Framing crown chakra dynamics in relation to symmetrical polyhedra

Global Insight from Crown Chakra Dynamics in 3D? (Part #4)

Chiliagon and chiliahedron? Reference here to the "1,000-petalled lotus" from an Eastern tradition invites dismissal of this argument as ridiculous from a Western perspective. It is therefore very curious to note the extent to which a range of Western philosophers have referred to the little-known chiliagon or 1000-gon. This is a polygon with 1,000 sides. Such philosophers refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Because 1,000 = 2³ × 5³, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon is not a "constructible polygon". Geometrically it can be constructed as a truncated 500-gon, or a twice-truncated 250-gon, or a thrice-truncated 125-gon.

In its extensive discussion of the chiliagon, Wikipedia notes:

René Descartes uses the chiliagon as an example in his Sixth Meditation to demonstrate the difference between pure intellection and imagination. He says that, when one thinks of a chiliagon, he "does not imagine the thousand sides or see them as if they were present" before him – as he does when one imagines a triangle, for example. The imagination constructs a "confused representation," which is no different from that which it constructs of a myriagon (a polygon with ten thousand sides). However, he does clearly understand what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon.

The entry notes reference to the chiliagon by other philosophers, including Immanuel Kant, David Hume, Gottfried Leibniz, John Locke, Henri Poincaré -- thereby inspiring 20th century philosophers to make such reference. By contrast there is little reference to the "chiliahedron", namely a polyhedron bounded by 1,000 plane surfaces -- posing even greater problems of construction, although there is mathematical speculation in that regard, notably with respect to the subdivision of the surface of a sphere. There is also the possibility that some such form might be constructed in four dimensions (or more).

With respect to the "myriagon" of 10,000 sides, it is appropriate to recall the "ten thousand things" of objective reality featured in the much-cited first stanza of the Tao Te Ching of Chinese Taoist philosophy. Surprisingly, reference to "myriahedron" is now the focus of a new class of mapping methods for the globe (Jarke J. van Wijk, Unfolding the Earth: Myriahedral Projections, The Cartographic Journal, 45, 2008, 1). The methods described suggest other approaches to configuring a chiliahedron (James Lane Conkling, Tools to generate and project a myriahedral grid onto the globe, GitHub, 2020).

An approximation to a 1,000-sided polyhedron is possible by generating a geodesic sphere, as shown below. Another approach is to consider the 20-rings of the "1,000-petalled lotus" to be great circles of a polyhedron. Ironically "20 great circles" features in a patent for the design of the common golf ball (Process for designing dimple pattern of golf ball, US patent 20130102417A1, 24 October 2012).

Great circles are of particular relevance in that they have been fundamental to the development of global navigation, as discussed separately with respect to the spherical geometry of the so-called Pentagramma Myrificum (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

Platonic polyhedra? Exploration of the manner of configuring such a set of great circles features in the argument of Gerald de Jong (The Geometry of the Time Star). The latter consists of five regular tetrahedra, configured so that their vertexes are located at the centers of the twenty triangular faces of the icosahedron -- in either a left-handed or right-handed configuration. As described, this gives rise to 20 great circles -- which the author notes as reducing to 10 because of pairing.

In quest of a pattern of 20 great circles and a 1,000-sided chiliahedron
Time Star configurations		From Icosahedron to 980-faced Geodesic sphere (7 freq.)

From Gerald de Jong (The Geometry of the Time Star)		Made with Stella Polyhedron Navigator

Particular design advantage can be taken of the 20-ring configuration of the "1,000-petalled lotus" in the light of the 20-fold ordering of two of the most fundamental symmetrical polyhedra, namely the dodecahedron (of 20 vertexes) and its dual the icosahedron (of 20 faces).

Treating the 20 rings of the crown chakra as great circles, they could possibly be given a dynamic independent of either of those polyhedra. This design metaphor, and the dynamic, is of relevance given the current launching of a configuration of thousands of satellites as previously described (Satellite Constellation and Crown Chakra as Complementary Global Metaphors?, 2020).

Another approach is to give the rings cn a dynamic from the centre of the polyhedron to each vertex or to each face -- possibly reversed back through the centre to the opposing vertex or face. The rings then emerge from the centre or withdraw into it -- suggestive of other cognitive implications. There is the additional possibility that their diameter might be reduced to zero as they reach the centre.

Dynamics of 20 rings of crown chakra configured in relation to dodecahedron and icosahedron ("1,000-petalled lotus")
Ring movement in relation to 20 vertex axes of dodecahedron	Rotation of dodecahedral array with ring movement in relation to vertex axes	Ring movement in relation to 20 faces of icosahedron

interactive web -- x3d	interactive web -- x3d	interactive web -- x3d

An earlier argument treated the 10 dodecahedral axes through the vertexes as the basis for an emphasis on the emergence and withdrawal of "spikes", as these might be relevant to an understanding psychosocial processes (Spike-endowed Global Civilization as COVID-19: humanity "bristles" as the world "burns", 2020). The two visual metaphors might then be combined as indicated below -- with the possibility of adjusting those dynamics to either coordinate their movement, to render them contrapuntal, or random.

Dynamics of 20 rings of crown chakra configured in relation to dodecahedron and icosahedron -- with spike dynamics indicated on left
Dodecahedron alone (20 spikes through 20 vertexes)	Dodecahedron with spikes (20 spikes through 20 vertexes)	Icosahedron with spikes (20 spikes through 20 faces)

interactive web -- x3d	interactive web -- x3d	interactive web -- x3d

The "spike-based" argument for such a 20-fold configuration derived in part from the as yet unexplained (unconscious?) enthusiasm for that pattern in the articulation of strategic practice (Requisite 20-fold Articulation of Operative Insights? Checklist of web resources on 20 strategies, rules, methods and insights, 2018).

The fundamental nature of that pattern is evident in that of the 20 protein amino acids vital to human life, suggesting a speculative comparison in memetic terms (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). The basis for that pattern merits particular attention (A. L. Weber and S. L. Miller, Reasons for the occurrence of the twenty coded protein amino acids, Journal of Molecular Evolution, 17, 1981, 5, pp. 273-284).