Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra

Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity (Part #8)

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Given the successful framing of helices in the previous section, the question is by what might this other set of n-tuple helices be comprehensibly framed? To what ways and manner of thinking does their asymmetry correspond? Do they imply and embody a peculiarly unsuspected form of "cognitive twist"? (Cognitive "twist" 2007, Enantiodromia: cycling through the "cognitive twist" 2007).

Twelve-fold helix: The Archimedean polyhedra do not include any with dodecagonal faces by which application of the great circle process could be further explored. The following animations of unusual polyhedra, derived by further truncation from the truncated cube, were discovered. 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 as a possible prelude to introducing a 12-fold helical pattern.

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 considerd 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 dynamiclly 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, as based on the truncated cube, is that of the drilled truncated cube, 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 .

Drilled truncated cube of 64 edges
Screen shot (faces non-transparent)	Animation with faces transparent (random attribution of codon combinations)

the 12-pointed dodecagram ***

Eleven-fold helix? Examples of use of the 11-pointed star hendecagram are rare. They include the Topkapi Scroll with an 11-pointed star Girih form used in Islamic art. Potentially far more intriguing are implication of the 11-dimensional form of string theory. Curiously the extra dimensions resulting from compactification are rendered comprehensible by suggesting that the common recognized dimensions can be understood as a line which, if observed close up, takes the form of a tube with whose circularity those dimensions are associated. The "great circles" of this argument might then be fruitfully understood as helical toroids.

Nine-fold helix?: The 9-pointed star ennegram features most notably in the enneagram of personality, a model of human personality which uses an enneagram figure, and in the Fourth Way enneagram, a diagram used in the teachings of G. I. Gurdjieff and others (A. G. E. Blake, The Intelligent Enneagram, 1996). The 9-pointed star is a common symbol of the Baha'i Faith representing unity and Bahá'í. As indicated separately (Imagining the nature of cognitive "flight" in terms of the enneagram, 2014), the polyhedral implications have been noted by management cybernetician Stafford Beer through recognition of the association between the enneagram and the icosahedron. This he describes as emerging from collaboration with Joseph Truss -- in a chapter on The Dynamics of Icosahedral Space (Stafford Beer, Beyond Dispute: the invention of team syntegrity , 1994, pp. 196-209). For Beer:

But it is a matter of great interest that in the whole of the literature... the enneagram occurs as a plane figure. Nowhere had there been the slightest hint that a three-dimensional manifestation existed... No wonder the search took so long, given that the diagram was discovered spread across four vertical planes... The icosahedron is the actual origin of the enneagram... (p. 206)

Various efforts have been made to depict the enneagram in 3D -- readily available on You Tube (Francisco Meana, Enneagram from 3d perspective, 2007; Francisco Meana, Sufi Enneagram, 2009; Chuck Middaugh, 3D Enneagram MOD 9, 2013), The degree of relationship to the icosahedron is not especially evident. Beer provided no depiction, but this is offered in subsequent documents (J. Truss, et al, The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003; J. Baldwin, BuckyWorks: Buckminster Fuller's Ideas for Today, 1996, p. 220). The Beer/Truss argument is also discussed by Andrew Pickering (The Cybernetic Brain: sketches of another future, 2010).

Since those promoting syntegrity are especially sensitive to copyright issues, a different depiction is offered in the following generated with virtual reality software..

9-fold enneagram embedded within an icosahedron with addition of an indicative central sphere (constructed by manual modification of a virtual reality model of the icosahedron generated by Stella Polyhedron Navigator software)
View of enneagram associated with only one pattern of vertices of icosahedron (view in 3D with virtual reality plugin)	View of enneagram from left image with the icosahedron framework hidden (view in 3D with virtual reality plugin)
Note the colour coding and positioning of the icosahedral vertices -- which offer guidance when rotating the above models in virtual reality in order to render visible the enneagram pattern. Green-Magenta links of the enneagram are the only links embodied within icosahedral edges (on the left), where they are invisible. Three of the 12 vertices, positioned on the vertical axis (of the image on the right), do not form part of the 9-fold enneagram pattern (Red, Cyan and Black). The various possibilities for rotating the models in three dimensions affect the proportions of the enneagram as portrayed and the relative visibility of the Cyan and Black vertices.

Of some relevance is the experimental configuration of helical toroids, as described separately (Concordian Mandala as a Symbolic Nexus Insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016; Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala, 2016) ***

Seven-fold helix? The 7-pointed star has been valued for the complex, magical interpretation of its vertices and by alchemists to represent the 7 elements of the world and the seven planets originally known. Christians have used that star to represent the 7 days of creation, as a symbol of perfection by some sects, and as a means for warding off evil. In Islam, the heptagram is used to represent the first seven verses in the Quran; the Cherokee people use a 7 pointed star to symbolizes peace. It is also termed the elven star or fairy star in neopagan traditions and druidism. The flags of Australia and Jordan incorporate such a star. It features in a variety of insignia of police forces.

Despite its importance, or because of it, the seven-fold signals a break in pattern. In endeavouring to apply the great circle process to frame emergence, and the limitations of any heptahedron, one possibiity is use of a heptagrammic pyramid as shown below. In such cases any great circle subtends lines of the heptagram as well as the apex of the pyramid -- indicative of a unique approach to symmetry.

Variants of heptagrammic pyramid

If great circles are considered to be primarily associated with the axes of a polyhedron, considerable significance is attributed by R. Buckminster Fuller to the seven great circles generated by the axes of the fourteen faces of the cuboctahedron:

Synergetics also discloses the foldability of each of these seven great circles into local bow-tie-like patterns, which act as local-circuit shunts and are reassemblable into whole-sphere integrities. Totally assembled, they reconstitute the whole great-circles patterning of the completed spheres. They demonstrate thereby that these great circles may act as local information-shunting and -holding circuits. (Cosmography: A Posthumous Scenario for the Future of Humanity, The Estate of R. Buckminster Fuller, 1992 p. 67)

Such reference to the cuboctahedron is especially intriguing in that its geometric dual, the rhombic dodecahedron has been used (as noted below) as a pattern to order Boolean connectives. This raises the question of how the framing by great circles might be applied from that perspective -- as each would then pass through the apex of the pyramid.

The break by the seven-fold in the sequence of patterning introduced by the spherically symmetrical polyhedra (above) can be explored otherwise through the strange form of the Szilassi polyhedron with 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 (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
Rotation (reproduced from Wikipedia)	12 types of the 21 edges (highlighted by colour)	7 Pairs of vertices with indication of question-pairs (coloured by vertex pair, as with 12 edge types; edge lengths not to scale in this rotated perspective of the polyhedron, with faces transparent)

Just as the enneagram is curiously embedded within the icosahedron, there is the strange possibility that the twisted cognitive transition associated with the 7-fold may be embodied within the cuboctahedron/rhombic dodecahedron dynamic through the Szilassi/Császár dynamic -- and the hole that the latter frames. The relationship between their characteristics in variously embodying seven-ness are suggestive of this, as indicated below.

Symbolic embodiment of seven-ness?
	Polyhedra		Corresponding duals
	Cuboctahedron	Szilassi polyhedron	Rhombic dodecahedron	Császár polyhedron
Faces	14	7	12	14
Edges	24	21	24	21
Vertices	12	14	14	7

Morphing animation between cuboctahedron and rhombic dodecahedron

Produced with Stella Polyhedron Navigator

Of some symbolic relevance:

• A tradition found in the longer recension of 2 Enoch 27.3-4 speaks of God creating seven great "circles" in the "foundation of light" (April D. DeConick and Grant Adamson, Histories of the Hidden God: concealment and revelation in western gnostic, esoteric, and mystical traditions, Routledge, 2016, p. 35)
• In the Vedic tradition, the sun is understood as describings seven great circles of latitude on the earth during 12 months of a year, known as the seven major chhandas. (Ravi Prakash Arya Psychology in Yoga DarshanIndian, Foundtion for Vedic Science)
• Ironically perhaps, given its current political significance at the highest level, a particular form of golf ball design is based on a pattern of seven great circles which, in the form of a truncated octahedron, form six squares each of which breaks down into four equal isosceles triangles. (U.S. Pat. No. 4,762,326 to William Gobush).
  

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