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**#34 How sacred geometries embody the heterotic superstring
gauge symmetry groups E _{8} & E_{8}×E_{8}**

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**a)** **1-tree**The 1-tree is the lowest Tree in any set of
overlapping Trees of Life. When its 19 triangles are Type A, it has 251 yods (see #6). Eleven of these (not shown in the diagram) coincide with SLs of
the 1-tree. As well as the 240 yods that are not SLs, there are four red yods below the apex of the 1-tree on
either side of the central Pillar of Equilibrium. Below the top of the 1-tree are

The trunk of the Tree of Life is the sequence:

point (Kether)

line (Chokmah-Binah Path)

triangle (Chesed-Geburah-Tiphareth)

tetrahedron (Netzach-Hod-Yesod-Malkuth)

When all triangles are Type A, the trunk contains **80** yods, where
**80** is the number of Yesod, i.e, 70 yods other than SLs. The "root" of the Tree of Life is
the pair of hexagonal yods on the Path connecting Daath and Tiphareth whose projection onto the plane of the
(7+7) enfolded polygons of the inner Tree of Life is their root edge. The root and trunk of the Tree of Life
contain **72** blue yods. They denote the **72** roots of E_{6}, the
rank-6 exceptional subgroup of E_{8}. The remainder of the 1-tree (its "branches") contains
**168** black yods. They denote the remaining **168** roots of
E_{8}. See further discussion here.

**b) Outer & inner Trees of Life**The outer Tree of Life with its 16
triangles turned into tetractyses contains 70 yods. Of these, 10 yods coincide with SLs at the corners of these
tetractyses, leaving 60 hexagonal yods. The inner Tree of Life with the 94 sectors of its (7+7) enfolded polygons
turned into tetractyses contains 444 hexagonal yods. Two pairs of hexagonal yods lying on each side pillar coincide
with hexagonal yods in the two hexagons when the outer Tree of Life is superimposed on its inner form. The number
of hexagonal yods in the combined outer & inner Trees of Life = 60 − 4 − 4 + 444 =

**c) (7+7) separate polygons of the inner Tree of Life**

With their sectors turned into tetractyses, the two sets of seven separate polygons contain 480 hexagonal yods
surrounding their centres. Four yods lie on the separate root edge, so that two hexagonal yods are added to this
straight line. Such transformation of the 14 polygons and a separate line denoting their root edge, therefore,
requires (480+14+2=**496**) *new* yods. **248** such yods are associated
with each set of seven polygons. They include the seven centres and one hexagonal yod in the separate root edge.
These eight yods (either black or white) denote the eight simple roots of E_{8}. The 240 red or blue
hexagonal yods added by the transformation of each set of polygons denote its 240 roots. The mirror symmetry of the
distribution of yods in the root edge and the two sets of polygons is responsible for the direct product nature of
E_{8}×E_{8}. This is how the inner Tree of Life in its expanded form encodes the root composition
of one of the two possible symmetry groups describing superstring forces.

**d) Five Platonic solids**

When their **50** faces are divided into their 180 sectors, their face-centres and vertices are
joined to their centres and the resulting internal triangles divided into their sectors, the five Platonic solids
are composed (other than their vertices) of 2480 points, lines & triangles that surround axes drawn through
their centres and two opposite vertices. On average, **496** geometrical elements other than
vertices surround the axis of a Platonic solid, **248** elements making up each half. See
here & here for detailed analysis.

**e) Inner Tree of Life & 10 Trees of Life**Ten overlapping Trees of Life
contain 520 yods when their triangles are tetractyses (see #24). This is the number of yods in the (7+7) enfolded polygons
outside their root edge when their 94 sectors are tetractyses. There are

**f) 3-torus & its inside-out version**The 3-torus can be constructed from
four triangular prisms and six square antiprisms. When the 56 triangles making up their faces are tetractyses, the
3-torus contains

**g) Square with 2nd-order tetractyses as sectors**When its sectors are
2nd-order tetractyses, a square contains

**h) Octagon with 2nd-order tetractyses as sectors**When its sectors are
2nd-order tetractyses, an octagon contains

**i) Disdyakis triacontahedron**The disdyakis triacontahedron has

**j) 2-d Sri Yantra**

j

According to Table 8 in Article 35, when its 43 triangles are transformed into Type A triangles, the
two-dimensional Sri Yantra contains 757 yods. 756 yods surround the bindu point at its centre. Of these, 16
green yods belong solely to the central triangle (its two upper corners are also corners of the innermost set of
eight triangles, so that green yods denoting yods exclusive to the central triangle are not assigned to them).
According to this table, 504 hexagonal yods line the 252 sides of the 126 tetractyses in the 42 triangles
surrounding the central one. The tips of some of these triangles touch 12 of their sides. From a strict,
mathematical point of view, these particular sides are not single, straight lines but *pairs* of lines
joined at the tip of another triangle. This leaves (252–12=240) true sides (i.e., single, straight lines) with
480 hexagonal yods. These yods cannot be divided into two sets of 240 that are mirror images of one another, if
the mirror is *horizontal*, because three of the four sides excluded from the layer of ten blue triangles
in Figure 1 of Article 35 (or see here) belong solely to its upper half, creating an imbalance. However, a mirror
symmetry persists after excluding the 12 sides if the mirror is *vertical*, because, then, 240 red
hexagonal yods and 240 blue hexagonal yods lie either on or, respectively, to the right and left of the vertical
axis passing through the centre of the central triangle, which is surrounded by (240+240=480) hexagonal yods on
true sides of triangles and 16 yods intrinsic to the central triangle, i.e., **496** yods that
symbolize the **496** roots of E_{8}×E_{8}.The 24 hexagonal yods lining sides that are touched by other
true triangles are coloured black. The two hexagonal yods in the side of the triangle touched by the lowest
corner of the central triangle are not black because, although it has the appearance of a triangle, it is,
mathematically speaking, not a true, triangular area, the bindu (a separate point) coinciding with its centre.
In the 3-dimensional Sri Yantra, the central triangle really *is* a triangle in the pure, mathematical
sense because the bindu hovers above it instead of being embedded in its surface. The 16 green yods in the
central triangular perimeter that surround the bindu are intrinsic to it because none of them is shared
with any of the 42 triangles distributed around it. They symbolize the 16 simple roots of
E_{8}×E_{8}. The (240+240=480) hexagonal yods in the 240 single lines making up the true sides
of these 42 triangles denote its (240+240=480) roots.

See
also the discussion in **The holistic pattern** of the Sri
Yantra under the heading "496 = 248 + 248".

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