Positive UPA negative UPA 
Positive UPA Negative UPA



Why the UPA is the real "God particle"




The UPA described by Besant & Leadbeater is the subquark state of the E8×E8 heterotic superstring.



The tetractys


Tree of Life

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 with the major whorls around the vertical axis, separating from the major whorls at the base of the UPA and twisting towards the top in an inner, narrower spiral 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.





1680 turns of whorl




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







       5  7

      25 27

     45 47

     65 67

 1680 =

    9  11 13      +

   29 31 33      +

   49 51 53      +

   69 71 73

15 17 19 21

35 37 39 41

55 57 59 61

75 77 79 81


=  1680 yods in 10-pointed star 


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

Number value 168 of Cholem Yesodeth

This is the number of points, lines & triangles below the top of the 1-tree constructed from 19 Type A triangles with 25 sides:


168 geometrical elements below top of 1-tree


Below the apex of the 1-tree are:


(10+19=29) corners of triangles;

(25+ 3×19 = 82) sides of triangles;

(3×19=57) triangles.

Total = 168.


7 orders of spirillae

Each 1st-order spirilla consists of 7 2nd-order spirilla spaced evenly around a circle; each 2nd-order spirillae comprises 7 3rd-order spirillae, and so on. The 7th-order spirilla is 7 "bubbles in koilon" spaced evenly around a circle. The 6 higher orders of spirillae represent the winding of curves around the 6 compactified dimensions predicted by superstring theory. It appears that the compactified space is (S1)6 = T6, namely, the 6-dimensional torus, which is a Ricci-flat Calabi-Yau space.


7 orders of spirillae


Equivalence of 2nd-order tetractys and Tree of Life

The Tree of Life and the 2nd-order tetractys are equivalent representations of holistic systems. The black hexagonal yods at the centres of the 10 1st-order tetractyses in the 2nd-order tetractys correspond to the 10 Sephiroth/corners of the 16 tetractyses in the Tree of Life; the 60 white hexagonal yods in the 2nd-order tetractys at the corners of 10 hexagons correspond to the 60 hexagonal yods in these tetractyses.


(Place cursor over image to enlarge) 

 3-d projection of 4-21 polytope


Two-dimensional & 3-dimensional projections of the 240 root vectors of the rank 8, exceptional Lie group E8 determined by the
vertices of the
Gosset polytope 421. This is one of 255 uniform 8-polytopes with 240 vertices, 6720 edges, 60480 faces,
241920 cells, 483840 4-faces, 483840 5-faces, 207360 6-faces & 19440 7-faces.




240 =  240 hexagonal yods in 7 separate polygons



The 240 hexagonal yods in the 48 tetractyses of the 7 separate polygons making up half of the inner Tree of Life denote, in context of superstrings, the 240 roots of the exceptional Lie group E8 that determines the forces between E8×E8 heterotic superstrings.




 240 yods other than Sephiroth in 1-tree




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.


240 gauge charges of E8 spread along 10 whorls


Interpretation 1
The three major whorls of the UPA correspond to the Supernal Triad of Kether, Chokmah & Binah (called in Theosophy the 'First, Second & Third Logoi'). 72 gauge charges of E8 are spread along the 3 major whorls, 24 per whorl. The 7 minor whorls are the manifestation of the 7 Sephiroth of Construction (corresponding to what is called in Theosophy the seven 'Planetary Logoi'). They
carry the remaining 168 gauge charges of E8. The 240 E8 gauge charges are spread along the 10 whorls of the UPA/subquark superstring, 24 per whorl. The difference in thickness between the major whorls and the minor whorls is the result of the breakdown of the symmetry of E8 into that of E6, the rank-6, exceptional subgroup of E8 that has 72 gauge charges.

Interpretation 2
Each whorl revolves 5 times around the axis about which the UPA spins as a spin-½ fermion. A helical whorl makes 10 half-revolutions around the axis, each half-revolution being made up of 168 turns. A half-revolution of all 10 whorls therefore contains 1680 turns — the same number as 5 revolutions of a single whorl. Suppose that 24 gauge charges of E8 are spread along such a half-revolution. Then all the 240 gauge charges of E8 are spread along the 10 half-revolutions of all 10 whorls, i.e., their 50 complete revolutions. The outer and inner halves of the UPA would represent the E8×E8 heterotic superstring manifestation of 120 gauge charges of E8, spread along 8400 turns. It corresponds to the compound of two 600-cells with 240 vertices that is the Petrie projection of the Gosset polytope whose 240 vertices determine the 240 root vectors in the 8-dimensional lattice space of E8. 70 turns (7 per whorl) represent a single charge, which is the source of a 10-dimensional gauge field. Seven turns are associated with each of the 2400 components of the 240 E8 gauge charge: 16800 = 2400×7. According to this alternative view, instead of the gauge charges being smeared out over single whorls, 24 to a whorl, they are spread out along all 10 whorls, 24 to a half-revolution of them all. Three whorls are different to the other 7 because they correspond to, or generate, the 3 spatial components of every 10-dimensional gauge field that are measured along the 3 large-scale dimensions of space. The 7 minor whorls represent the time and spatial components of each gauge field measured along the 6 compactified dimensions.

72:168 division in sacred geometries

whorl is 10-dimensional 




Each whorl in the UPA is a helix with 1680 circular turns. It is 10-dimensional, the six higher orders of spirillae being closed curves that wind around the six compactified dimensions (actually progressively smaller circles) predicted by superstring theory. 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. This means that a whorl can be mapped by either  a single Tree of Life or 10 Trees of Life.


ADONAI prescribes the 10-tree

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 and mapped by 10 Trees of Life. 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. 



ADONAI prescribes 1680 yods in 10-tree



Each of the 10 whorls spirals five times around the axis of the UPA. Each revolution of the 10 whorls comprises 3360 helical turns (1st-order spirillae), 336 per whorl. An outer or inner half-revolution of a whorl comprises 168 turns and a quarter-revolution comprises 84 turns.


336 turns in one revolution of helical whorl



3360 yods in 7 enfolded polygons


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.


336 yods line triangles in 3-d Sri Yantra

The 3-d Sri Yantra  is shaped by the same number (336) as the number of turns in one revolution of a helical whorl of the UPA/subquark superstring. The 168 yods lining each half of the Sri Yantra denote the 168 turns in an outer or inner half-revolution of a whorl.


3360 geometrical elements in inner form of 10-tree outside root edges are unshared with its outer form


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.

 Parallelogram representation of 16800

16800 yods surround the centre of the 7-pointed star

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




Representations of 8400 & 16800

The Godname ELOHIM with number value 50 prescribes the 16800 turns in the 50 revolutions of the 10 helical whorls of the UPA/subquark
superstring.  The Godname ADONAI MELEKH with number value 155 prescribes the 8400 turns in an outer or inner half of these 10 whorls.

Superstring theory requires the symmetry group of the unified interaction between heterotic superstrings to be either SO(32) or E8×E8, both with dimension 496. In the latter case, heterotic superstrings of ordinary matter have a unified force that is described by the symmetry group E8 with dimension 248.


Square embodies dimension 248 of E8


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:

Square representations of 496




248 yods below top of 1-tree symbolize 248 roots of 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.









3-d projection of rotating 600-cell 

 3-d projection of rotating 600-cell


3-d projection of rotating 600-cell


3-d projection of rotating 600-cell




720+240+720 vertices & edges of two 600-cells


It is no coincidence that sacred geometries reproduce the 720:240:720 pattern in the vertices & edges in the compound of two 600-cells that is the Petrie projection of the 421 polytope. Rather, what is appearing in the polychorons is a universal archetype that is embodied in ancient sacred geometries and which manifests in space-time as the whorls making up the UPA/subquark superstring.










Superstring structural parameter 1680 embodied in some sacred geometries


Correspondence between the geometrical or yod compositions of the first four Platonic solids,

the disdyakis triacontahedron, the inner & outer Trees of Life and the inner form of the 10-tree

5040 as of odd integers assigned to 70 yods in Tree of Life


412 − 1 = 3 + 5 + 7 +... + 81 = 1680.
712 − 1 = 3 + 5 + 7 +... 83 +... + 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.

5040 embodied in sacred geometries 



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

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, surrounding the axis are 1680 geometrical elements comprising 180 corners of sectors in the faces, 180 edges & 1320 geometrical elements in the internal triangles created by edges. This totals 1680 elements, leaving 3360 elements.


Zome model of 600-cell as compound of five 24-cells

Zome model of the 600-cell as a

 compound of five 24-cells. 


3-d projection of rotating 600-cell


3-dimensional projection of a rotating 24-cell



3-d projection of rotating 600-cell


3-dimensional projection
a rotating 600-cell


UPA embodies properties of a compound of two 600-cells

The 240 vertices of the 421 polytope coincide with the positions of the 240 roots of E8, the rank-8, exceptional Lie group. This polytope is a compound of two 600-cells. The 120 vertices of a 600-cell can be partitioned into those of five disjoint 24-cells. As each vertex defines one of the 240 root vectors of E8, there is a geometrical basis for dividing its 240 gauge charges into 10 sets of 24. The outer half of the UPA is the counterpart of one 600-cell, the 120 gauge charges denoted by the 120 vertices of the five 24-cells being spread out along the five half-revolutions of the 10 whorls in the outer half. The inner half of the UPA is the counterpart of the other 600-cell. The 2½ revolutions (five half-revolutions) of the whorls that make up each half are the counterpart of the five 24-cells in each 600-cell. The 840 vertices & edges in each 600-cell are the geometrical counterpart of the 840 circular turns in the five half-revolutions of the outer or inner half of each helical whorl. The 1680 vertices & edges belonging to the compound of two 600-cells in the Gosset polytope are the counterpart of the 1680 circular turns in each helical whorl of the UPA. 70 turns "carry" an E8 gauge charge: 1680 = 24×70.

This correlation is irrefutable evidence that the UPA paranormally described over a century ago is a state of the E8×E8 heterotic superstring (see Article 62 for more details).

Comparison of inner form of 10 Trees of Life and 421 polytope


The 421 polytope is the 8-dimensional polytope counterpart of the inner form of 10
overlapping Trees of Life when the (70+70) regular polygons enfolded in the latter are
Type B.
 Each set of (7+7) enfolded polygons embodies the number 137 that determines
the fine-structure constant because its yod population is that of 137 tetractyses.
The topmost corners of the two hexagons coincide with
 the lowest corners of the
hexagons enfolded in the next higher Tree. This leaves 1368 yods intrinsic to
Tree, i.e., 13680 yods are intrinsic to 10 Trees of Life. They correspond to the
yods lining the 6720 edges of the 4
21 polytope with tetractyses as its 60480 faces:

                               240 vertices + 6720×2 hexagonal yods = 13680.

The 7 black yods in each hexagon coincide with yods on the two side pillars of the outer Tree
of Life.  Six other black yods are centres of the other polygons in each half of the inner Tree of
Life. 26 black yods in both halves are either centres of polygons or shared with the outer Tree.
Of these, two are shared with the hexagons enfolded in the next higher Tree. This means that
24 black yods per set of (7+7) enfolded polygons are either centres or shared yods. There are
240 yods in the (70+70) polygons enfolded in 10 Trees of Life that are either centres or shared
yods. T
here are 13440 yods (6720 pairs) intrinsic to these 140 polygons that surround their
centres. They correspond to the
13440 hexagonal yods on the 6720 edges of the 421 polytope.
Each such yod and its mirror image
in the other set of 7 polygons can be joined by a horizontal
straight line. There are 6720
 of these lines in the (70+70) polygons enfolded in 10 Trees that do
not connect either
shared yods or centres of polygons; each line corresponds to an edge in the
421  polytope. The 10 sets of 24 yods that are either centres or yods shared with the outer Trees
are the counterpart of the vertices of the 10 sets of 24-cells (each with 24 vertices) in the
of two 600-cells making up the 421 polytope.

Discounting coincidence as too improbable, what do these correspondences imply?
1. E8×E8 heterotic superstrings exist;
2. their unified forces are represented by the geometry of the 421 polytope that has its
2-dimensional counterpart in the inner form of
10 overlapping Trees of Life. This has been
proved elsewhere on this website to
be isomorphic to other sacred geometries and to
the UPA structural parameter 1680, which is the number of vertices & edges of
compound of two 600-cells making up the 421 polytope;
3. the 421 polytope conforms to the archetypal patterns exhibited by sacred geometries
4.  the UPA described by Besant & Leadbeater is a state of the E8×E8 heterotic superstring
because its structure is characterized by numbers which appear against the run of chance
in the geometry of the 421 polytope and the 600-cell;
5. the UPA is the microscopic manifestation of both the sacred geometries of religions and
the polytope representation of the roots of E8 known to professional mathematicians. It is
the yet-to-be-discovered bridge between physics and the transcendental.



Examples of how Godnames prescribe superstrings