Imagining further implications: helical coil, toroidal knot or crown?

Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (Part #13)

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]

---

Reminiscent of an archetypal crown? The argument was introduced with reference to the Knights of the Round table, engendered according to legend by King Arthur -- as a symbolic echo of the Last Supper. The crown symbol is implied both with respect to the strategic role of Arthur and to the tragic irony of the Crown of Thorns immediately following the Last Supper. The latter has come to be associated ambiguously with a Crown of Martyrdom and a Crown of Immortality (Mike Aquilina, The Crown of Martyrdom, Catholic Exchange)

The dual of the simplest torus -- the star torus -- could be seen as recalling the form of the simplest crown, whether adorned with "jewels" (at the vertices of the form), or their recognition as "thorns".

A crown can be understood as indicative of the strategic integration with which royalty has been traditionally associated as the integrative focus of governance. It is traditionally "packed" with symbolic elements -- whose nature is readily forgotten -- even long forgotten, following the deprecation of that function in a secular world. The metaphor continues to be valued through the process of "crowning" those appointed to positions of leadership -- or through the laurel wreath -- in recognition of excellence (Game-playing, bull-leaping and laurel wreaths, 2014). The emblem of the United Nations, for example, continues to recall this.

There is therefore a case for decoding any toroidal crown or circlet in terms of the integrative strategic significance it can carry, as separately argued (Engaging with Globality through Cognitive Crowns, 2009; Implication of Toroidal Transformation of the Crown of Thorns: design challenge to enable integrative comprehension of global dynamics, 2011). Given reference to "Knights", it is appropriate to note the analysis of the strategically relevant "knight's move" of chess in the latter (Predictability and pattern-breaking: the Knight's move, 2011).

The point was made above that the star torus as discussed here is with some difficulty to be distinguished from the so-called crown polyhedra. To further clarify the distinction, the following are presented. A principal distinction is that the crown polyhedra are self-intersecting and self-dual in contrast with the star torus, as noted above. Further exploration might establish, for that reason alone, that crown polyhedra (being more noble in geometric terms) could indeed be more appropriate for mapping a set of strategies.

Indications of crown polyhedra as distinguished from forms of the star torus
4-fold	4-fold (faces transparent)	Rotation of 4-fold	6-fold
Models variously produced with the aid of Antiprism, Meshlab, X3D-Edit and Stella Polyhedron Navigator

Helical coil or toroidal knot? First inspection of the simplest torus in its dual form raises the question of whether the pattern of edges forms a circular coil. Use of software to explore the interior of its toroidal "tunnel" offers the illusion of traversing a 3-walled tube. There are in fact 4 interlaced lines of edges (suggestive of being interwoven) -- as rendered obvious by some images above. The question remains as the extent to which the pattern constitutes a form of coil or knot of the simplest kind -- whether in topological terms or as an effect of constrained perception.

Possibilities for consideration include:

• Understood as a kind of helical coil, this recalls current preoccupation with the Triple Helix model of innovation, as discussed separately (Embedding the triple helix in a spherical octahedron, 2017; Contrasting the implications of "triple helix" -- cognitive and otherwise, 2017). The interlaced rhombic forms of the star torus (shown above) suggest the extension of this approach to such patterns beyond the 4-fold, possibly in the light of the experimental approach to nonagons (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)
  
  
• Given that work on the Triple Helix model has inspired exploration of a quadruple variant of strategic relevance, the interweaving of the four rhombic forms could be considered in that light (Systemic closure: fourth helix -- and beyond?, 2017; Embedding the quadruple helix in a spherical cube, 2017).
  
  
• Given current references to the strategic challenge of the times as meriting comparison with the riddle of the Gordian knot, could the polyhedral form be recognized as such a knot (Mapping grossness: Gordian knot of governance as a Discordian mandala? 2016; Engaging globally with knots and riddles -- Gordian and otherwise, 2018).
  
  
• The paradoxical nature of such a knot might be usefully explored in the light of the study by Douglas Hofstadter (I Am a Strange Loop, 2007). Rather than "I" being such a loop, does the world merit recognition as a "strange loop" -- if only in terms of the many vicious cycles of violence, and the inability to encompass cognitively and strategically the cyclicity of the adaptive cycle?Â

Toroidal dynamics? It is potentially remarkable that the form of a torus is so significant to astrophysical dynamics, whether with respect to toroidal stars, planets, or the shape of the universe. This should suggest an approach to understanding the "shape of civilization" other than as simply global. More immediately intriguing is the possibility that strategic mappings onto an N-fold star torus could derive insights into the dynamics of governance from the extensive studies of astrophysical models. Curiously, with a significant shift of time scale, the spherical planet through its annual cycle could well be perceived as of toroidal form -- if slightly elliptical.

There is a degree of irony to such possibiities in that the consideration of N-fold sets of strategies at global summits (G7, G20, and the like) could be understood from a toroidal perspective -- given both the circular seating arrangements and a caricaturization of their outcomes as taking the form of smoke rings. The irony is all the greater in that participants are framed as the "stars" of the global system -- typically faced with a strategic "back hole" (at least in financial terms) around which they gravitate.

Dynamics of strategic round tables as suggested by stellar dynamics?
Magnetic field from a charged rotating torus	Section of a rotating toroidal black hole	Rotating electrically charged torus
Reproduced from Robert Spivey, Quasars: a supermassive rotating toroidal black hole interpretation, Monthly Notices of the Royal Astronomical Society, 316, 2000, 4)

Such dynamics merit consideration in the light of the remarkable insights of Nikola Tesla into the rotation of magnetic fields (Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014). Related insights can be sought from the design constraints of the International Thermonuclear Experimental Reactor currently under construction (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006). Consideration could notably be given to the fact that 18 toroidal field coils are considered appropriate to the containment of plasma in order to achieve fusion. It is highly probable that 18 were chosen on the basis of very careful investigation of the dynamics of the system -- a degree of understanding unrecognizable in the case of the sets of gobal strategic goals currently on the table.

Toroidal "wave" implication? The pattern of faces and edges around the dual of the simplest torus could be explored as indicative of one or more spiral waves interweaving around the form -- as such emblematic of cognitive dimensions of strategic integration. This could follow from the implications of the study by Alexander Wendt (Quantum Mind and Social Science: unifying physical and social ontology, 2015), as discussed separately (Quantum consciousness implications of fundamental symbol patterns, 2017; On being "walking wave functions" in terms of quantum consciousness? 2017).

In emphasizing dynamics rather than statics, does global strategic integration call for a much more extensive exploration of wave patterns -- of which a toroidal form might be a particular approximation (Encountering Otherness as a Waveform -- in the light of a wave theory of being, 2013; Being a Waveform of Potential as an Experiential Choice: emergent dynamic qualities of identity and integrity, 2013). One justification is the insightful use of "wave" in relation to public opinion and change -- "waves of change"?

Cognitive implications of aesthetic options? The remarkable opportunities offered by software include facilities to modify the aesthetics of visual renderings -- possibly interactively -- suggesting a variety of alternative ways of understanding any pattern of strategies. Examples are illustrated by the following. That on the left is a variant of the crown polyhedron presented above, but with faces rendered one-sided, as is particularly evident on rotation. The others are variants of a 16-fold model presented above, exaggerating features reminiscent of a fortress of times past.

Indications of aesthetic options of cognitive significance for representation of strategic configurations
4-fold (one-sided faces)	Alternative representations of 16 Sustainable Development Goals
Video of rotation (mp4)	Modifications made with facilities of Stella Polyhedron Navigator

Hexagram and tetragram encoding? Of further interest is the manner in which the 6 great circles could be "turned on" or "turned off" in an animation , by emphasizing the lighting or rendering them partially transparent. This technique could instead be variously applied to the 4 rhombic forms with which 4 of the great circles are associated. The traditional lines of a hexagram would thereby be converted into stacked cylindrical rings -- if not stacked tori.

This would give the option of encoding the 2⁴ possibilities which are a feature of 16-fold organization. Extending the approach to the green circles would of course then give 2⁶ possibilities, namely the pattern held by the 64 hexagrams of the I Ching. Such a combination of 8-foldness is necessarily consistent with auspicious appreciation in Chinese culture.

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]