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11. Interior & vertex angles of the triacontagon conform to the I Ching pattern of 64 hexagrams

 The 168:24:24:168   pattern of lines/broken lines in the 64 hexagrams is identical to that in the angles of either one or all 8 triacontagons.

Each of the 64 hexagrams used in the I Ching system of divination is a 6-fold combination of lines (denoting Yang) or broken lines (denoting Yin). It consists of a pair of trigrams (triplets of lines/broken lines). The basic set of 8 trigrams comprises two subsets of 4. The 64 hexagrams are the conjunctions or intersections of 8 rows and 8 columns of the 8 trigrams, i.e., 8 pairs of a complete set of trigrams. Therefore, they comprise 8 pairs of two subsets of 4 trigrams, each subset comprising 12 lines/broken lines.

Compare this with the 8 triacontagons that are the Petrie polygons of the 421 polytope. The vertex angle of a sector of a triacontagon is 12° and the interior angle subtended by two sides of a pair of adjacent sectors is 168°, whilst the sum of their vertex angles is 24°. As the 64 hexagrams consist of 8 pairs of two subsets of 12 lines/broken lines and the 8 triacontagons comprise 8 types (differing in size) of pairs of adjacent sectors and their diametric opposites, the following correspondence:

angle of one degree → line/broken line

is implied. It means, considering only vertex angles, that:

• a sector (S) with a 12° vertex angle → to a subset of 4 trigrams with 12 lines/broken lines;
• two adjacent S (S') with a combined 24° vertex angle → to a complete set of 8 trigrams with 24 lines/broken lines;
• a pair (P) of S' and its diametric opposite with a combined 48° vertex angle → to a set of 8 hexagrams with 48 lines/broken lines;
• the 8 types of P in the 8 triacontagons with a combined 384° vertex angle → to the 8 rows of hexagrams with 384 lines/broken lines.

We see that a single P in a triacontagon with 8 half-sectors corresponds to (or is equivalent to) one row of hexagrams, whilst a set of 8 P's in the 8 triacontagons with (8×8=64) half-sectors corresponds to the 8 rows of 8 hexagrams, a total of 64 hexagrams. A half-sector with a vertex angle of 6° corresponds to a hexagram (pair of trigrams).

If we now consider the interior angles as well as the vertex angles, then the 4 generic sectors in a triacontagon have a combined vertex angle of 48° and a combined interior angle of (84°+84°+84°+84°=336°). This corresponds to the 48 lines/broken lines in the 8 diagonal hexagrams and to the 84 lines and 84 broken lines in the 28 off-diagonal hexagrams in each diagonal half of the 8×8 array. One base angle of 84° corresponds to the 84 lines; the base angle in the adjacent sector corresponds to the 84 broken lines. Their reflected counterparts correspond to the 84 lines and 84 broken lines in the opposite half of the 8×8 array. P displays the 24:168, 84:84 & 12:12 divisions that are characteristic of holistic systems (see The holistic pattern). The sum of the vertex angles for one triacontagon is (24+24= 48) and the sum for the other 7 is (7×48=336=168+168). which is the sum of the two diametrically opposite interior angles. This compares with the 8 diagonal hexagrams having 48 lines/broken lines and the 56 off-diagonal hexagrams having 336 lines/broken lines (168 in each half). It is as though one triacontagon (we need not specify here which one) corresponds to the 8 hexagrams forming the diagonal and the 7 other triacontagons correspond to the 7 copies of these two similar sets of 8 trigrams that make up the 56 hexagrams in the remainder of the 8×8 array.

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