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**#50 The 3:69:168 pattern in seven sacred
geometries**

__1-tree__The 1-tree (lowest Tree of Life) with its 19 triangles turned into tetractyses
has

__2-d Sri Yantra__At the centre of the 2-dimensional Sri Yantra is a point, or bindu,
that symbolizes the Absolute, the source of all existence. It is bounded by the three green sides of a triangle,
although, strictly speaking, this does not count as a triangular

The trimûrti of the three Hindu Gods: Brahmā, Vishnu,
and Shiva (left to right) at Ellora Caves, near Aurangabad, in India. They are
expressed in sacred geometries

by the three generic, structural components that are their seed or source elements, namely, the "3" in
the 3:69:**168** pattern that is common to these
geometries.

(preserver or maintainer) and Shiva (the destroyer or transformer). They are surrounded by 42 red triangles with 68 black
corners and 126 red sides. Including the lowest corner of the central triangle, there are 69 black
points, three green lines and (42+126=**168**) red lines & triangles in the 240
geometrical elements that surround the centre of the Sri Yantra.

__Two separate sets of 7 enfolded polygons__120 yods line the sides of the seven enfolded
polygons of the inner Tree of Life. They comprise

__Two separate Type B dodecagons__Divided into their sectors, the two Type A dodecagons
have

__Disdyakis triacontahedron__The disdyakis triacontahedron has

__First three Platonic solids__When the 60 sectors of their 18 faces are Type A
triangles, the tetrahedron, octahedron & cube have 240 hexagonal yods. The tetrahedron has

black hexagonal yods that are either on its edges or at the centres of the nine tetractyses in
the other three faces and 24 red hexagonal yods on internal sides of the 60 tetractyses. The 24 tetractyses in the
eight faces of the octahedron contain 24 black hexagonal yods at their centres and **72** red
hexagonal yods on their sides. The 24 tetractyses in the six faces of the cube contain 24 black hexagonal yods at
their centres and **72** red hexagonal yods on their sides. There are three green hexagonal yods,
69 black hexagonal yods and **168** red hexagonal yods in the faces of the first three Platonic
solids.

__The {3,7} tiling of the 3-torus with tetractyses__When their square faces are stuck
together, four triangular prisms and six square antiprisms create an object that is topologically equivalent to a
3-torus. Its 56 hyperbolic triangles are the {3,7} tiling on the 3-torus of the

X^{3}Y + Y^{3}Z + Z^{3}X = 0

where X, Y & Z are complex variables (see Article 43). Transformed into tetractyses, they contain
**248** yods (see the table in the diagram below). They symbolize the
**248** roots of the rank-8 Lie group E_{8}. Eight of these are white yods at the
centres of the triangular faces of the triangular prisms (the diagram shows only one for the sake of clarity).
The remaining 240 yods comprise **72** yods at either the centres or the corners of the 56
tetractyses and **168** red hexagonal yods on their 84 sides. The yods at the

corners of one triangular face are coloured green. The **21** other vertices
of the prisms & antiprisms and the centres of the **48** tetractyses in the faces of the
antiprisms (i.e., 69 yods), are coloured black. Intuitively speaking, it makes sense to regard the three green
corners of one face of one triangular prism as the natural starting point for the construction of the 3-torus from
56 triangles. It is simply like starting with a point, calling it the first vertex of a tetrahedron and then
creating the rest of it from this point, only this time we start with a triangle, namely, one of the faces of one
of the four triangular prisms, and proceed to stick them to the antiprisms. The conversion into tetractyses of the
56 hyperbolic triangles in the {3,7} tiling of the 3-torus displays the same 3:69:**168** pattern
as that found in other sacred geometries. This is because the Klein quartic and the 3-torus play a fundamental role
in the physics of the E_{8}×E_{8} heterotic superstring, which this website demonstrates
conforms to different sacred geometries. The embodiment in the 3-torus of *both* the numbers
**168** and **248** as basic structural and dynamical parameters of the
superstring cannot be a coincidence!

According to the table opposite, the 192 yods lining the 56 tetractyses comprise the 24 vertices
of the four triangular prisms and **168** red hexagonal yods. This
24:**168** division (actually a 3:**21**:**168** division) is
characteristic of sacred geometries (see **The holistic pattern**). Notice
also that the **36**:**48** division of the 84 sides of the triangular prisms
& square antiprisms conforms to the pattern of holistic systems. For example, it is found in the 2nd-order
tetractys as the **36** yods lining its sides and as the **48** yods inside
it that surround its centre, or, alternatively, as the **36** yods either in the three
tetractyses at its corners or at corners of the other tetractyses and as the **48** hexagonal
yods in the seven other tetractyses that surround its centre (see here).

Here is a clear, remarkable illustration of the correspondence between
*seven* objects that possess sacred geometry. Not only do they contain 240 structural components — yods
or geometrical elements — but this number also divides up in a *natural* way into analogous sets of
three, 69 and **168** components. See also here.

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