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#49 Correspondence between the 384 geometrical elements in the outer & inner Trees of Life and the 384 lines & broken lines in the 64 hexagrams
It was shown in #12 at Sacred geometry/Correspondences that the 192 lines & broken lines in each diagonal half of the table of 64 hexagrams used in the ancient Chinese system of divination called I Ching symbolize the 192 corners, sides & triangles that surround the centres of each of the two sets of seven separate, regular polygons making up the inner Tree of Life. It will now be proved that another correspondence exists, namely, the 384 geometrical elements that make up the outer and inner Trees of Life when they are superimposed on each other.
The seven enfolded polygons that make up half of the inner Tree of Life have 47 sectors with 41 corners and 88 sides. They comprise 176 geometrical elements. Of these, two corners are the endpoints of their shared root edge, three corners (shown as green points in the diagram opposite) coincide with Sephiroth of the outer Tree of Life and two sides (indicated opposite by green lines) are Paths. Eight geometrical elements in the inner Tree of Life are therefore either shared with its outer form or outside the root edge, leaving 168 geometrical elements (36 corners, 85 sides & 47 triangles) outside the latter that are intrinsic to the seven enfolded polygons. Both sets of polygons have (168+168=336) such elements. The outer Tree of Life comprises 16 triangles with 10 corners and 22 sides, i.e., 48 geometrical elements, where 48 is the number value of Kokab, the Mundane Chakra of Hod. Superposed on each other, the outer & inner Trees of Life contain (48+336=384) geometrical elements.
Compare this with the 384 lines & broken lines making up the 64 hexagrams. The eight diagonal hexagrams contain 48 lines & broken lines and the 28 off-diagonal hexagrams in each diagonal half contain 168 lines & broken lines. The 48 lines & broken lines in the eight diagonal hexagrams symbolize the 48 geometrical elements making up the outer Tree of Life, whilst the 168 lines & broken lines in the 28 off-diagonal hexagrams in each half of the 8×8 array of hexagrams symbolize the 168 geometrical elements intrinsic to each set of seven enfolded polygons that are needed to generate their form beyond their shared root edge.
Combined, the outer & inner Trees of Life consist of (10+36+36=82) corners, (22+85+85=192) sides and (16+47+47=110) triangles, that is, (82+110=192) corners & triangles and 192 sides. This is remarkable because their combined geometrical composition displays the same 192:192 division as found in both the table of 64 hexagrams, which have 192 (Yang) lines and 192 (Yin) broken lines, and the two sets of seven separate polygons, which comprise 192 corners & triangles and 192 sides surrounding their 14 centres. It re-appears in the first (6+6) enfolded polygons as the 192 yods (24 polygonal corners & 168 yods) associated with each set of the first six polygons enfolded in successive, overlapping Trees of Life. This pattern characteristic of holistic systems (see here) manifests in other sacred geometries (see #36 in Wonders of sacred geometries). The 48 corners intrinsic to the two sets of the first six enfolded polygons correspond to the 48 geometrical elements making up the outer Tree, whilst their (168+168) remaining yods correspond to the (168+168) geometrical elements outside the root edge that were found above to be intrinsic to the two sets of seven enfolded polygons. The 168:48:168 division characteristic of holistic systems gets reproduced in smaller and smaller subsets of the (7+7) enfolded polygons of the inner Tree of Life provided that these subsets themselves constitute a holistic system (as in the case of the first six polygons).
This pattern is discussed in greater detail in #31 at Sacred geometry/Correspondences.
Including the root edge, the outer & inner Trees of Life have (2+82=84) corners and (1+192=193) sides, that is, 277 corners & sides. Of these, six corners and four sides are shared between the outer and inner Trees (depicted by the green points & lines in the two side pillars of the outer Tree). This leaves 267 corners & sides that are unshared, so that there are 264 such corners & sides outside the root edge. This number is a parameter of holistic systems, being the number of yods that make up the seven enfolded polygons of the inner Tree of Life when their 47 sectors are tetractyses:
The 47 tetractys sectors of the seven enfolded polygons contain 264 yods.
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