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**#20 How the first four Platonic solids embody the
superstring structural parameter 1680**

Consider a Platonic solid with V vertices, E edges & F faces that are regular n-gons. Dividing their faces into their nF sectors generates F corners, so that the Platonic solid has (V+F) corners & (E+nF) sides of nF triangles in its faces. Joining its V vertices to its centre creates V sides of E internal triangles. When the latter are Type A, there are also E corners & 3E sides of 3E sectors of these internal triangles. Joining the F centres of faces to the centre of the solid generates nF internal triangles & F internal sides. When these triangles are Type A, nF corners & 3nF sides of 3nF sectors are added. The numbers of geometrical elements surrounding its centre are:

CornersSidesTrianglesTotalFaces: V + F E + nF nF 2 + 2E + 2nF Interior: E + nF V + F + 3E + 3nF 3E + 3nF 2 + 8E + 7nF Total: 2 + 2E + nF 2 + 5E + 4nF 3E + 4nF 4 + 10E + 9nF

(Euler's polyhedral formula for a simply-connected polyhedron:

V – E + F = 2

has been used to simply the expressions). The number "2" in the expression for the total
geometrical composition of the faces can be interpreted as the two vertices lying on an axis that passes through
the centre of the polyhedron. The number "2" in the expression for the composition of the interior denotes the two
sides shared by some internal triangles that form this axis. The number "4" in the expression for the total number
of geometrical elements denotes the two internal sides that form the axis and the two vertices at their ends. The
number of corners, sides & triangles surrounding the axis of the Platonic solid = 10E + 9nF. For the first four
Platonic solids with 60 edges & 38 faces with 120 sectors, the number of
geometrical elements that surround their axes = 10×60 + 9×120 = 1680. The dodecahedron with 30 edges & 12 faces
with 60 sectors has (10×30 + 9×60 = 840) elements surrounding its axis. This is very remarkable, for the number
1680 is the number of 1st-order spirillae in each whorl of the UPA, whilst the number 840 is the number of
1st-order spirillae in an outer or inner half of a whorl (see here)! Embodied, therefore, in what the
ancient Greeks believed are the shapes of the particles of the Elements Fire, Air, Water & Earth is the very
structural parameter 1680 of the subquark state of the E_{8}×E_{8} heterotic superstring
remote-viewed over a century ago by Annie Besant & C.W. Leadbeater. The 840 geometrical elements in each
half of the first four Platonic solids that surround their axes correspond to the 840 circular turns in either
an outer or an inner half of these regular polyhedra. This is
one of the most important discoveries discussed on this website that relate sacred geometries to superstring
physics. It compares with the 1680 geometrical elements that surround an axis of the disdyakis
triacontahedron — the outer form of the polyhedral Tree of Life (see here & here) — when its internal triangles are not divided into their
sectors and centres of faces are not joined to its centre. When they *are* joined (noting that E =
180, F = 120, n = 3 and nF = 360), the number of geometrical elements surrounding the axis of the disdyakis
triacontahedron = 10×180 + 9×360 = 5040. This fact is just as remarkable, for the number 5040 (=3×1680) is the
number of 1st-order spirillae in the three major whorls of the UPA/heterotic superstring! So the basic
construction of this polyhedron from triangles creates 1680 geometrical elements around its axis, whilst
addition of internal triangles, some of whose sides divide its faces into their sectors, requires 2×1680
elements; each element corresponds to a 1st-order spirilla in the three major whorls of the UPA.

The numerical counterpart of this 1680:2×1680 division in the Tree of Life was discussed in
#15. It arises from the distinction between the region outside the
Lower Face of the Tree of Life and the latter when the first 70 odd integers after 1 are assigned to the 70 yods
of its 16 tetractyses. Whereas the first four Platonic solids embody only the superstring structural parameter
1680, the disdyakis triacontahedron embodies the superstring structural parameter 5040 *as well*. This is
because the subquark state of the E_{8}×E_{8} heterotic superstring is the physical
realisation of the cosmic blueprint which, whilst expressed in the Platonic solids collectively, has its
*single*, polyhedral representation in the disdyakis triacontahedron.

The number of sides surrounding the axis of a Platonic solid = 5E + 4nF. The number of sides
surrounding the axes of the first four Platonic solids with 60 edges & 120 sectors of 38 faces = 5×60 + 4×120 =
780 = 78×10. The number of corners & triangles = 1680 – 780 = 900 = 90×10. This is remarkable confirmation of
the result obtained above because **168** is the number value of
*Cholem* *Yesodoth*, the Mundane Chakra of Malkuth, the number value of *Cholem* is
78 and the number value of *Yesodoth* is 90:

It is highly implausible that subsets of geometrical elements would make up the first four Platonic solids that, by mere chance, are the number values of these two Hebrew words. The same division appears in the inner Tree of Life as the 78 yods associated with the triangle, pentagon & octagon other than corners and as the 90 yods in the square, hexagon & decagon other than corners (see here). This gematria-based division manifests in the simplest Platonic solid — the tetrahedron, as shown in the table below listing the numbers of geometrical elements surrounding the axes of the five Platonic solids:

Platonic solid |
Parameters |
Corners |
Sides |
Triangles |
Total |

Tetrahedron | E = 6, n = 3, F = 4 | 24 | 78 | 66 | 168 |

Octahedron | E = 12, n = 3, F = 8 | 48 |
156 | 132 | 336 |

Cube | E = 12, n = 4, F = 6 | 48 |
156 | 132 | 336 |

Icosahedron | E = 30, n = 3, F = 20 | 120 | 390 | 330 | 840 |

Subtotal = |
240 | 780 | 660 | 1680 | |

Dodecahedron | E = 30, n = 5, F = 12 | 120 | 390 | 330 | 840 |

Total = |
360 | 1170 | 990 | 2520 |

78 sides & (24+66=90) corners & triangles surround its axis. It demonstrates in an unambiguous way how the
simplest regular polyhedron embodies not only the number value **168** of the Mundane Chakra of
Malkuth but also the number values of the two words that compose it. What better evidence could one have that this
is, indeed, the correct Kabbalistic name of the Mundane Chakra of Malkuth? The geometrical compositions of the
octahedron and its dual — the cube — are the same, with 336 geometrical elements surrounding each axis
(**168** elements in each half). Surrounding the axes of the tetrahedron, octahedron & cube
are 840 geometrical elements — the same as for the icosahedron. The 840:840 division in the outer 2½ and inner 2½
revolutions of each whorl of the UPA/heterotic superstring corresponds to the distinction between the first three
Platonic solids and the fourth one. Alternatively, it corresponds to the 840 geometrical elements in the upper
halves of the four Platonic solids and to the 840 elements in their lower halves. The dodecahedron has the same
geometrical composition as the icosahedron. Indeed, they have the same numbers of corners, sides & triangles.
The 840:840 division of geometrical elements for this pair reflects the fact that each is the dual of the other, so
that they have the same value 30 of E (edges) and the same value 60 of nF (face sectors) appearing in the formulae
listed above.

The table indicates that the entries in each column for the subtotal row for the first four
Platonic solids are ten times those for the tetrahedron, whilst the entries for the total row are
**15** times the corresponding ones for the tetrahedron. They depend on the value of E (number of
edges) and nF (number of sectors in all faces). Taking the values E = 6 & nF = 12 for the tetrahedron as the
base values, their values for the five Platonic solids increase according to ×1, ×2, ×2, ×5 and ×5. We see that, in
terms of these base values, the sums of the values for E and nF for the first four Platonic solids (and therefore
for their numbers of corners, sides & triangles) = 1 + 2 + 2 + 5 = 10, whilst the sums for all five solids = 10
+ 5 = **15**. This reflects the Godname YAH (Hebrew: YH = 10 + 5 = **15**). The factor of
10 appears only in the first four Platonic solids. For historical reasons, the dodecahedron is regarded as the
fifth Platonic solid, being associated with the fifth Element, Aether. As it is dual to the icosahedron, it has the
same number of edges and the same number of face sectors as the icosahedron. This means that these two Platonic
solids are composed of the same numbers of corners, sides & triangles. Moreover, the first three solids have
the same total numbers as the icosahedron or dodecahedron because they have the same number of edges and the same
number of sectors. Arithmetically speaking, it makes no difference whether the superstring structural parameter
1680 is regarded as compounded from the icosahedron and the dodecahedron, from the first three solids and the
fourth solid or from the first three solids and the fifth one. However, although the modern mathematician or
physicist may regard as arbitrary which choice is correct, the fact that the dodecahedron displays the Golden Ratio
in its pentagonal faces — a proportion that has been believed for 2500 years to embody the divine ideal of beauty —
distinguishes it from the other Platonic solids. It is, therefore, fitting that the dodecahedron should be viewed
as the *last* in the mathematical sequence of regular polyhedra, quite apart from being the one with
the most vertices. The Tetrad Principle formulated in Article 1 states that numbers of universal significance like 1680 are always
determined by either the *fourth* member, or the first *four* members, of a class of
numbers or mathematical objects. The principle indicates that, of the three combinations of solids with 1680
geometrical elements surrounding their axes, the embodiment of this number by the first
*four* Platonic solids is the correct one. It is a remarkable confirmation of how this principle
yields numbers of universal significance — both to Earthlings and to ETs, in this case the number of circularly
polarized waves that run around each of the 10 closed curves of the E_{8}×E_{8} heterotic
superstring constituents of the up and down quarks in the protons and neutrons of atomic nuclei.

The 1680 geometrical elements surrounding the axes of the first four Platonic solids consist of 240 corners and 1440 sides & triangles (720 sides & triangles in the first three solids & 720 sides & triangles in the fourth solid, or, alternatively, 720 sides & triangles in each half of the four solids). This 240:720:720 pattern manifests in the outer & inner forms of the Tree of Life as the 240 yods other than Sephiroth in the 1-tree and the 720 yods that surround the centres of each set of seven separate, Type B polygons:

The tetrahedron & octahedron have **72** corners and the cube &
icosahedron have **168** corners. This **72**:**168** division is
characteristic of holistic systems. In terms of the root structure of the superstring gauge symmetry group
E_{8}, its 240 roots consist of the **72** roots of its exceptional subgroup
E_{6} and the **168** roots of E_{8} that are not roots of
E_{6}. That this is not a coincidence is indicated by the fact that the 240 corners of the first four
Platonic solids surround the eight endpoints of their axes, which correspond to the eight simple roots of
E_{8}. Amazingly , these Platonic solids embody not only the superstring structural parameter 1680 but also
the superstring dynamical parameter **248** as the dimension of E_{8}!

The number of corners surrounding the axes of the five Platonic solids = 360 =
**36**×10. This is also the number of geometrical elements in the faces of the first four Platonic
solids that surround their axes. The Divine Name ELOHA of Geburah with number value
**36** prescribes both all five Platonic solids and the first four solids. The number of sides
& triangles surrounding the axes of the five solids = 2160 = **216**×10, where
**216** is the number value of Geburah. Here is an example of the conjunction of two or more
number values referring to the same Sephirah. The average number of corners surrounding the axes of the five
Platonic solids is **72**. This is the number value of Chesed, the Sephirah preceding Geburah in the
Tree of Life. Including the centres, there are 375 corners, that is, (375 – **50** = 325) corners
other than the **50** polyhedral vertices, making an average of **65** extra
corners. This demonstrates how ADONAI, the Godname of Malkuth, prescribes the average additional number of points
in space needed to construct the five Platonic solids out of triangles.

The number of corners & sides surrounding the axes of the five Platonic solids = 1530 =
**153**×10. This shows how ELOHIM SABAOTH, the Godname of Hod with number value **153**,
prescribes the geometry of the five Platonic solids. Including the 20 corners & sides making up the five axes
that surround their centres, (1530 + 20 = 1550 = **155**×10) geometrical elements surround their
centres. ADONAI MELEKH, the complete Godname of Malkuth, prescribes the complete geometrical composition of the
five Platonic solids. As 1550 = **31**×10×5, the average number of geometrical elements surrounding
their centres = **31**×10, showing how EL, the Godname of Chesed with number value
**31**, prescribes the five Platonic solids. (310/2 = **155**) geometrical elements on
average in each half of a Platonic solid surround its centre, again showing how ADONAI MELEKH prescribes the
regular polyhedra.

The number of sides & triangles surrounding the axes of the five Platonic solids = 2160 =
**216**×10, where **216** is the number value of Geburah. Each half of a Platonic
solid on average has **216** such elements.

__Fine-structure number 137__The number of corners & triangles surrounding the axes
of the five Platonic solids = 1350. Including the 10 endpoints of their axes, the number of corners & triangles
surrounding the centres of the five Platonic solids = 1360

10 | 20 | 30 | 40 | |

50 | 60 | 70 | 80 | |

= | ||||

90 | 100 | 110 | 120 | |

130 | 140 | 150 | 160 . |

This is the sum of the gematria number values of Malkuth, its Godname ADONAI, its Archangel
*Sandalphon*, its Order of Angels *Ashim*, and its Mundane Chakra
*Cholem* *Yesodoth*:

**496** + **65** + **280** +
**351** + **168** = 1360.

The sceptic will, no doubt, attribute this to chance. But it would mean that it is also just
chance that the number **155** of the complete Godname of Malkuth is the average number of
corners & sides in half a Platonic solid that surround its centre. It is, surely, stretching common sense too
far to believe that the *same* Sephirah could appear twice by accident in this context?! The average
number of corners & triangles surrounding the centres of the five Platonic solids = 1360/5 =
**272**. This is the number value of *Cherubim*, the Order of Angels assigned to Yesod. The
average number of corners & triangles in each half of a Platonic solid that surround its centre = 136.
Therefore, including the centre, 137 corners & triangles (38 corners, 99 triangles) are needed on average to
create half a Platonic solid. This show how the scientifically mysterious number 137, which determines the
approximate value of the fine-structure constant α = e^{2}/ħc ≅1/137 at the heart of atomic physics, measures the geometry of the five
Platonic solids (see also #16 in **Sacred geometry/Platonic
solids**). Its geometrical basis indicates that it divides into (1+136). It is predicted that any
theoretical derivation of the fine-structure constant will replicate in some analogous way this geometrical
distinction between the centre of a Platonic solid and the 136 corners & triangles needed on average to
shape half of it.

__Dimensions 248 of E_{8} & 496 of
E_{8}×E_{8}__The table above indicates that 2520 corners, sides & triangles surround the
axes of the five Platonic solids. Of these, 40 are vertices. Hence, (2520−40=2480) geometrical elements other than
vertices surround their axes. On average, (2480/5 =

The axis of each Platonic solid consists of its two endpoints, its middle & two straight
lines, i.e., five geometrical elements. Therefore, (5+**496**=501) geometrical elements other than the
8 vertices surrounding the axis are needed on average to build a Platonic solid, that is, 500
(=**50**×10) elements, starting with its centre. This demonstrates another way in which the Godname
ELOHIM with number value **50** prescribes the five Platonic solids with
**50** vertices & **50** faces. Indeed, as was seen in #9 of this section, **50** overlapping Trees
of Life comprise 2520 yods when their 604 triangles are tetractyses and this is, precisely, the number of
corners, sides & triangles that surround the axes of the five Platonic solids! Here is the way in which the
Tree of Life encodes parameters of analogous, holistic systems such as the five Platonic solids. The 40 yods in
the **50**th Tree next above the **49**-tree with 2480 yods correspond to the 40
vertices of the five Platonic solids that surround their axes.

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