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#43 The dodecagonal connection between the superstring structural parameter 16800 and the 240 nonzero roots of E_{8}
Spread along the 16800 circular turns or circularly
polarised waves in the 10 whorls of the UPA/E_{8}×E_{8} heterotic
superstring are the 240




A wide variety of sacredgeometrical contexts discussed in this website has revealed that holistic systems embody certain numbers. Among these are the numbers 1680 (the number of circular turns in each whorl of the UPA), 16800 (the number of turns in its 10 whorls) and the number 240, which is the number of roots of the rank8 exceptional Lie group E_{8}. The identification of the UPA as the subquark state of the E_{8}×E_{8} heterotic superstring led to the interpretation of the 16800 turns as the string manifestation of the 240 E_{8} gauge charges associated with these roots, 24 of them being spread out over the length of each of the 10 halfrevolutions of the UPA. Sometimes, the number 1680 is explicitly related to the number 240. For example, as the seven enfolded Type A polygons have 264 yods, of which 69 yods outside the root edge belong to the dodecagon, the first six enfolded polygons have (264−69=195) yods, i.e., 193 yods are associated with each set of the first six enfolded polygons, including 25 corners. The number of yods associated with the 6n regular polygons enfolded in n Trees of Life = 192n + 1, of which (24n+1) are corners of polygons and 168n yods are not such corners. The 60 polygons enfolded in 10 Trees of Life have 241 corners and 1680 other yods. The topmost corner of the hexagon enfolded in the tenth Tree coincides with the lowest corner of the hexagon enfolded in the eleventh Tree. This means that 240 corners and 1680 other yods are intrinsic to the polygons enfolded in the lowest 10 Trees (and in every 10 Trees) (see Figure b in #33). In the case of the 20 dodecagons enfolded in 10 Trees, when each sector is a 2ndorder tetractys, their 240 sectors have 240 central yods (the white yods shown above) that are surrounded by 16800 yods outside the 10 root edges separating pairs of dodecagons. The proof of this is as follows:
Number of yods in a 2ndorder tetractys = 85.
Number of yods lining each side of a 2ndorder tetractys = 13.
Number of yods per sector of an ngon with 2ndorder tetractyses as sectors = 85 − 13 = 72.
Number of yods in an ngon = 72n + 1.
Number of yods outside one side of an ngon that surround the yods at the centres of its 2ndorder tetractyses = 72n + 1 − 13 − n = 71n − 12.
Number of yods outside the root edge of two ngons that surround their 2n central yods = 71n − 12 + 71n − 12 = 142n − 24.
Number of yods outside the root edge of two dodecagons (n=12) that surround their 24 central yods = 142×12 − 24 = 1680.
Number of yods outside the root edges of the 20 dodecagons enfolded in 10 Trees that surround the 240 yods at the centres of their 240 2ndorder tetractyses = 1680×10 = 16800.
We see that, when they are constructed from 2ndorder tetractyses, the 20 dodecagons making up the inner form of 10 overlapping Trees embody both the numbers 16800 and 240. The 10 Trees represent the 10 halfrevolutions of the whorls of the UPA; the 240 centres of all the 2ndorder tetractyses in the 20 dodecagons represent the 240 gauge charges of E_{8}; the 16800 yods that surround these centres represent the 16800 circularly polarised oscillations of the whorls making up the UPA as the subquark state of the E_{8}×E_{8} heterotic superstring. This indicates that the (12+12=24) centres of the 2ndorder tetractys sectors of each pair of dodecagons denote the 24 E_{8} gauge charges that are spread along each halfrevolution of the 10 whorls of the UPA. It is shown in 4d sacred geometries that the 240 vertices of the 4_{21} polytope that represent the 240 roots of E_{8} project in 4dimensional space onto the (120+120) vertices of a compound of two concentric, but different sized, 600cells (see text below Fig. 4 here). The 120 white central yods in each set of 10 dodecagons enfolded in 10 Trees of Life correspond to the 120 vertices of each 600cell. As a 600cell is a compound of five 24cells, the 4dimensional projection of the 4_{21} polytope comprises 10 24cells. Each pair of dodecagons corresponds to a 24cell, the 24 white yods at the centres of their 2ndorder tetractyses corresponding to the 24 vertices of each 24cell.
Elsewhere in this website (e.g., see here), evidence has been presented that the dodecagon is the polygonal counterpart of the outer Tree of Life. As the UPA is the microscopic realisation of this blueprint, it should come as no surprise that the dodecagons enfolded in the inner form of 10 Trees of Life embody the 240 gauge charges of E_{8} and the 16800 circularly polarised oscillations in the 10 whorls of the E_{8}×E_{8} heterotic superstring that carry these charges.
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