynamics of discord anticipating the dynamics of concord
Associating significance with a dodecahedron
Increasing the dimensionality of the archetypal Round Table?
Necessity of encompassing a "hole" -- with a dodecameral mind?


The challenge of the "headless hearts" to the "heartless heads" was used in an earlier argument as a means of framing the response to the refugee/migration crisis (Systematic Humanitarian Blackmail via Aquarius? Confronting Europe with a humanitarian Trojan Horse, 2018). The conclusion pointed to the possibility of new mnemonic aids to discourse, as developed here -- possibly to be understood as an annex to that argument.

The earlier argument, with its exercise in "astrological systematics", derived from recognition of unexamined preferences for 12-fold strategic articulations (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011). Briefly it can be asserted that any such 12-fold pattern is typically developed without any effort to determine the systemic relations between the 12 elements so distinguished. The astrological system, however deprecated, is a major exception to this -- and therefore merits consideration, if only for that reason.

The surprising absence of systemic insight seemingly dates back to the failure to distinguish the relations between the 12 deities in the pantheons of Greece and Rome -- other than through well-known fables. Curiously this is as true of the 12 Apostles of Christianity, as of their analogues in other faiths. The omission is similarly evident in the case of the UN's set of Millennium Development Goals and its revision as the Sustainable Development Goals. They are "systemically handicapped".

There is therefore a case for discovering mnemonic aids to systemic comprehension, as previously argued (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007). There is a provocative challenge to mapping any 12-fold set within which the Aquarius function may deemed to be systemically embedded, aside from other exercises to that end, using computer technology and virtual reality (Visualization Enabling Integrative Conference Comprehension: global articulation of future-oriented 3D technology, 2018).

The concern here is to reframe the apparent tendency of the Aquarius modality to dominate the archetypal Round Table of humanitarian discourse -- as the only valid voice at the table. To remedy this, consideration is given to moving beyond the flat "table" in 2D to a "Global Table" -- from a table of static positions to a dynamic configuration of dancing patterns of discourse between the variously "heartless" and the variously "headless". Unfortunately the traditional Round Table is now overly reminiscent of a "flat Earth" mentality inappropriate to a global society.

An unusual contribution to this possibility is the dynamics associated with the Schatz cube inversion, as presented here. This may be understood as a way of moving beyond the static structure of the cube so characteristic of both the physical architecture within which people live and with the knowledge architecture fundamental to a global knowledge-based civilization. It suggest a new approach to visual representation of "thinking outside the box".

Roman dodecahedron, Chinese puzzle balls and Rubik's Cube?

One provocation derives from the mysterious function of the so-called Roman dodecahedron. This is a small hollow object made of bronze or stone (4-11 centimeters in diameter). Of dodecahedral form, it has twelve flat pentagonal faces, each face having a circular hole of varying diameter in the middle, the holes connecting to the hollow center. Over a hundred of these have been found across Europe, dating from the 2nd or 3rd centuries AD.

They have evoked speculation of every kind as to their function -- with no conclusion. They can be usefully compared with carved stones balls (petrospheres) dating from the Neolithic period, and with the Chinese carved ivory puzzle balls (Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds", 2015).

Roman dodecahedron Chinese ivory puzzle ball Neolithic carved stone ball
Romann dodecahedron Chinese ivory puzzle ball Neolithic carved stone ball
By Lokilech [GFDL, CC-BY-SA-3.0 or CC BY-SA 2.5 ],
from Wikimedia Commons
British Museum [CC BY 2.0 ],
via Wikimedia Commons
National Museums of Scotland,
via Wikimedia Commons

The absence of any explanatory closure would seem to give the form a similarly intriguing status to the many 12-fold sets of unrelated concepts -- and therefore to constitute a suitably tangible catalyst for reflection on systemic connectivity. How is the relation of the "headless hearts" to the "heartless heads" to be understood otherwise -- reframing the strident polarization noted in the earlier argument?

In that spirit it could be speculated that the Roman dodecahedron had a function analogous to Rubik's Cube -- as a hand-held puzzle inviting engagement. Separately the use of such a cube or its variants was explored as a means of reconciling systemically the UN's sets of goals (Interplay of Sustainable Development Goals through Rubik Cube Variations: engaging otherwise with what people find meaningful, 2017). One of its many variants, the Megamix, is of dodecahedral form; the Rubik 360 is spherical. Note also the comparison: Chinese Puzzle Balls: the Rubik’s Cube of the Ancient World (2012)

Aside from the many other speculations as to the use of such a device, Roman culture was necessarily especially familiar with both the 12-fold pattern of the Zodiac and the 12-fold pattern of the Dii Consentes, possibly arranged in the form of a banquet of the gods in gender balanced pairs (Lectisternium). Clearly a deity could be associated with each face of the dodecahedron (as a form of Denkmodell).

Interweaving disparate insights?

The conceptual challenge is who is positioned where? The holes in the device suggest that in a puzzle-solving mode, as with Rubik's Cube, the holes could be variously connected by passing one or more coloured threads between them. The prongs at the 20 vertices could be used to change the direction of threads to enable them to wind around the form in aesthetically appreciable patterns. The feasibility of their use for knitting has been variously demonstrated (Knitting with the Roman dodecahedron, YouTube, 1 July 2014; Martin Hallett, Has The Roman Dodecahedron Mystery Been Solved? YouTube, 3 June 2014). The possibility is somewhat reminiscent of the use of qhipu as the mode of communication in the Andean civilization, a pattern of knots now in process of decoding by the Harvard Khipu Database Project.

The approach can be understood as one of conceptual weaving, as can be otherwise argued (Warp and Weft of Future Governance: ninefold interweaving of incommensurable threads of discourse, 2010). As a means of evoking different stories, this then suggests a guide to tales regarding the systemic relationships between the deities and the value functions they represent. Plato is alleged to have indicated that the dodecahedron was the shape used for embroidering the constellations on the whole heaven. Necessarily involving both "heart" and "head", the 12-fold challenge would appear to be one of thematic interweaving (Interweaving Thematic Threads and Learning Pathways, 2010).

Curiously (as with the deities of Rome), there is little reference to the mapping of the 12 signs of the Zodiac onto the faces of a dodecahedron -- as might otherwise be expected. Arguably their functional relationship is not to be understood as a simple 3D mapping. There is however a rare image of a 12-sided dice indicating incompletely such a mapping -- possibly arbitrary. Roman culture may well have recognized any such mapping, including that of deities, to be dynamic in some way -- rather than static.

Further pointers to such a dynamic are provided by Paul Schatz (Rhythm Research and Technology: the evertible cube / polysomatic form-finding, 2013) as indicated by the following images.

Exploring a possible wave pattern relating zodiac functions on a dodecahedron
Wave pattern on dodecahedron Wave pattern on dodecahedron
Images of Paul Schatz (Verlag Freies Geistesleben, 1975, p.38) from Dieter A.W. Junker (The Zodiac Dodecahedron, flyping-games, 2004)
Original image on left modified with the addition of colours corresponding to those determined below

Inversion of the cube and related forms: configuring discourse otherwise?

Paul Schatz was seemingly frustrated by the conclusion of the above approach. It did however lead to a much more fruitful exercise with respect to inverting or everting the cube for which he is widely known, as illustrated by a number of videos (Charles Gunn, Schatz Cube Eversion, Vimeo, 25 April 2017; Daniel Wall, The Schatz Cube, or inverting cube, YouTube, 8 October 2010; Ryser Andreas, Invertible Cube, 30 March 2013; Invertible cube; Dolf Perenti, Inverting Cube, YouTube, 13 July 2011). Flexible card models are also marketed with commentaries (model; model) as with many wireframe models known as Hexyflex.

This approach is described by the Paul Schatz Foundation in the following terms:

On 29 November 1929, Paul Schatz set in motion the cube, the solid until then representative of everything that was earthbound, rigid -- static, even. In the course of his studies of the pentagonal dodecahedron, that to some degree most noble of the Platonic solids, he discovered that the cube can initially be subdivided into two stellated solids and one cube belt. It is the cube belt liberated from its interlocking parts that illustrates so impressively the dynamic qualities inherent in the cube. The model turns the richly varied movements of the cube into a tactile experience. Moreover, the pulsating movement inspired Paul Schatz to build several machines, of which the most famous are the Turbula, the Inversina and the Oloid.

The potential of this approach is consistent with that widely framed in terms of the need for "thinking outside the box". Reference to "inside the box" is considered analogous with the current, and often unnoticed, assumptions about a situation. The associated dynamics are consistent with arguments for fluidity in creative thinking (Douglas Hofstadter, Fluid Concepts and Creative Analogies: computer models of the fundamental mechanisms of thought, 1995). There is a case for recognizing the analogy implied by literal use of "the box" as a widely employed method of punitive solitary confinement, as vividly described by Shruti Ravindran (Twilight in the Box: what does solitary confinement do to the brain? Aeon, 27 February 2014). Widespread conventional dependence on a cuboid framework may well constitute an analogous form of a solitary confinement.

Other polyhedral inversions and mapping applications? The inversion movement recalls that which fascinated Buckminster Fuller with respect to the cuboctahedron of 12 vertices (H. F. Verheyen, The complete set of Jitterbug transformers and the analysis of their motion, Computers and Mathematics with Applications, 17, 1989). Many videos of that "jitterbug" movement exist (Buckminster Fuller's Jitterbug, YouTube, 5 May 2007; Joe Clinton, R. Buckminster Fuller's Jitterbug: its fascination and some challenges, YouTube, 15 March 2008; Bucky's "Jitterbug" -- Vector Equilibrium, 16 October 2008).

Projective geometry of discourse: points, lines, frames and "hidden" perspectives

Commonalities to discourse terminology: Typically missing from use of polyhedral mapping possibilities is the manner in which the contents of discourse get associated with the chosen geometry. It could be considered remarkable that the process of discourse uses terminology which is typical of descriptions of the geometry. Most obvious is reference to "making a point", a "line of argument", or the "frame of a discussion".

Any effort to transcend the polarization typical of discourse on refugees (the trigger for the current argument) merits reflection in that light. These issues can be explored more extensively (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009).

Locus of "values" in projective geometry? A primary characteristic of divisive and polarized discourse is associated with reference to values and contrasting perspectives with regard to them. It could be argued that the elusively implicit nature of values is usefully located by being "framed".

In the case of a regular polygon the configuration of positions of edges or lines frames a centre which is an implicit feature of the polygon. The point with which the centre is associated is not an explicit feature of the geometry. The same applies to a (semi)regular polyhedron, although in this case it is the "framing" provided by the configuration of the sides which locates the implicit centre.

Associating a primary value with such am elusive centre is consistent with reference to the "central" nature of values, and to any reference to "axis" -- implicit in its own way. Especially intriguing with respect to implicit values are those axes of symmetry which pass through features on opposite sides of the polyhedron and are typically associated with an implicit great circle.

With respect to a polyhedron in 3D, any regular polygons forming the sides can then each be understood as defining (or defined by) "secondary" values -- which together imply the primary value at the centre of the polyhedron. potentially far more intriguing is the even "subtler" values which are framed by polyhedra in 4D, and higher (4-polytopes and n-polytopes), also known as polychora. Their (virtual) existence, extensively studied, offers scope for further investigation in terms of their strategic implications, as separately discussed (Four-dimensional requisite for a time-bound global civilization?; Comprehending the shapes of time through four-dimensional uniform polychora; Five-fold ordering of strategic engagement with time, 2015).

Interplay of mapping possibilities: In the quest for a 12-fold pattern of significance, it is appropriate to note that the focus on the dodecahedron emphasizes a mapping on 12 faces, in contrast with the cube where such a mapping could be envisaged with respect to the 12 edges. Another approach, of considerable relevance to this argument, was taken from a cybernetic perspective by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994). That focused on the icosahedron of 12 vertices, as indicated with respect to videos of the resulting syntegration process (Team Syntegrity International, Syntegration -- for achieving solutions, YouTube, 3 November 2013; Olaf Brugman, Syntegration accelerates problem solving in complex systems: the case of Responsible Soy, YouTube, 27 May 2016 and text).

The argument here relates to the challenge of interrelating -- through the geometry -- the distinctive points and lines of discourse in order to honour the manner in which the discourse can be variously framed. Of particular importance, as in the case discussed above, is how different constituencies identify with different parts of the geometry in opposition to other portions (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018). The latter notes the relevance of geometry in 3D and 4D.

Dynamic mapping possibilities as "dancing patterns": It is useful to recall the properties of some of the polyhedra discussed here in order to recognize how the number of elements (with which mapping may be required) may be shifted by using one polyhedron rather than another. This is notably the case through transformation into the dual of the polyhedron -- as between dodecahedron and icosahedron.

Polyhedra potentially used for mapping purposes
(with indication in parentheses of the maximum number of features visible without rotation)
  Faces Edges Vertices
Cube 6 (3) 12 (6) 8 (4)
Dodecahedron 12 (6) 30 (15) 20 (10)
Icosahedron 20 (10) 30 (15) 12 (6)
Cuboctahedron 14 (7) 24 (12) 12 (6)
Schatz cube 24 (12) 30 (15) 12 (6)
Drilled truncated cube 32 (16) 64 (32) 32 (16)
Szilassi polyhedron 7 (of 4 types) 21 (12 types) 14 (7 types)

Of particular relevance to this argument is that endeavouring to view any of these forms limits recognition to a limited number of elements -- typically half in each case -- the others being "hidden", as with whatever is mapped onto them (unless the faces are rendered transparent, or the form is rotated). The sociopolitical and cognitive implications then merit consideration.

Many of the regular and semi-regular polyhedra can be transformed into one another through geometrical operations, most notably through duality. The following simplified map offers a sense of particular transformational pathways between patterns of order -- in which prime numbers appear to play a determining role as argued separately (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015, and annex of Changing Patterns using Transformation Pathways, 2015).

The colouring of the "routes" in the map serves to highlight pathways of contrasting significance. Arguably some of the features derive simply from design choices, although the degree of symmetry calls for future comment.

Map highlighting distinctive relationships pathways between spherically symmetrical polyhedra
(regular and semi-regular)
F=faces, E=edges, V=vertices (Total of these in parenthesis)
[Total reduced to prime number, other than 2, in square brackets]
Route maps of psychosocial life suggested bysymmetrical  polyhedra

The dancing metaphor with respect to such patterns is explored separately with respect to classic Chinese encoding systems (Sustainability through Magically Dancing Patterns: 8x8, 9x9, 19x19 -- I Ching, Tao Te Ching / T'ai Hsüan Ching, Wéiqí (Go), 2008). Those classics have the considerable merit in that they indicate both how content can be distinctively associated with the elements distinguished and how the relationships between the elements (with their content) can be understood systemically.

Is the dancing metaphor relevant to the switch from the 8-fold Millennium Development Goals of the UN to its current 16-fold Sustainable Developmet Goals (the 17th being the coordination of the 16)? In relation to this argument regarding the 12-fold, it could be considered curious that the former was 4 less than the 12-fold, and the latter is now 4 greater than the 12-fold, with the latter being twice the former.

Points, ball-games and "passing patterns"? Not only is "point" of value in discourse, it is also extremely evident in ball-games. Use of "ball" could even be considered as competitive play with a "point", or vice versa. "Line" is similarly important in relation to ball "passing patterns" (many videos), recognition of openings, and framing the boundary of a game. Technology is increasingly used in games, most obviously in football, and in relation to scoring. Perhaps the most extreme irony is that the basic design of an association football is that of a truncated icosahedron -- ironic in that the familiarity with the ball kicked around in games worldwide has in no way contributed to reframing global discourse more fruitfully.

It is indeed the case that divisive discourse is highly reminiscent of a competitive ball game with each endeavouring to score points and to downgrade the other within any ranking with other teams (Nature of the "ball" in game-playing and governance, 2016). The pattern is echoed in competitive point-scoring in parliamentary discourse. Curiously little effort is made in the latter case, and in the case of discourse more generally, to track points and passing patterns. Technology is limited to recording votes and voting patterns.

More fundamentally, there is a degree of irony to the manner in which the geometry of point and ball echo that of "global" -- especially in the case of discourse of global issues. To the extent that (semi)regular polyhedra are approximations to a sphere, the relevance to tracking global discourse (and its integrity) clearly merits exploration.

The argument can be further developed with respect to the contrast between bipolar discourse (as a fallback modality) with the multipolar discourse characteristic of a more complex global society. In particular it can be fruitfully asked why investigation of multi-sided games is so rare, as discussed separately (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016). It is appropriate to suggest that an unexplored reason for non-convergence of global discourse on the purported ideal of a unified perspective is the inherent interest in the dynamics of games -- whether as spectator, participant or gambler -- and familiarity with the default format of 2-sided games. People "like to watch" conflictual interaction -- to be understood in several connotations.

Given the familiarity with football, no effort has seemingly been made to experiment with four-team football, however three-sided football (also known as 3SF or d3fc) has indeed been developed. The three teams play over a hexagonal pitch (Geoff Andrews, The Three Sided Football Revolution: football's new idea, Philosophy Football, 9 June 2013; Sachin Nakrani, Three-sided football gives players something to think about. The Guardian, 7 May 2013; A game of three halves, Philosophy Football; see video and video and d3fc blog).

The point to be emphasized is that in each case the other side has a different "view" of the game. None see it whole in the absence of other forms of visualization and mapping.

Eliciting the dynamics of the cube: reframing discourse dynamics

The argument of Paul Schatz is especially relevant in that 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. 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 (see image below left).

The question is therefore whether the form that 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.

Cubical representation
of BaGua pattern of I Ching

Rotation of views of a phase
in inversion of cube
Animation of selected phases
in inversion of cube
Cubical representation  of BaGua pattern of I Ching Rotation of views of a phase  in inversion of cube Cube inversion animation
Reproduced from Z. D. Sung, The Symbols of Yi King or the Symbols of the Chinese Logic of Changes (1934, p. 12) Images derived from Charles Gunn (Schatz Cube Eversion, Vimeo, 25 April 2017)
with the assistance of the author; interactive vrml version of centre model adapted by Sergey Bederov (Cortona3D)

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 (solid and wireframe screen shot images) prior to inversion
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
Schatz cube inversion

Association of the Szilassi polyhedron with cube inversion

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
(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)
Rotation of Szilassi polyhedron Szilassi polyhedron 7 Pairs of vertices in Szilassi polyhedron with question-pairs
Reproduced from Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity (2017)

Animating a ring configuation 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)
Animation of 6 Szilassi polyhedra in circular Schatz linkage Animation of 6 Szilassi polyhedra in circular Schatz linkage Wireframe animation of 6 Szilassi polyhedra in circular Schatz linkage
animation of 6 Szilassi polyhedra in circular Schatz linkage

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 Scatz 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
Cyclic movement of parallels in drilled truncated cube
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)

Dynamics of discord anticipating the dynamics of concord

Mapping discord processes? Use of Szilassi polyhedra as indicative of the dynamics of concord, and the possibility of mapping its dynamics, usefully frames the possibility of mapping the dynamics of discord prior to any such concord. The unusual visual and mathematical frameworks elaborated by Charles Gunn and Sergey Bederov lend themselves to slight modification to explore conditions prior to the integration variously illustrated above.

The 6 Szilassi polyhedra that form the final pattern can be usefully recognized as contrasting frames of references through which the requisite variety of a higher order of integration is comprehended. Indicative in this respect is the work of Edward de Bono (Six Frames For Thinking About Information, 2008; Six Thinking Hats: an essential approach to business management, 1985). Somewhat appropriate to this argument, the former primarily uses polygons (and a heart) as mnemonic aids. In any discourse the question is how these frames of references are "oriented" to one another in the dialogue process.

Prior to their reconciliation, such frames of references "dancing" around each other can also be understood as territories or intellectual properties for those identified with the perspective each offers. This recalls a provocative implication of fundamental physics (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). To the extent that any frame of reference implies a form of arrogance (especially when presented with gravitas), the provocation can be taken further (Understanding models otherwise -- as centres of "gravity"; Arrogance as an analogue to gravity -- equally fundamental and mysterious, 2015).

"Disorientation" and discord? Through modification of the Szilassi depiction and the degree of orientation of one to the other, visualization of the resultant dynamics offers a means of elaborating different "stories" regarding the discord process anticipating any potential concord. For that experiment, the 21 edges (of 12 types) of the polyhedron can be rendered visually explicit, as well as the 14 vertices -- and potentially distinctively coloured -- in contrast the animations above. Especially interesting is the visualization of the dynamics between the 6 polyhedra when they are more or less "disoriented" to one another -- in contrast to the images above where their interlocking is a feature of the model and its dynamics. The degrees of disorientation, and the associated dynamics, then offer a valuable "language" for illustrating visually the complex processes in the quest for concord between distinctive frames of reference -- or in resisting it.

Whereas the dynamics of the model above derive from linking the 6 Szilassi polyhedra by their edges at right angles to each other, consideration could also be given to achieving a form of collective bonding by linking the longer edges. As a distinctive basis for interlocking frames of reference into a circular configuration, the result could probably only be rigid. It would however frame a circular tunnel around that configuration through the holes of the polyhedra so joined. Rather than a circular configuration, the polyhedra linked by the longer edges could constitute a form of serpentine chain -- another understanding of consensus. Consideration could also be given to interlocking the polyhedra by their matching faces.

Of particular interest from a discourse mapping perspective are the distinctions that could be associated with the 14 vertices of a Szilassi polyhedron in constituting a frame of reference -- the reference "points" by which it is framed (notably as key points in an argument). When bonded through the Schatz linkage illustrated above, each frame of reference shares 4 of its vertices with the two contiguous polyhedra -- 2 for each. Thus 10 vertices ("points") remain unique to each polyhedron, whereas in the configuration of 6 Szilassi polyhedra as a whole, each is effectively associated with 12 distinctive vertices in a pattern of 72. It is the pattern of 72 "points" which defines concord in a larger and more fundamental sense -- articulated in greater detail.

Theocracies: angels and demons? However controversial, the leadership of the world's acknowledged superpower is now recognized to be intimately associated with an evangelical perspective (Evangelicals’ White House meetings illegal, church-state watchdog says, Religion News Service, 31 August 2018). Evangelism is understandably associated with an angelic worldview (Angel, Baker's Evangelical Dictionary of Biblical Theology; What does the Evangelical Church teach about Angels? Spokane FAVS; Saints and Angels, Evangelical Times, 2010). The pattern in the US is more general (Religiosity reigns in US, on the wane in Western Europe, Deutsche Welle, 7 September 2018). A very specific case can be argued (David Ray Griffin, The American Trajectory: Divine or Demonic? 2018).

The Abrahamic religions have traditionally cultivated a belief in angels. It is however somewhat extraordinary to note the extent to which the leading governments in these times, notably the Permanent Members of the UN Security Council (with one exception), could be understood to be "theocracies" of a kind -- or more accurately, to be under the influence of Christian religions to an unquestionable degree. Whilst a number of countries have political parties using "Christian" in their names, the case of the UK is perhaps the most obvious in that a significant proportion of the unelected members of the House of Lords are bishops of the Church of England. As noted by Polly Toynbee:

Our 26 bishops in the House of Lords seem a quaint anachronism compared with Iran’s ayatollahs, but only Iran and the UK are still theocratic, with faith in their legislature. (The culture of respect for religion has gone too far, The Guardian, 28 August 2018).

The divisive conflicts of the world could then be understood as engendered together with governments similarly unwilling to disassociate themselves from other Abrahamic religions. The governments of all these countries are effectively unable to undertake initiatives contrary to the dominant faiths of their countries. This is especially the case when the strength of such influence may frame decisions to engage violently with the "evil" so widely declared to exist (Evil Rules, 2015; Existence of evil as authoritatively claimed to be an overriding strategic concern, 2016).

To whatever degree this may be appreciated or regretted, there is clearly a need to find a means of engaging with that worldview rather than assuming naively that it can be dismissed as meaningless.

Patterns of angels and demons? Deprecated or not, much has been made in the past of the number of angels and the manner in which they are ordered. Evangelism offers relatively little insight into the manner in which their qualities are interrelated or how they function systemically together. Given the extent of the conflicts engendered with religious complicity, there is a case for exploring their theologies in a proactive manner as ordered emanations of any comprehension of unity, as argued separately (Mathematical Theology: Future Science of Confidence in Belief, 2011).

Given the tendency to frame "problems" with "evil", potentially intriguing is the possibility that the "fallen angels" cast out of heaven could be discussed through visualization of the dynamics of disagreement and "hellish" disassociation from concord. "Problematique" as "Demonique"? Can configurations of problems be recognized as configurations of demons -- and a challenge to visualization (Transcending the wicked problem engendered by projecting negativity elsewhere, 2015)? In the quest for dominon over the Earth, has Genesis 1-28 been paradoxically misinterpreted: Be fruitful and divide, rather than Be fruitful, and multiply? (Risk-enhancing Cognitive Implications of the Basic Mathematical Operations, 2013).

In the surreal condition of global society, the pattern of 72 "points of light" -- framed by the dynamic pattern of Szilassi polyhedra -- offers one language through which engagement with such "hyperreality" might be discussed, if only in metaphorical terms (Engaging with Hyperreality through Demonique and Angelique? 2016; Mnemonic clues to 72 modes of viable system failure from a demonic pattern language, 2016; "Angelique": evangelisation of the resolutique in the light of angelology?, 2016). As discussed with respect to Hyperbolic reframing of the Demonique and Angelique of tradition (2016):

The focus here on 72 constitutes an interesting challenge to the conventional articulations of sustainability through sets of factors of a much more limited size and a far lesser degree of systemic organization. Mathematically 72 has particular properties, notably as 23 x 32, which suggest that the set as a whole might then be especially conducive to comprehension. Clearly smaller sets may be even more conducive, but without necessarily offering the requisite variety for understanding the dynamic nature of "heaven" -- or sustainability. This suggests exploration of optimum set size as a balance between comprehensibility and distinguishable variety. The size of some circlets of prayer beads as mnemonic aids extends to 108, for example...

The traditional set of 72 is described in the Wikipedia entry on Shemhamphorasch with reference to a "hidden name of God" in Kabbalah (including Christian and Hermetic variants), and in some more mainstream Jewish discourses. It is composed of either 4, 12, 22, 42, or 72 letters (or triads of letters), the last version being the most common as the 72-fold angels of the Shemhamphorash. It is derived from Exodus 14:19-21 read boustrophedonically to produce 72 names of three letters.

Screen shots of animations with 6 Szilassi polyhedra variously oriented to each other
(showing 84 angelic "points of light", but which are visible from what perspective?)
Animations created through slight modification of those produced above by Sergey Bederov and Charles Gunn based on the Schatz linkgage
Interactive 3D variants (vrml; x3d); videos (solid mp4 ; wireframe mp4).

Emergence of fruitful dialogue? Beyond the difficulties of the disorientation visualized above, a different design metaphor may be used for for representing a dialogue process. As frames of reference the 6 Szilassi polyhedra can be presented as rotating each on their own vertical axis within a ring -- in cognitive terms, each "within its own world". Getting any dialogue into a coherent pattern is of course already a major challenge (implicit in the visualization programming). The dynamics of the configuration can then be used to imply that at particular moments the communication between the frames "flows" -- exemplifying a higher order of ("magical") coherence for that dialogue process -- whereas at others that possibility is "blocked".

Indication of various dysfunctional conditions in dialogue
(particular "points" could be represented with disproportionate size to others, with all changes in size occurring dynamically)
Excessive extension of
"lines of "argument"
Conflation of two frames
of references
Inversion of some frames
relative to others
Representation of dysfunctional conditions in dialogue Representation of dysfunctional conditions in dialogue Representation of dysfunctional conditions in dialogue

In the following preliminary experiments, the "flow" between frames is indicated by distinctive circular rings (as illustrated below). The emergence of such moments could be indicated more clearly in improvements to the timing of the dynamics -- notably to be associated with "alignment" of particular vertices, whether at the top of the Szilassi polyhedra, or the bottom. Such alignment could be presented as bondng, reducing the total number of vertices to 72. When otherwise aligned, those vertices framing the holes in the polyhedra could similarly frame the larger ring when they are aligned -- a ring absent when they are not. This metaphor offers a sense of the systemic function of 84 vertices (whether or not 12 are to be excluded as not contributing actively to any coherence).

Screen shots of experimental animations of rotation of 6 Szilassi polyhedra oriented to each other in a ring
(showing 84 angelic "points of light"; but which are visible from what perspective -- given the absence of transparency of the wireframe version on right)
Rotation of 6 Szilassi polyhedra oriented to each other in a ring Rotation of 6 Szilassi polyhedra oriented to each other in a ring Rotation of 6 Szilassi polyhedra oriented to each other in a ring Rotation of 6 Szilassi polyhedra oriented to each other in a ring
Interactive 3D variants (vrml; x3d); videos (solid mp4; wireframe mp4).

Increasing the rate of the animation offers a visualization reminiscent of the experiments of Nikola Tesla with rotating electromagnetic fields -- an inspiration for the fruitful operation of dialogue (Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014).

Discourse as projective geometry? As argued, the strange numerical relationships between the polyhedra noted above provide a transformative framework for various stories about how to dance between configurations of perspectives. As noted, there is a certain charm to the common terminology of geometry and discourse -- points, lines, frames, and the like. It is remarkable that both discourse and geometry refer to "sides" or "faces" -- namely what is framed by a 2D polygon as the sides of a polyhedron. The challenge in divisive discourse is one of reconciling sides. Much is then made of "face", especially in the personalization of discourse and in the need for "face-to-face" encounters, and in the consequences of "loss of face" the means of avoiding it. It is intriguing that as the number of faces of a polyhedron increases -- approximating more closely to a sphere -- that immediately neighbouring faces are less readily visible from any given face (as in conference seating).

Especially provocative for the dyslexic is the relation between "angels" and "angles" -- as a confluence of perspectives and linear pathways. Formally, an angle is formed by two rays, termed the sides of the angle, sharing a common endpoint, termed the vertex. Through the transformations of polyhedra, the number of angles increases or decreases in an ordered way -- dancing curiously in relation to any 12-fold pattern. Faces and sides are characteristically oriented or angled with respect to the centre of a polyhedron.

More provocative is the deprecation of debate in the past on "the number of angels which can dance on a pin head". The phrase is now used to deprecate certain modes of discourse -- but with little indication of more frutiful alternatives. Curiously, however, it is pins which may be used in mapping ("bullet") points made during some dialogues -- possibly with relational lines between them. The question might then be reframed in terms of the number of such lines which can meaningfully converge on a point. There are constraints to this in the geometry of polyhedra. Cognitively these constraints can even be considered in the light of the much-cited study of George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956).

Relation of 7-fold discourse to 12-fold dscourse? Much hs been made of the optimum size of a working group -- consistent with the "magical number seven". It is therefore of interest to use the unique combination of 7-fold and 12-fold characteristics of the Szilassi polyhedron to take the argument further. It is indeed the case that 7 Szilassi polyhedra (rather than the 6 used above) can be configured in an unexpected way to form an unusual structure indictive of the kind of coherence that might be expected of 7-fold discourse. This is illustrated below.

Screen shots of animations of 7 Szilassi polyhedra forming a ring indicative of dialogue modalities
Interactive vrml; Video mp4 Interactive vrml; Video mp4

It is intriguing to note that although the necessarily static images are somewhat suggestive of the complexity of the dynamics, and that the videos offer further insight, the selection required for both images and video obscures further insights. These are more evident in the interactive virtual reality models, although these in turn are based on particular selections of parameters which can be readily modified in the programs for such models (speed, colouring, axes of rotation, and the like).

Associating significance with a dodecahedron

As discussed separately, efforts are made to associate logical significance with a variety of polyhedra (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018). Polyhedra may be used for experimental mappings of disparate concepts of which their use for configuring articles of various human rights charters are most relevant to the above argument (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008). In this respect it is appropriate to stress again that such sets of values are only otherwise configured as checklists lacking ny systemic organization. Indeed it could be asked how fundamental human values are organized other than in that manner -- if there is any consensus on what such values are.

The speculative challenge is whether the considerable interest in the pattern of the Zodiac, given the value-related meanings held to be associated with it, suggests the possibility of some such mapping, notably in the light of the preliminary exploration of Paul Schatz. One approach to the development of the argument gave rise to a variant of the image on the left below, in the light of the work on the Geometry of Meaning (1976) of Arthur Young, aa described separately (Geometry of meaning: an alchemical Rosetta Stone? 2013). This concluded a discussion of Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships (2013).

The pattern of the image on the left below can be seen as framing the question as to whether it can be understood as a 2D projection of a form in 3D (or more). In particular does its 12-foldness relate in any way to that of the cube -- especially as inverted by Schatz (as indicated above).

In the absence of any other indication as to how a set of 12 concepts or functions might be mapped onto a dodecahedron (as one logical option), use was made of an early dice which had such a mapping of signs of the Zodiac (Silberbeschlagener Würfel mit den Namen der Sternzeichen, gefunden Genf, 1982). This gave rise to the mapping in the image and animation (centre and right below). Images of 12-sided dice used for various purposes are offered by Wikipedia, although these do not include any with the Zodiac. There is some discussion of Board games with dodecahedral dice. Use of dodecahedral dice in relation to Dungeous and Dragons is the subject of an extended commentary.

The centre image is a development of that of Schatz (as reproduced above, to which the colours below were added). It derives from unwrapping the 3D dodecahedron into a 2D net. Of particular interest is that the similarly coloured pentagons correspond to opposite positions in the traditional configuration image on the left (to which the corresponding "measure formulae" have been added according to the argument of Arthur Young). He presented that configuration of learning cycles -- as a Rosetta stone of meaning.

Traditional pattern of Zodiac
with associated "measure formulae"
Zodiac signs on dodecahedral
net according to dice attribution
Animation of dodecahedron
with Zodiac signs from dice
Zodiac with measure formulae from  Arthur Young Zodiac signs on dodecahedral  net according to dice attribution Animation of dodecahedron with Zodiac signs from dice
Following the attributions of Artthur Young (1976) Animations generated using Stella Polyhedron Navigator

Curiously the signs which are opposite in the image on the left are contiguous as coloured in the central pattern (when folded into the 3D form on the right). There is however one exceptional set of opposites -- coloured turquoise. Although opposite in the image on the left, they are on opposite sides of the 3D version and are therefore not contiguous.

Sociophysics? Young's amendment highlights a progression in temporal reciprocation (or inversion) which features in an appropriately titled subsequent study (Nested Time, 2004). He successively distinguishes: production capacity, is effectively timeless as a product of 1/T0; change over time as a product of 1/T1 (or T-1); rate of change as a product of 1/T2 (or T-2); and a measure of control as a product of 1/T3 (or T--3).

This could now be explored within the controversial framework of sociophysics (divisive in its own right), usefully summarized by Frank Schweitzer (Sociophysics, Physics Today, 71, 2018). A commentary with respect to Young's presentation features in the work of Paris Arnopoulos (Sociophysics: Cosmos and Chaos in Nature and Culture, 1993):

Since power is the rate of applying force, controlling this rate is of utmost importance. Control has been identified as the capacity to modify the rate of change, ie to speed it up or slow it down. Therefore, power control is a necessary ingredient of any orderly social change. The mathematical definition of power, and its algebraic equivalents show that:

P = W/t = Fv = ma(s/t) = m(s/t2)(s/t) = ms(s/t3) = msc

This last parenthesis (s/t3) has been defined by Young as control (c), and translates as the rate of change of acceleration. It will be recalled that since v=s/t and a=s/t2, control becomes the third derivative of velocity....

Since power is directly proportional to the rate of energy conversion or information flow, dynamic systems require a great degree of control. As people become more energetic or informed, they tend to get out of control; so in order to avoid that, dynamic societies must become more regulated. It may therefore be said that the kind of government that a system has depends on the amount of power it disposes. (p. 82)

He develops the argument otherwise as a means of engaging with the Triple Helix model of innovation thesis (Braiding the Triadic Codex and Triple Helix: the sociophysics of nature-culture-nurture and academy-industry-polity, 2000). There he notes:

... this short paper interfaces with the triple helix paradigm by weaving its triadic social focus-locus with the power-wealth-data flows among its state-market-school centers. In this way we can concentrate on the most significant influence-finance-science transactions of the polity-industry-academy triangle.... In doing any job, force performs work: W = Fs =mas= mv2. This means that some work must be done in order to bring about social change. If that change is needed fast then one must exert a lot of power: P = W/t =mav = Fv. By this mathematical transformation, we have arrived at this crucial notion of power politics as well as physics. Social power however, unlike physical power, does not move inanimate objects but human masses to act far and fast.... Informative societies are negentropic because they increase systemic organization and decrease environmental degradation. Accumulating human knowledge also improves social control (C = a/t), since it regulates social change in a more enlightened manner. For that reason the exercise of responsible
social power requires strict political control (P = msC)....

Unusually Arnopoulos presents a synoptic overview in schematic form (p. 84) of the interrelationship between 15 fundamental concepts deriving from a triadic hypothesis (space-time-existence) correlating space curvature, material density and universal time (p. 5).

These suggest an interesting relationship to Young's 12-fold Rosetta stone of meaning, as depicted above (especially if 3 are omitted or conflated in some way). Otherwise it would take the form of three pentagons, thereby suggestive of arguments in relation to the Chinese understanding of the 5-fold Wu Xing (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012) and to the 15 transformations of Christopher Alexander, as discussed separately (Tentative adaptation of Alexander's 15 transformations to the psychosocial realm, 2010).

Tentative preliminary amendment of the dodecahedral configuration

using both Arthur Young (1976) and Paris Arnopoulos (2000)

Of relevance to the triadic hypothesis articulated by Arnopoulos is the subsequent argument of T.N. Palmer (The Invariant Set Hypothesis: a new geometric framework for the foundations of quantum theory and the role played by gravity, Electronic Notes in Theoretical Computer Science, 2011).

Also calling for integration with such insights is the extensive work on systematics by John Bennett (The Dramatic Universe, 1955-66), as introduced by Anthony Blake (Systematics, Duversity), notably with respect to 12-fold sets (Overview of 12 Systems).

Increasing the dimensionality of the archetypal Round Table?

The 12-seated Round Table of Arthurian legend, echoing that of the Last Supper of the Apostles, is readily associated with the distribution of signs of the Zodiac, as argued by Ralph Ellis (Arthur, his Round Table and the Zodiac, Passion for Fresh Ideas, 22 August 2014; Astrology and King Arthur, Passion for Fresh Ideas, 20 August 2014). Although the argument can be developed, it must be emphasized that it constitutes an interpretative exploration of myth and legend variously challenged by other variants of that narrative. This is especially clear with respect to the actual existence of such a Round Table, and the number of Knights of the Round Table.

As noted in the extensive Wikipedia entries, different stories had different numbers of Knights, ranging from only 12, through 24, 36, 72 to 150 -- a distinction being potentially made between "major knights" and others. As in the past, the myth of configuring synthesis, wisdom and transcendence continues to be cultivated through use of "Round Table". A curious feature of the term is that "table" implies two-dimensional flatness whereas "round" is ambiguous in obviously referring to its circular nature, but potentially implying the global roundness of a ball. The flat interpretation for a global society merits challenge, as argued separately (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008). Is there need for some form of "Global Table" recognizing the merits of both 3D and 2D perspectives?

Irrespective of the variations and the veracity of accounts, the point to be stressed is the influence of the 12-fold pattern in modern articulations of functions, most notably strategic patterns (as noted above). It is of course extremely unfortunate that it is only in the patterns of Ancient Rome and Greece that the 12-fold pattern included both genders. The images are also consistent with the over-emphasis on founding myths of the Christian culture -- implicit in use of "Round Table"..

Arthurian Round Table Depiction of Last Supper with Apostles Knights envisioning the Holy Grail
Arthurian Round Table Last Supper by Da Vinci Knights envisioning the Holy Grail
By Evrard d'Espinques
(Original at Bibliothèque nationale de France)
via Wikimedia Commons
Leonardo da Vinci
via Wikimedia Commons
Attributed to Maître des cleres femmes
via Wikimedia Commons

Rather than focus directly on the dodecahedron as a means of redistributing the associated functions, the approach here is is to consider a "table" of positions based on the cuboctahedrom and the dodecagon -- in 3D.

Cuboctahedron rather than dodecahedron? The 12 vertices of the cuboctahedron have been used as a means of distributing 12 of the 13 Archimedean polyhedra by Keith Critchlow (Order in Space: a design source book, 1969). He explores the relationship between the 5 Platonic forms and the Archimedean forms which are so fundamental to many conventional patterns (Examples of Integrated, Multi-set Concept Schemes: Annexes to Patterns of N-foldness, 1984). It is the representation of the configuration of them which is especially relevant to this argument -- notably the potentially controversial "reconciliation" of 12 and 13 implicit in the archetypal configurations.

The 13 distinct Archimedean polyhedra in which similar arrangements of regular, convex polygons of two or more different kinds meet at each vertex of the polyhedron [which can itself be circumscribed by a tetrahedron, with 4 common faces]. Such semi-regular polyhedra are defined by the fact that all their vertices lie on a circumscribing sphere. Critchlow configures 12 of them, within their circumscribing spheres, in a closest packing configuration around the circumscribing sphere of the 13th -- a truncated tetrahedron -- as shown below. The truncated tetrahedron is the only semi-regular solid with 12 independent axes passing through its vertices from its centre. Removal of the central sphere allows the 12 other spheres to close into a more compact icosahedral configuration.

Archimedean polyhedra

Successive truncations of octahedron
2, 3, 4-fold symmetry

Successive truncations of icosahedron
2, 3, 5-fold symmetry

  1. truncated octahedron (14 polygons: 4 / 6 sided)
  2. cuboctahedron / vector equilibrium (14: 3 / 4)
  3. truncated cuboctahedron (26: 4 / 6 / 8)
  4. snub cube (38: 3 / 4)
  5. rhombicuboctahedron (26: 3 / 4)
  6. truncated cube / hexahedron(14: 3 / 8)
  1. truncated icosahedron (32 polygons: 5 / 6 sided)
  2. icosidodecahedron (32: 3 / 5)
  3. truncated icosidodecahedron (62: 4 / 5 / 10)
  4. snub dodecahedron (92: 3 / 5)
  5. rhombicosidodecahedron (62: 3 / 4 / 5)
  6. truncated dodecahedron (32: 3 / 10)

truncated tetrahedron (8 polygons: 3 / 6 sided)

Arrangement of the 12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron
Arrows indicate the succession of truncations from 1 to 6 in each case.
Numbers as in the table above
Rotation of cuboctahedral array of 12 polyhedra
(around an omitted 13th at the centre; totalling 984 edges, 558 vertices, 452 faces)
12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron
(from Keith Critchlow, Order in Space, 1969, p. 39). Interactive virtual reality variant (.wrl)

Other variants of the animation on the right are accessible and discussed separately, including wireframe versions (Packing and unpacking of 12 semi-regular Archimedean polyhedra, 2015). The approach lends itself to exploration of an analogue to the Chinese puzzle balls cited above (Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds", 2015).

Nesting polyhedra: The Platonic and Archimedean polyhedra may also be dynamically "nested" within one another, as illustrated by other animations (Embodying Global Hegemony through a Sustaining Pattern of Discourse: cognitive challenge of dominion over all one surveys, 2015; Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra, 2015). The following animations depict the "collapse" of 12 distinctive Archimedean polyhedra into a common cuboctahedral centre.

Screen shots of animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre
(without indication of the 13th at the centre: the truncated tetrahedron)
Contextual cuboctahedron rendered partially transparent
Video animation (.mov); virtual reality (.wrl; .x3d)

Wireframe version with all faces transparent
Video animation (.mov); virtual reality (.wrl; .x3d)
Animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre Animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre
Animations prepared with the aid of Stella Polyhedron Navigator

Dodecagonal table in 3D? Given the implied 12-sided Round Table, consideration can be given to a dodecagonal table and its projection into 3D -- and what this might then imply for enhanced modes of discourse . The Archimedean polyhedra do not include any polyhedra with dodecagonal faces.

The following animations of unusual polyhedra, derived by further truncation from the truncated cube, were discovered in relation to the communication implications of great circles in connection with different polyhedra (Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity, 2017). However, in order to reproduce that configuration so as to explore the great circle process, it proved necessary to construct in 3D a cubic arrangement of dodecagonal faces (right-hand image below).

Animations of variants of truncated cube with dodecagonal faces Framework of dodecagonal faces
Animations of variants of truncated cube with dodecagonal faces Animations of variants of truncated cube with dodecagonal faces 3D Framework of dodecagonal faces
Reproduced with permission from The Truncated Cube, with Two Variations Featuring Regular Dodecagons (RobertLovesPi's blog, 2016) Constructed by use of Stella Polyhedron Navigator and X3D-Edit

Given the importance conventionally accorded to a 12-fold patterns of dialogue, most notably in round tables of the wise and in juries, the question explored by the great circle process was the potentially implied pattern of interactions. Three sets of 12 great circles were therefore applied to the dodecagonal framework. This was done as a possible prelude to introducing a 12-fold helical pattern as discussed in relation to the Triple Helix model of innovation and suggestions for a Quadruple and Quintuple variants (Embedding the triple helix in a spherical octahedron, Embedding the quadruple helix in a spherical cube, Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum, 2017). The more complex variants necessarily address strategic issues of greater complexity ( (Elias Carayannis and David F. J. Campbell, Triple Helix, Quadruple Helix and Quintuple Helix and How Do Knowledge, Innovation and the Environment Relate To Each Other? International Journal of Social Ecology and Sustainable Development, 1, 2012).

Arguably any singular modality, as implied by Aquarius in relation to complex humanitarian considerations, merits embedding in patterns of analogous complexity.

As indicated below, the 36 great circles create a complex interweaving pattern in their own right, possibly precluding addition of helical patterns (or implying them in some way). As to any emergent symbol, this might be better understood as taking a 3D form (rather than 2D, as in the cases above). Given that any of the Kepler-Poinsot star polyhedra could be considered too complex, a better symbol might be the 8-vertex compound of two tetrahedra (otherwise known as Stella Octangula), and discussed separately with respect to the Merkabah as a 3D variant of the Star of David (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017).

Successive addition of 36 great circles to dodecagonal-faced cubic framework (above-right)
Application of 1st set of 12 great circles Application of 2nd set of 12 great circles Application of 3rd set of 12 great circles
36 great circles to dodecagonal-faced cubic framework 36 great circles to dodecagonal-faced cubic framework 36 great circles to dodecagonal-faced cubic framework
36 great circles to dodecagonal-faced cubic framework 36 great circles to dodecagonal-faced cubic framework 36 great circles to dodecagonal-faced cubic framework
Patterns dynamically combining red / green / blue circles are shown in the animation. Interactive 3D versions: x3d; wrl/vrml. Video: mp4 (7mb)

Use of a dodecagonal-faced truncated cube pattern is especially interesting for mapping purposes in that 72 edges are subtended by the 36 great circles. However 8 of these edges are associated with two great circles, offering 64 edges for distinctive mapping. A further 24 edges are excluded from this encirclement. The pattern of 72 edges recalls the traditional symbols articulated as the contrasting qualities of the angelic order on the one hand, and the demonic order on the other, as discussed separately (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016; Variety of System Failures Engendered by Negligent Distinctions: mnemonic clues to 72 modes of viable system failure from a demonic pattern language, 2016).

Structurally consistent with the 3D structure of the dodecagonal configuration (based on the truncated cube) is that of the drilled truncated cube (discussed above), unique in its pattern of 64 edges (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). As discussed there, this offers a 3D mapping surface for the 64 distinctions made by the I Ching encoding or the genetic codon combinations.

Necessity of encompassing a "hole" -- with a dodecameral mind?

Significance of a "hole"? Of potential significance is the manner in which some form of hole may be depicted at the centre of the various round tables -- a hole which continues to feature in many summit seating configurations. This relates to consideration of a higher value being associated with the centre of a dodecahedron, of the drilled truncated cube, and of the dodecagonal configuration.

The mysterious nature of a hole is explored by Roberto Casati and Achille C. Varzi (Holes and Other Superficialities. 1994) and can be provocatively extended to current strategic challenges (Is the World View of a Holy Father Necessarily Full of Holes? Mysterious theological black holes engendering global crises, 2014).

The nature of a hole can also be explored in terms of "something missing" and "necessary incompleteness", as discussed with respect to the work of biological anthropologist Terrence Deacon (Incomplete Nature: how mind emerged from matter, 2012; The importance of what is missing, New Scientist, 26 November 2011). The argument can be explored with respect to the Szilassi polyhedron and its unfolding (Reframing nothing as a vital focus for sustainability, 2014).

The matter is of relevance to the question of any 13th presence at a table (such as at the Last Supper), but is usefully exemplified by the role of the truncated tetrahedron at the centre of the 12-fold configuration of Archimedean polyhedra. The tetrahedron may be similarly understood as at the centre of a 4-fold configuration of Platonic polyhedra. The map above indicating distinctive relationships pathways between spherically symmetrical polyhedra usefully highlights their distinctive positions -- and the manner in which the terahedron and the truncated tetrahedron are "transcendent".

Given the historical point of departure to this argument with the mysterious Roman dodecahedron (depicted above), there is an aesthetic irony to the possibility that the implication of configuration of any cognitive mapping device around a hole merits greater attention -- as an integrative necessity for 12-fold variety. The argument is potentially of greater relevance in the case of the significance attached to the Chinese puzzle balls -- especially in the depth of focus required to carve them. There is of the curious correspondence in English between "hole" and "whole", potentially to be understood in terms of "wholth" (Wholth as Sustaining Dynamic of Health and Wealth: cognitive dynamics sustaining the meta-pattern that connects, 2013).

Drilled truncated dodecahedron: On that basis, of potential interest is the manner in which each of the 12 positions can be understood as centred on a hole through the centre of a drilled truncated dodecahedron -- which appropriately reflects the structure of the Roman dodecahedron with its 12-holes of unknown functionality.

Animation of drilled truncated dodecahedron
Faces=260 (5 types); Edges=420 (8 types); Vertices=140 (3 types)
All faces non-transparent Outer faces transparent
Animation of drilled truncated cube Animation of drilled truncated cube
Animations generated using Stella Polyhedron Navigator

Again it should be stressed that this is an argument for any pattern of 12 distinctions, with the Zodiac serving primarily as a familiar example. Deprecated or not by some, it should also be emphasized that few if any 12-fold patterns currently articualated endeavour to be more than a checklist; they do not make any effort to articulate systemic relations among the 12.

Dynamics around a hole? Given the argument above with respect to achieving cognitive liberation from the static structure of the cube -- and "entrapment" therein -- it might then be asked whether the complex structure depicted above is simply "a better mousetrap". As a challenge of suggestive (if not provocative) design, several approaches to rendering such structures dynamic can be envisaged.

Screen shots of animation of 12 spheres in icosahedral configuration emerging from the drilled truncated dodecahedron
Icosahedral configuration of spheres  emerging  from the drilled truncated dodecahedron Icosahedral configuration of spheres emerging from the drilled truncated dodecahedron Icosahedral configuration of spheres emerging from the drilled truncated dodecahedron Icosahedral configuration of spheres emerging from the drilled truncated dodecahedron
Video mp4; Interactive variants of animation wrl, x3d (can be switched to wireframe rendering)
Animations generated using Stella Polyhedron Navigator

Beyond the use of colours to distinguish the pattern of 12, these may be appropriately replaced by the actual Archimedean polyhedra (not to scale), as shown below.

Screen shots of animation of 12 Archimedean polyhedra in icosahedral configuration emerging from the drilled truncated dodecahedron
Icosahedral configuration of Archimedean polyhedra spheres emerging from the drilled truncated dodecahedron Icosahedral configuration of Archimedean polyhedra spheres emerging from the drilled truncated dodecahedron Icosahedral configuration of Archimedean polyhedra spheres emerging from the drilled truncated dodecahedron Icosahedral configuration of Archimedean polyhedra spheres emerging from the drilled truncated dodecahedron
Video mp4; Interactive variants of animation wrl, x3d (can be switched to wireframe rendering)
Animations generated using Stella Polyhedron Navigator

Psychosocial implications of a "dodecameral mind"? Much is made of the bicameral nature of the human mind with its two hemispheres. Much is also made of two-hemisphere division of global society: East-West, North-South, "two cultures", and the like. The integration of such hemispheres is a challenge (Engendering Viable Global Futures through Hemispheric Integration: a radical challenge to individual imagination, 2014; Corpus Callosum of the Global Brain? Locating the integrative function within the world wide web, 2014).

In Greek mythology, the Twelve Olympians deities, were also collectively known as the Dodekatheon (Ian Rutherford, Canonizing the Pantheon: the Dodekatheon in Greek Religion and its Origins, 2010). In Plato's dialogue -- Phaedrus -- a relation between these 12 gods and the 12 signs of the Zodiac is recognized (Carlos Albuquerque, The Twelve Olympians in the Zodiac, 13 May 2013; Ken Gillman, Twelve Gods and Seven Planets, Considerations, 11, 1996, 4).

To the extent that those deities, as with the signs of the Zodiac, can be understood as external projections of a potential internal psychological organization, there is therefore also a case for imagining the operation of a "dodecameral mind" (Internalizing a "dodekatheon" to inform the "dodecameral mind", 2009). This is consistent with the approach explored by Thomas Moore (The Planets Within: the astrological psychology of Marsilio Ficino, 1993), as separately discussed (Composing the Present Moment: celebrating the insights of Marsilio Ficino interpreted by Thomas Moore, 2001). It is also consistent with the argument of Joseph Campbell (The Inner Reaches of Outer Space: metaphor as myth and as religion, 1986).

There is a case for recognizing the contrasting qualitative modalities as characteristic of multiple intelligences, as articulated by Howard Gardner: Frames of Mind: the theory of multiple intelligences (1984). These can be understood as modalities or worldviews of different "orientation", readily distinguished as cognitive or cultural biases (Systems of Categories Distinguishing Cultural Biases, 1993). Gardner has since articulated some 10 "intelligences", fulfilling 8 citeria, with tentative suggestions for "additional intelligences".

The relation to astrological types is the subject of commentary (The intelligence quotient of the zodiac natives, 8 May 2015). The theory is considered controversial, especially in the light of the prevailing theory of general intelligence and the lack of empirical evidence. Unfortunately for those deprecating the lack of evidence for such variety in favour of a singular intelligence, their preference has not engendered more fruitful approaches to the variety of perspectives which give rise to the intractable differences in discourse -- now effectively tearing global society apart. An attitude typical of "heartless heads"?

The Zodiac, as with any set of deities, might be usefully explored as a case study of the controversial interplay of cultural heritage, deprecation of the past, popular familiarity, and potentially simplisitic comprehensions of intelligence (proving "unfit for purpose" in terms of engaging with the challenges of society).

The number-based formalism of patterns of polyhedra suggest a more fruitful approach to mapping the variety of 12-fold strategic perspectives in play -- and to rendering them comprehensible through animations. Rather than deprecating patterns of insights because of their content or antiquity, greater insight could be derived from exploring the patterns in their own right, as argued separately (Representation, Comprehension and Communication of Sets: the Role of Number, 1978). Such exploration is consistent with the cognitive issues raised by George Lakoff and Rafael E. Nunez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).


Paris Arnopoulos:

Stafford Beer. Beyond Dispute: the invention of team syntegrity, 1994

A. G. E. Blake. The Intelligent Enneagram. Shambhala, 1996

Joseph Campbell. The Inner Reaches of Outer Space: metaphor as myth and as religion. Alfred van der Marck Editions, 1986 [summary]

Roberto Casati and Achille C. Varzi. Holes and Other Superficialities. MIT Press, 1994

Keith Critchlow. Order in Space: a design source book. Thames and Hudson, 1969

Edward de Bono:

J. François Gabriel. Beyond the Cube: the architecture of space frames and polyhedra. John Wiley, 1997

Serge Galam:

Howard Gardner. Frames of Mind: the theory of multiple intelligences. Heinemann, 1984

H.-C. Hege and K. Polthier. Mathematical Visualization: algorithms, applications and numerics. Springer, 2013

Douglas Hofstadter. Fluid Concepts and Creative Analogies: computer models of the fundamental mechanisms of thought. Harvester Wheatsheaf, 1995

Geert Hofstede. Culture's Consequences: international diffrences in work-related values. Sage, 1984

W. T. Jones:

George Lakoff and Rafael E. Nunez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000

Will McWhinney. Paths of Change: strategic choices for organizations and society. Sage, 1991

Thomas Moore. The Planets Within: the astrological psychology of Marsilio Ficino. Lindisfarne Books, 1993

Paul Schatz. Rhythm Research and Technology: the evertible cube / polysomatic form-finding. Niggli, 2013

Z. D. Sung. Symbols of Yi King -- or symbols of the Chinese logic of changes. China Modern Education, 1934

Arthur M. Young. Geometry of Meaning. Delacorte Press, 1976

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