Increasing the dimensionality of the archetypal Round Table?

Time for Provocative Mnemonic Aids to Systemic Connectivity? (Part #10)

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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
By Evrard d'Espinques (Original at Bibliothèque nationale de France) via Wikimedia Commons	Leonardo da Vinci via Wikimedia Commons	Attributed to Maitre 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.

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)
(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)
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
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
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.

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