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**10. The correspondence between the 24-cell and the seven
separate Type B polygons**

The number of corners, sides & triangles in a Type B n-gon = 10n + 1, where "1" denotes its
centre. The number of corners = 2n. The number of sides = 5n. The number of triangles = 3n. The number of corners
& sides = 7n. The number of corners & sides of the 144 triangles surrounding the centres of the seven Type
B polygons with **48** corners = ∑7n = 7∑n
= 7×**48** = 336:

They comprise 96 corners and 240 sides, the latter being a parameter of holistic systems that
manifests in the 24-cell as its 240 vertices, edges, faces & octahedral cells. This demonstrates *par
excellence* the holistic character of the number 336, in particular, the Tree of Life nature of the
24-cell, which is composed of 336 geometrical elements. The seven polygons can be divided into two sets in two
possible ways so as to generate the **168**:**168** division that other sacred
geometries exhibit:

triangle+square+pentagon+dodecagon + hexagon+octagon+decagon

triangle+pentagon+hexagon+decagon + square+octagon+dodecagon

Each set displays the 24:24 division of the **48** corners of the seven
polygons, a division that is also part of the universal pattern displayed by holistic systems (e.g., the 24 Yang
lines and 24 Yin lines of the eight diagonal hexagrams in the 8×8 square array of
**64** hexagrams). One of the combinations in either possibility displays the further 84:84
division exhibited by sacred geometries that embody the holistic parameter **168**. The numbers of the
types of geometrical elements in the 24-cell are: 24, 96, 96, 24 & 96. As none of them is divisible by 7 (the
number of corners & sides per sector of a polygon), nor any combination thereof, no subset of polygons can have
a number of corners & sides equal to any of these numbers. This means that each of the numbers characterising
the geometry of the 24-cell must correspond to either all corners, all sides or partial combinations of both. Given
that all these numbers are integer multiples of 24 — the number of sectors of each set of polygons — it seems
natural to associate the two sets of polygons, each with **168** corners & sides, with the
two halves of the 24-cell, each having **168** geometrical elements. Indeed, sacred geometries
embodying the number 336 always have two "halves" (not necessarily physical), each embodying this number; the
**168** turns in a half-revolution of a whorl of the UPA are the superstring example of this
fundamental division. However, this leads to a problem, namely, neither possibility shown above has
*both* sets comprising polygons with 12 corners and 12 sides to match the 12 vertices and 12 internal
sides in each half of the 24-cell. The 24 vertices and 24 internal sides in the 24-cell can, therefore, correspond
only to the two sets of 24 corners of the polygons, whilst its 96 edges correspond to the 96 sides of the sectors,
its 96 faces correspond to the 96 sides that end on corners of polygons and its 96 internal triangles correspond to
the **48** other corners and **48** sides of triangles that join to the centres
of polygons. Remember that, in comparing two holistic systems, like does not have to correspond quantitatively to
like. Indeed, this is, rarely, the case. All that is necessary in demonstrating their equivalence is to show the
same pattern of *numbers* of geometrical elements. It is only that which counts, not what the numbers
denote.

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