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#64 Correspondence between the five Platonic solids and the outer & inner forms of the 1-tree
When each of the F faces of a Platonic solid with V vertices and E edges is an Type A n-gon, V vertices and 2E
hexagonal yods line its edges. Turned into tetractyses, the nF sectors in its faces have (inside the n-gons) F
corners, 2nF hexagonal yods on their sides and nF hexagonal yods at their centres. The number of yods in the
faces of a Platonic solid = V + 2E +F + 2nF + nF = V + 2E + F + 3nF = 3E + 3nF + 2, using Euler's formula for a
simply-connected polyhedron:
V − E + F = 2.
"2" in the derived formula denotes two vertices at the opposite ends of an axis passing
through its centre. Edges can be regarded as sides of internal triangles, each with one corner located at the
centre of the Platonic solid. They may be regarded as Type A triangles, each having two hexagonal yods on a
side. This means that four hexagonal yods lie on the axis that passes through two opposite vertices and the
centre of the Platonic solid, for the axis consists of two of these internal sides. The number of yods in
either the faces or the axis that surround its centre = 3E + 3nF + 2 + 4 = 3E + 3nF + 6. The number of such
yods in the five Platonic solids = 3∑E + 3∑nF + 5×6. The five Platonic solids have 90 edges and 180 sectors.
Therefore, the number of yods surrounding their white centres = 3×90 + 3×180 + 30 = 840 = 10×84 =
(1+2+3+4)(1^{2}+3^{2}+5^{2}+7^{2}).
This number is the number of turns in an outer or inner half of each whorl of the UPA/subquark remote-viewed by
Besant & Leadbeater. It includes the 50 vertices and (2×90=180) hexagonal yods lining
the edges of the five Platonic solids, as well as the (5×4=20) hexagonal yods on their axes, i.e., 250 yods.
Within the edges of their faces are (840−250=590) yods. (590/2=295) such yods are in their upper halves and 295
yods are in their lower halves.
Suppose that the five Platonic solids are nested inside one another so that they share the same centre but none
of their vertices coincide — either with one another or with other yods. Their axes and faces contain 841 yods.
Ten blue vertices and one white yod (their centres) lie on their axes, which are surrounded by 240 red yods
lining their edges and 590 black yods (not shown in the diagram opposite for the sake of clarity) in their
faces. Compare this with the 1-tree and the (7+7) separate polygons that make up its inner form. When its 19
triangles are Type A, there are 10 blue Sephirothic points at their corners, one white yod located at Daath
(also a corner of triangles) and 240 red yods generated by the transformation of the triangles into Type
triangles. Each set of seven separate, Type A polygons contains 295 yods.
The following correspondences are established:
We see that the edges and axes of the five Platonic solids are the counterpart of the outer form of the 1-tree, the 10 hexagonal yods on their axes being analogous to the 10 Sephiroth, and that their faces correspond to its inner form. One may argue about whether it would be more appropriate (as far as the analogy is concerned) to make their shared centre correspond to Kether of the 1-tree rather than Daath because these points are the origin or source of the geometries in both cases. But the issue is irrelevant to what is being demonstrated here, namely, that the five nested Platonic solids are the regular polyhedral counterpart of the 1-tree and its inner form not merely because they contain the same number of yods (which is remarkable in itself) but because the pattern of distribution of these yods is analogous as well — a feature that discredits chance as a plausible explanation for why they should have exactly the same yod population.
The Pythagorean Decad determines the number 841 because 841 = 29^{2}, where 29 is the tenth prime number. How fitting it is that the structural parameter 840 of the 10-fold subquark superstring should be arithmetically determined by the tenth prime number! As 29^{2} = 1 + 3 + 5 + ... + 57, 840 is the sum of the first 28 odd integers after 1:840 = 3 + 5 + 7 + ... + 57.
The number of turns in the 10 helical whorls in the outer or inner half of the UPA, namely, 8400 (=840×10), is the sum of 280 odd integers, where 280 is the number value of Sandalphon, the Archangel of Malkuth. The total number of turns in the UPA = 16800 = 168×10×10, where 168 is the number value of Cholem Yesodoth, the Mundane Chakra of this Sephirah. The centre of a Platonic solid on average is surrounded by (840/5=168) yods in its faces & axes. This demonstrates how the superstring structural parameter 168 (the number of circularly polarized oscillations made by a whorl in a half-revolution about the spin axis of the UPA/subquark superstring) is embodied in the five Platonic solids. The UPA has the following analogies with the five Platonic solids:
Platonic solids UPA
5 types; 5 revolutions of each whorl about the axis of spin; 840 yods in axes & faces surround their centres; 840 helical turns in 5 half-revolutions of each whorl; 168 yods on average in axis & faces surround centre; 168 helical turns in a half-revolution of each whorl; 84 yods on average in axis & faces in half a Platonic solid surround centre; 84 helical turns in a quarter-revolution of each whorl. They exist because, just as the five Platonic solids have 10 halves, each with 84 yods on average, so each whorl of the UPA spirals five times around its axis, making 10 quarter-revolutions, each with 84 turns, in an outer or inner half.
The embodiment in the Platonic solids of the superstring structural parameter 840 is also discussed here. The embodiment in the first four Platonic solids of the structural parameter 1680 is analysed here and here. The average number of yods in their faces is 168; this is discussed here.