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#35 The inner Tree of Life with 4th-order polygons embodies the global, holistic parameter 384
According to Table 8 here, the number GnN of geometrical elements in the nth-order N-gon is
GnN = (3n+1)N + 1,
where "1" denotes its centre. Therefore, the number of geometrical elements surrounding the centres of the seven separate types of 1st-order polygons with 48 corners that make up the inner Tree of Life = ∑G1N = (31+1)∑N = 4×48 = 192. Surrounding the centres of the two sets of seven separate 1st-order polygons are (192+192=384) geometrical elements (see here). According to Table 8, the number of corners surrounding the centre of an 1st-order N-gon = N, the number of sides = 2N and the number of triangles = N. The 384 geometrical elements therefore comprise 192 corners & triangles and 192 sides. This 192:192 division is characteristic of the number 384 as the global parameter of holistic systems (see here). It reappears in the combined outer & inner forms of the Tree of Life as their 192 corners & triangles and 192 sides (see here). According to Table 10 here, it reappears in the 14 enfolded polygons of the inner Tree of Life as the 3840 sides of triangles that are outside the root edge shared by these polygons when they are 4th-order, each set of seven enfolded polygons having 1269 triangles with 1920 sides outside this root edge:
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Including the root edge, the inner Tree of Life with 4th-order polygons contains 2538 triangles with 3841 sides. This number can be represented by a 10-fold array of the squares of the integers 1-10 because
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12 |
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| 22 | 32 | |||
| 385 = |
42 |
52 |
62 |
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| 72 | 82 | 92 | 102 |
so that
3840 = 10×384 = 10×(22+32 + ... +102)
and
3841 = 12 + 10×(22+32 + ... +102).
The representation comprises 91 squares, where
91 = 12 + 22 + 32 + 42 + 52 + 62.
The central square 12 corresponds to the root edge. The sum (1920) of the 45 squares of 2-10 in each pentagram is the number of sides of the 1269 triangles in each half of the inner Tree of Life outside the root edge shared by the two sets of seven polygons. This is the arithmetic counterpart of a geometrical object that has a holistic character because it embodies the same patterns and parameters as other sacred geometries. The total number of sides of triangles in the seven enfolded, 4th-order polygons is 1921, which is the sum of the 46 squares of 1-10 in either pentagram. The number 46 is the number of yods in a Type B triangle (see here). It is also the human diploid number, being the number of types of chromosomes in the human cell.
The number 3840 manifests in the disdyakis triacontahedron as the number of hexagonal yods in the faces (constructed from tetractyses) of the 29 Platonic, Archimedean & Catalan polyhedra whose vertices can fit the 62 vertices of this polyhedron. For discussion of this amazing evidence that the disdyakis triacontahedron has archetypal status as the polyhedral counterpart of the inner Tree of Life, see #6, #7 & #9 in Superstrings as sacred geometry/disdyakis triacontahedron; see also here. The number manifests in the inner form of the 10-tree as the number of hexagonal yods and intrinsic corners belonging to the 120 polygons of the first six types that are enfolded in the 10-tree (see #9). There are 384 such yods repeated in the first (6+6) polygons making up the inner form of successive Trees of Life. They comprise (24+24=48) corners of polygons and (168+168=336) other yods. This is the pattern found in the 384 lines & broken lines of the 64 hexagrams used in the ancient Chinese divinatory system known as I Ching, in the intervals between the notes making up the seven diatonic scales (see here) and in the 384 occurrences of the four nitrogen bases in the 64 codons and 64 anticodons that encode for DNA (see here). What is revealed here in the geometry of the inner Tree of Life composed of 4th-order regular polygons is a universal pattern both found in nature and embodied in certain sacred geometries of ancient philosophies and mystical traditions. As the fourth order of polygon has this property, it conforms to the Tetrad Principle formulated in Article 1, whereby parameters of holistic systems (in this case, their global parameter 384) are always embodied in, or otherwise expressed by, either the first four members of a class of mathematical object or its fourth member (in this case, the 4th-order polygons making up the inner Tree of Life).
Question: How can one explain the appearance of the number 384 and the similar pattern of its division in such diverse contexts like genetics, the musical diatonic scales, I Ching and sacred geometries? Is it due to just chance, or is it archetypal design by a transcendental intelligence? Wanting to avoid having to accept the possibility of "intelligent design", a notion that academics shun at all costs because it contradicts their belief in a purposeless universe, some people will embrace the cold comfort that this number and its pattern of division appear by chance in all these contexts. By doing so, they blind themselves to the obvious absurdity of their position, wherein they seem to think that believing in fortuitous miracles is more rational than believing in a universe that has mathematically analogous phenomena because it has been intelligently designed at all levels in accordance with a universal blueprint. However much they try to dismiss ostensible evidence for hidden design as nothing more than "confirmation bias," such a psychological explanation evades the crucial fact that these mathematical analogies have not been manufactured, for they really do exist. Nor have they been concocted by cherry-picking random details from topics that have no conceptual connection. Is it cherry-picking to point out, simply, that the number of geometrical elements surrounding the centres of the seven separate polygons in the inner Tree of Life is the same as the number of lines and broken lines in the 64 hexagrams, as well as the number of instances of the four nitrogen bases in the 64 codons and the 64 anticodons of RNA? Of course not! Well, in that case, is it plausible in the last resort to dismiss as mere chance not only the appearance of the holistic parameter 384 amongst all these supposedly unconnected topics but also its detailed, similar division into its various subsets? No? Well, if one is unwilling to believe in the possibility of it all being just a long chain of miraculous coincidences, what logical alternative can one appeal to as an explanation of such similarities other than that hitherto unsuspected, mathematically analogous phenomena exist because they are manifestations of universal design by a transcendental intelligence that religions call "God"?
