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18. The disdyakis triacontahedron embodies the number of edges of the 4_{21} polytope
The 4_{21} polytope has 6720 edges. |
(A) 6720 yods other than vertices are needed to
construct the faces & interior of the disdyakis triacontahedron from Type A and Type B
triangles; |
There are two possible ways in which the faces and interior of the disdyakis triacontahedron can be constructed from Type A and Type B triangles in order for it to embody the number 6720:
A Type A triangle contains 19 yods. Nine yods line its sides and 10 yods lie inside its boundary (one corner & nine hexagonal yods in three tetractyses). A Type B triangle contains 46 yods. Nine line its sides and 37 are internal (four corners and 33 hexagonal yods in nine tetractyses).
Case A. Type B triangular faces & Type A internal triangles extended by edges
Faces: The 120 Type B triangles have (120×9=1080) tetractyses with
(120×4=480) internal corners and (120×33=3960) internal hexagonal yods. They have 60 external corners surrounding
an axis passing through two diametrically opposite vertices.
Edges: (180×2=360) hexagonal yods line 180 edges.
Interior: The 180 Type A internal triangles have 180 internal corners and (180×9=1620)
hexagonal yods. Their sides have (60×2=120) hexagonal yods.
The number of corners = 60 + 480 + 180 = 720. The number of hexagonal yods = 3960 + 360 + 1620 + 120 = 6060. The total number of yods surrounding the axis = 720 + 6060 = 6780. Therefore, (6780−60=6720) yods other than vertices surround the axis.
Case B. Type A triangular faces & Type A internal triangles extended by sides of sectors of faces
Faces: The 120 Type A triangles have (120×3=360) tetractyses with 120
internal corners and (120×9=1080) internal hexagonal yods. They have 60 external corners surrounding an axis
passing through two diametrically opposite vertices.
Edges: (180×2=360) hexagonal yods line 180 edges.
Interior: The 180 edges extend 180 Type A triangles with 180 internal corners of (180×3=540)
tetractyses and (180×9=1620) hexagonal yods. Their internal sides have (60×2=120) hexagonal yods. The (120×3=360)
sides of tetractyses in the faces extend 360 Type A triangles with 360 internal corners of (360×3=1080) tetractyses
and (360×9=3240) hexagonal yods. Their sides have (120×2=240) hexagonal yods as well as the 120 hexagonal yods
generated by sides joining the centre of the polyhedron to vertices.
The number of corners = 60 + 120 + 180 + 360 = 720. The number of hexagonal yods = 1080 + 360 + 1620 + 120 + 3240 + 240 = 6660. The total number of yods surrounding the axis = 720 + 6660 = 7380. Therefore, (6660+60=6720) yods other than new corners of tetractyses are needed to construct the disdyakis triacontahedron. This is how the disdyakis triacontahedron embodies the holistic parameter 6720 that manifests in the 4_{21} polytope as its 6720 edges.
It can be argued that Case A is illegitimate because it treats the exterior and interior triangles inconsistently, the faces being regarded as Type B triangles and interior triangles being treated as Type A. It would be more correct to say that such an analysis would be incomplete (and, therefore, misleading on its own) because it would be describing part of a holistic system constructed solely from Type B triangles. In Case B, all faces and interior triangles are of the same type, and so it is mathematically complete. For this reason, discussion will focus on this case alone.
Comments
2, 1, 3, 4, 7, 11, 18, 29, ....
where the nth Lucas number L_{n} = L_{n-1} + L_{n-2} if n>1. As L_{n} = Φ^{n} + (-Φ)^{-n}, where Φ is the Golden Ratio, 123 = Φ^{10} + Φ^{-10}. Assigning the sum of the tenth power of Φ and the tenth power of its reciprocal to the 60 yods surrounding the centre of a Type A decagon (symbol of the number 10) generates the number of yods needed to construct the disdyakis triacontahedron (observe that this is the number of yods in the polyhedron that surround its axis, which comprises the seven yods lining the two vertical sides shared by those triangles that are formed by the North & South poles of the polyhedron, their nearest neighbour vertices & its centre):
This demonstrates in a remarkable way how the Decad determines both arithmetically and geometrically the yod population of this polyhedral version of the Tree of Life. It also determines the number (720) of corners of all the tetractyses making up the disdyakis triacontahedron because this is the number of yods in a decagon with 2nd-order tetractyses as sectors:
* This Godname also prescribes the inner form of the Tree of Life in an analogous way because the (7+7) enfolded polygons have 153 corners & sides.
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