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The same number 240 is embodied in the Platonic solids as the 240 hexagonal yods in the
faces of the tetrahedron, octahedron & cube and as the 240 hexagonal yods in the faces of the icosahedron or
dodecahedron when their faces are constructed from the template of the tetractys by transforming their sectors into
the latter. This 240:240 division of hexagonal yods making up the first four Platonic solids is the regular
polyhedral manifestation of a fundamental pattern displayed by holistic systems that corresponds to the 240 roots
of each group E_{8} in E_{8}×E_{8}, one of the two symmetry groups describing the unified
force between superstrings. It is the potent Pythagorean tetractys that makes this connection between sacred
geometry and the group mathematics that underlies superstrings. The 480 hexagonal yods in the faces of
the first four Platonic solids denote the 480 roots of E_{8}×E_{8}. The 240 hexagonal yods in
one half of them correspond to the 240 non-zero roots in one group E_{8} and the 240 hexagonal yods in
their other halves correspond to the 240 non-zero roots in the second, similar group. The tetrahedron has 24
hexagonal yods in two of its four faces and the octahedron (or cube) has **48** hexagonal yods in half
its faces. There are (24+**48**=**72**) hexagonal yods in half their faces, leaving
**168** hexagonal yods. This **72**:**168** division is the counterpart of
the 720:1680 division encountered on page 1 in the disdyakis triacontahedron when 720 geometrical elements are
added to the 1680 geometrical elements surrounding its axis when its 120 faces are each divided into three sectors.
The division is characteristic of *all* holistic systems (see The holistic pattern). It corresponds to the
division of the 240 non-zero roots of E_{8} into the **72** roots of its exceptional
subgroup E_{6} and **168** other roots (see the second picture on #13 of Superstrings as sacred geometry/Tree of
Life).

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