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 Counterpart of Gosset polytope 421 in lowest Tree of Life

The E8 root system consists of 240 roots represented by 240 vectors in an eight-dimensional space. These vectors are the position vectors of the vertices of an eight-dimensional object called the Gosset polytope, or the 421 polytope. In the 1960s, Peter McMullen drew (by hand) a 2-dimensional representation of 421. In 2007, a four-year collaboration between mathematicians from Europe and the USA announced the results of their calculations of the mathematical structure of E8, using a supercomputer. The image shown here was computer-generated by John Stembridge, based on McMullen's drawing. (Credit: Image courtesy of American Institute of Mathematics). A 300 dpi PDF of this complex image is available here. A 3-dimensional model of it can be seen here.

The edges join vertices that are nearest neighbours. Every point has 56 such neighbours, although some of their projections onto the 2-dimensional plane lie on top of one another, so that fewer lines appear in the diagram to emanate from some points.

The counterparts in the outer Tree of Life of the 240 vertices of 421 are the 240 new yods in the lowest Tree of Life generated by turning the 57 sectors of its 19 triangles into tetractyses, for it contains 251 yods in total (see footnote on previous page), of which 11 are corners of these triangles, and 251 − 11 = 240. Although depicted here in 2-dimensions, this object is really 3-dimensional — hence, some yods depicted in the diagram as circles are either directly in front of or behind yods denoted by black dots. Its "trunk" consists of the point (Kether), the four yods in the line joining Chokmah & Binah, the 19 yods in the triangle with Chesed, Geburah & Tiphareth at its corners and the 56 yods in the faces of the tetrahedron whose vertices are located at Netzach, Hod, Yesod & Malkuth. Hence, the trunk is composed of 80 yods, where 80 is the number of Yesod. There are (80−10=70) yods other than SLs of the lowest Tree. The remainder of this Tree outside the trunk comprises (251−80=171) yods. They include three yods located on the Path connecting Daath and Tiphareth, The projection of this line onto the plane of the (7+7) enfolded polygons of the inner form of the Tree of Life is their shared root edge. This leaves 168 yods outside both the line and the trunk, where 168 is the number value of Cholem Yesodeth, the Mundane Chakra of Malkuth. They constitute what will be called the "branches" of the lowest Tree of Life. Just as the arrangement in space of the branches of a tree characterises its shape, so this number expresses the form of the microscopic Tree of Life. This is because 168 is the number of circular turns in a half-revolution of each helical whorl of the UPA (see here), identified as the subquark state of the E8×E8 heterotic superstring. The Path connecting Daath and Tiphareth is what may be called the "root" of the lowest Tree. Two hexagonal yods lie on it, whilst some yods are behind the Path and others are directly in front of it in the line of sight. The number of non-SL yods in the root and trunk is 72. Together with the 168 yods in its branches, they make up the 240 yods that denote the 240 roots of E8. Those in the root and trunk denote the 72 roots of E6, the rank-6, exceptional subgroup of E8; those making up the branches denote the 168 roots in E8 that do not belong to E6.




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