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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 4_{21}
polytope. In the 1960s, Peter McMullen drew (by hand) a
2-dimensional representation of
4_{21}. 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 4_{21} 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* *Yesodoth*, 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 E_{8}×E_{8}
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
E_{8}. Those in the root and trunk denote the **72** roots of E_{6}, the rank-6,
exceptional subgroup of E_{8}; those making up the branches denote the **168** roots in
E_{8} that do not belong to E_{6}.

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