The UPA (subquark state of the
E₈×E₈ heterotic superstring)

The tetractys

The Tree of Life

• The UPA is composed of 10 closed curves, or "whorls."
• Three ("major") whorls are thicker than the rest.
• Seven "minor" whorls spiral around the vertical axis in the opposite sense to the three major whorls. 

• The tetractys is a triangular array of 10 dots, or "yods."
• Three white yods are at the corners of the tetractys.
• Seven coloured yods are at the centre and corners of a hexagon.

• Ten Sephiroth (Divine qualities) compose the Tree of Life.
• The Supernal Triad comprises Kether, Chokmah & Binah.
• Daath ("knowledge") is the "Abyss" separating the Supernal Triad from the seven Sephiroth of Construction.

Whorl

Each whorl is a closed helix with 1680 circular turns, or 1st-order spirillae.

=   

168 is the gematria number value of Cholem Yesodeth, the Mundane Chakra of Malkuth:

This is the number of points, lines & triangles below the top of the 1-tree constructed from Type A triangles:

The Tree of Life and the 2nd-order tetractys are equivalent representations of the Wholeness of holistic systems.

(Place cursor over image to enlarge) 

Two-dimensional projection of the 240 root vectors of the Lie group E₈ in the Gosset polytope 4₂₁.

240 =  

The 240 hexagonal yods in the 48 tetractyses of the 7 separate polygons making up half of the inner Tree of Life denote the 240 roots of the superstring gauge symmetry group E₈.

The 1-tree with 19 Type A triangles contains 240 yods other than Sephirothic corners of these triangles. They denote the 240 roots of the superstring gauge symmetry group E₈.

The three major whorls of the UPA correspond to the Supernal Triad of Kether, Chokmah & Binah (in Theosophy the First, Second & Third Logoi). 72 gauge charges of E₈ are spread along them, 24 per whorl. The seven minor whorls correspond to the seven Sephiroth of Construction (in Theosophy the seven Planetary Logoi). They carry the remaining 168 gauge charges of E₈.

Each helical whorl in the UPA is 10-dimensional. A dimension is represented by a Tree of Life, so that a whorl is geometrically represented by 10 overlapping Trees of Life. As the microscopic Tree of Life, the UPA has 10 whorls corresponding to the 10 Sephiroth. Each Sephirah can be represented by a Tree of Life with 10 Sephiroth, each of the latter by a Tree of Life, and so on.

The Godname of Malkuth — the physical manifestation of the Tree of Life blueprint — is ADONAI. Its number value is 65, which is the number of Sephirothic levels in the 10-tree. This is equivalent to a tetractys-divided decagon that is enclosed in a square. ADONAI prescribes the 10 dimensions of space-time predicted by superstring theory. EL ("God"), the Godname of Chesed with number value 31, also prescribes them because the 10-tree has 127 triangles, where 127 is the 31st prime number. EHYEH ("I am"), the Godname of Kether wiith number value 21, prescribes the 10-tree because 21 Sephirothic levels are on each side pillar of it. 

Each of the 10 whorls spirals five times around the axis of the UPA. Each revolution of the 10 whorls comprises 3360 1st-order spirillae.

The seven enfolded polygons of the inner Tree of Life contain 3360 yods when their sectors are 2nd-order tetractyses.

 84 = 1² + 3² + 5² + 7².

      336 = 4² + 8² + 16² = 2²×84

         = 2² + 6² + 10² + 14².

 

Divided into their sectors, the (70+70) polygons enfolded in 10 overlapping Trees of Life are composed of 3360 points, lines & triangles that are unshared with the outer Trees (shared geometrical elements are coloured green). Each set of (7+7) enfolded polygons has (168+168=336) geometrical elements that are unshared with its outer Tree of Life.

16800 yods surround the centre of the 7-pointed star

  Number of white yods = 3×1680 = number of turns in 3 major whorls of UPA.
  Number of coloured yods = 7×1680 = number of turns in 7 minor whorls of
  UPA.

There are 248 hexagonal yods in a square with 2nd-order tetractyses as its sectors. Each yod symbolizes a root of E₈, the rank-8 exceptional group.

The square also provides an arithmetic representation of the dimension 496 of the two possible superstring symmetry groups SO(32) & E₈×E₈:

There are 248 yods below the top of the 1-tree with its triangles turned into Type A triangles. The eight yods outside the 1-tree denote the eight simple roots of E₈ and the 240 yods other than Sephiroth denote its 240 roots.

41² − 1 = 3 + 5 + 7 +... + 81 = 1680.

71² − 1 = 3 + 5 + 7 +... + 141 = 5040 = 3×1680.

5040 − 1680 = 3360 = 83 + 85 + 87 +... + 141 = 2×1680.

The sum of the 70 odd integers after 1 assigned to the 70 yods of the Tree of Life = 5040. This is the number of turns in the three helical major whorls of the UPA/subquark superstring. It is the sum of the first 40 odd integers after 1 (blue) assigned to the 40 yods outside the (red) Lower Face and the next 30 odd integers 83-141 (green) assigned to the 30 yods in the Lower Face.

The subquark state of the E₈×E₈ heterotic superstring remote-viewed by the Theosophists Annie Besant & C.W. Leadbeater over a century ago consists of 10 closed curves, or "whorls." They bear a correspondence to the 10 Sephiroth of the Tree of Life. The three major whorls correspond to the Supernal Triad and the seven minor whorls are the counterpart of the seven Sephiroth of Construction. Each whorl is a helix with 1680 circular turns. The three major whorls have (3×1680=5040) turns.

Sacred-geometrical embodiment of 504 & 5040

Heptagon
When the seven sectors of a heptagon are 2nd-order tetractyses, 504 yods surround its centre.

Type C dodecagon
When the 12 sectors of a Type C dodecagon are constructed from tetractyses, 504 yods surround its centre. They can be grouped into three sets of 168 yods (coloured red, green & blue).

Disdyakis triacontahedron
The disdyakis triacontahedron has 62 vertices (60 vertices surrounding an axis passing through two opposite vertices). Its 120 triangular faces have 180 edges. Their (120×3=360) sectors have (360+180=540) sides. The number of geometrical elements in the faces that surround the axis = 60 + 120 + 540 + 360 = 1080.

Each edge and each side of a sector in the green faces of the disdyakis triacontahedron are sides of internal grey triangles with the centre of the polyhedron as their shared corner. The (180+360=540) internal triangles have (540×3=1620) sectors with (60 + 120 + 540×3 = 1800) internal sides & 540 internal corners surrounding the centre, i.e., 3960 geometrical elements. The number of geometrical elements in the faces and interior that surround the axis = 1080 + 3960 = 5040. They include 1680 elements (red cells) either in the faces (1080) or sides (600) of sectors of internal triangles created by the edges of the disdyakis triacontahedron, leaving 3360 elements (1680 elements in each half of the polyhedron).* This is the polyhedral counterpart of the 1680 helical turns in the first major whorl and the 3360 turns in the second & third major whorls.

* Alternatively, there are 1680 geometrical elements comprising 180 corners of sectors in the faces, 180 edges & 1320 geometrical elements in the internal triangles created by edges, totalling 1680 elements, leaving 3360 elements.