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**#20 Holistic properties of the
dodecahedron**

A Type B n-gon is an n-gon whose n sectors are Type A triangles. It has (2n+1) corners & 5n
sides of 3n triangles, i.e., (10n+1) geometrical elements, the "1" referring to its centre. The Type B pentagon
(n=5) is special in that the number of geometrical elements surrounding its centre is **50**, which
is the number of ELOHIM, the Godname of Binah, the "Great Mother" source of all divine archetypes of Form. The
number of yods in a Type B n-gon = 2n + 1 + 5n×2 + 3n = **15**n + 1, the number of hexagonal yods
being 13n. So the pentagon is also special because its Type B version has **76** yods, of
which **65** are hexagonal yods, that is, it is the only regular polygon whose geometrical
composition and two yod populations are all Godname numbers (it also contains
**15** triangles, **26** corners & triangles and
**36** corners & sides, where **15** is the number value of YAH, Godname
of Chokmah, **26** is the number value of YAHWEH, the full Godname of this Sephirah, and
**36** is the number value of ELOHA, the Godname of Geburah). Being so richly prescribed by
Divine Names, it should not be surprising that, as a face of the dodecahedron, the pentagon endows this regular
polyhedron with unique properties, in keeping with its traditional assignment by the ancient Greeks to Aether,
their fifth Element, although it is really the *first,* progenitor Element in terms of the sequence
of generation of the five Elements. These holistic properties are now analyzed.

For brevity, the geometrical composition of an object constructed from triangles will be
written: (A, B, C), where A is the number of corners of its C triangles with B sides. The Type B pentagon (see
diagram opposite) has 41 internal geometrical elements: (6, 20, **15**). The dodecahedron has 12
pentagonal faces with internal geometrical composition: 12×(6, 20, **15**) = (**72**,
240, 180). It has 20 vertices & 30 edges, so that its faces have the geometrical composition:
(20+**72**, 30+240, 180) = (92, 270, 180). The faces of the dodecahedron have (92+270+180=542)
geometrical elements that surround its centre — a total of **543** geometrical elements. This
is the number value of EHYEH ASHER EHYEH ("I am that I am"), which is the *complete* Godname of
Kether:

אהיה אשר אהיה

** 21** + **501** + **21** =
**543**

As there are 30 edges and 12 corners that are centres of pentagonal faces,
**21** edges & face centres prescribed by EHYEH (אהיה) are in each half of the dodecahedron, leaving
**501** geometrical elements prescribed by ASHER (אשר) that consist of its centre and 500 (=**50**×10) elements
with the composition: (**80**, 240, 180), i.e., they comprise (**80**+180=260
=**26**×10) corners & triangles and 240 sides. This demonstrates how EHYEH, YAHWEH &
ELOHIM prescribe the shape of the dodecahedron (see also below).

The number **543** was shown in #13 to be the number of yods intrinsic to the seven enfolded Type
B polygons of the inner Tree of Life that line their 141 tetractyses. As the 91 overlapping Trees of Life in the
Cosmic Tree of Life have 550 SLs (see here), the number **543** is the number of SLs down to
Daath of the lowest Tree of Life (Yesod of the second Tree).

Now consider the 20 vertices of the dodecahedron being joined to its centre so that its 30 edges
become sides of 30 different, internal triangles. When the latter are Type B, each 3-gon has the geometrical
composition (7, **15**, 9), i.e., the internal composition (4, 12, 9). The joining of vertices to
the centre creates 20 sides of triangles, so that the geometrical composition of the 30 internal triangles is:
(1 + 30×4, 20 + 30×12, 30×9) = (121, 380, 270). Therefore, 500 (=**50**×10) corners & sides and
650 (=**65**×10) sides & triangles in the interior of the dodecahedron surround its centre,
showing how ELOHIM, the Godname of Binah with number value **50**, and ADONAI, the Godname of
Malkuth with number value **65**, prescribe its internal geometrical composition. ELOHIM also
prescribes its 12 faces, which have **50** vertices & edges.

The geometrical composition of the dodecahedron with Type B faces and interior triangles is:

Faces: (92, 270, 180);

Interior: (121, 380, 270);

Total = (213, 650, 450).

The dodecahedron is composed of 1313 geometrical elements. Surrounding its centre are 1312
elements, i.e., (1312−12−30=1270=127×10) elements other than vertices & centres of faces, where 127 is the
**31**st prime number and **31** is the number of EL, the Godname of Chesed.
Surrounding an axis passing through two opposite vertices and its centre are (213−3=210=**21**×10)
corners and (650−2=648) sides of 450 triangles, i.e., 1308 geometrical elements. EHYEH with number value
**21** prescribes the number of points that surround the axis. ADONAI prescribes the 650
straight lines in the dodecahedron. The number of lines & triangles = 650 + 450 = 1100, so that each half of
this Platonic solid consists of 550 lines & triangles. In other words, the dodecahedron embodies the same number as the number (550) of SLs in the Cosmic Tree of
Life! Plato asserted that the dodecahedron represented the celestial sphere because, of all the
regular polyhedra, it most resembled the sphere. We see now that its geometry embodies the number that measures
all levels of reality! In fact, it is embodied in *all* five Platonic solids because they have 550
corners, sides & triangles in their **50** faces when the latter are Type A (see here).

As 550 = 10×(1+2+3+4+5+6+7+8+9+10), i.e., 550 is *ten* times the
*tenth* triangular number, we see how the Decad determines this archetypal number as a parameter of
holistic systems. It does so geometrically as well as arithmetically because 550 corners & sides of 270
triangles surround a Type D decagon (see here).

Of the 1313 geometrical elements, 20 are vertices of the dodecahedron and three are elements
making up the axis, leaving 1290 (=**129**×10) elements other than vertices surrounding the axis.
This shows how YAHWEH SABAOTH, the Godname of Netzach with number value **129**, prescribes the
geometry of the dodecahedron. It has (20+30=**50**) vertices & edges, so that
(1313−**50**−3=**1260**) geometrical elements other than vertices & edges surround
the axis. **1260** is the number value of *Tarshishim*, the Order of Angels assigned to
Netzach. Here is a remarkable conjunction in the same context of two number values referring to the
*same* Sephirah. **630** such geometrical elements surround the axis in each half
of the dodecahedron. **630** is the number value of *Seraphim*, the Order of Angels
assigned to Geburah. The **630** elements include **36** corners, where
**36** is the number value of ELOHA, the Godname of Geburah (another example of conjunction!).
Of the 650 sides of the 450 triangles, 30 are edges, leaving **620** sides that are not edges.
This is the number value of Kether, which was shown in #5 to be the number of hexagonal yods in the decagon with
2nd-order tetractyses as its sectors. The triangles in each half of the dodecahedron have 310
(=**31**×10) sides that are not its edges. This shows how EL, the Godname of Chesed with number
value **31**, prescribes the dodecahedron.

Now let us consider the yod composition of the dodecahedron when its 450 triangles are tetractyses. Each side of a tetractys has two hexagonal yods. Writing the yod composition of an object as [A, B, C], where A is the number of yods at corners of its C tetractyses, B is the number of hexagonal yods on their sides and C is the number of hexagonal yods at their centres, the yod population of the dodecahedron is:

Faces: [92, 540, 180];

Interior: [121, 760, 270];

Total = [213, 1300, 450].

It is composed of 812 yods in its faces and 1151 yods in its interior — a total of 1963 yods.
1300 hexagonal yods line the 650 sides of 450 tetractyses, where 1300 = 1^{5} +
2^{5} + 3^{5} + 4^{5}, demonstrating the beautiful properties of this Platonic
solid. There are (540+180=720) hexagonal yods in its faces. This is the number of yods that surround the centres
of the seven separate Type B polygons, as well as the number of yods surrounding the centre of a decagon with
2nd-order tetractyses as sectors (see #6) and the number of yods outside the root edge in the combined outer
& inner Trees of Life (see #19). The dodecahedron contains (1300+450=1750) hexagonal yods. As we
found in #19, this is the number of hexagonal yods outside the root edge that are unshared by the combined Trees
of Life when their triangles & polygons are Type B.

The dodecahedron with 20 vertices and 30 edges is lined by (2×30 + 20 = **80**)
yods. This is the number of Yesod. The number of yods in the 12 faces other than their centres = 812 − 12 = 800
= **80**×10. The 180 tetractyses in the faces have (92+180=**272**) corners and
central hexagonal yods. **272** is the number value of *Cherubim*, the Order of Angels
assigned to Yesod. Here is yet another example of conjunction of two different numbers associated with the same
Sephirah. The number of yods in faces other than on their edges & at their centres = 800 −
**80** = 720. There are 1151 yods inside the dodecahedron, i.e, 1150 internal yods surround
its centre, so that (720+1150=1870=**187**×10) yods other than those on edges or at centres of
faces surround the centre. **187** is the number value of *Auphanim*, the Order of
Angels assigned to Chokmah.

The 450 triangles in the dodecahedron have 212 corners surrounding its centre, of which 20 are
vertices. This leaves 192 corners (**72** in faces, 120 internal). Therefore, 192 extra points
have to be added in order to construct the dodecahedron from triangles. This number is a defining parameter of
holistic systems (see here). The numbers of geometrical elements surrounding the axis other than
vertices are shown below:

Faces: (

72, 270, 180);

Interior: (120, 378, 270);Total = (192, 648, 450).

(192+648=840) sides & non-polyhedral corners of 450 triangles surround the axis. This is the
number of circular turns in an outer or inner half of each helical whorl of the UPA. *Embodied in the dodecahedron is this structural parameter of the subquark state of the
E _{8}×E_{8} heterotic superstring.* It appears in the sacred geometry of
the inner Tree of Life as the 840 intrinsic corners associated with each of the two sets of 70 Type B polygons
enfolded in ten overlapping Trees of Life (see here). It also appears in the pair of joined hexagons with their
sectors turned into 2nd-order tetractyses as the number of yods outside their shared side (see #17 at Wonders of sacred
geometry/superstrings).

Let us now consider the dodecahedron with Type A pentagonal faces and internal triangles. Its geometrical composition is:

Faces: (32, 90, 60);

Interior: (

31, 110, 90);Total = (63, 200, 150).

The faces are composed of 182 geometrical elements and the interior has 231 elements, making a
total of 413 elements. The number 182 is prescribed by YAHWEH with number value
**26** because

2 + 4 + 6 + .... + **26** =
182.

The number 231 is prescribed by EHYEH, the Godname of Kether with number value
**21**, because

1 + 2 + 3 + ... + **21** =
231.

YAH with number value **15** prescribes the dodecahedron because it has 150
(=**15**×10) triangles. There are 263 corners & sides of 150 triangles. The seven enfolded Type
A polygons of the inner Tree of Life have 264 yods (see last diagram in #46 here). The topmost corner of the hexagon coincides with the lowest
corner of the hexagon enfolded in the next higher Tree of Life. This means that 263 yods are intrinsic to the
set of seven polygons enfolded in successive Trees of Life. That this is not coincidence is indicated by the
fact that the triangle, square & pentagon have 63 yods outside the root edge and the hexagon, octagon,
decagon & dodecagon have 200 yods, which correspond to the 63 corners and 200 sides of the 150 triangles in
the dodecahedron. 258 corners & sides surround its axis, which comprises three corners and two sides of
triangles. 258 is the **129**th even integer, showing how YAHWEH SABAOTH, the Godname of Netzach
with number value **129**, prescribes the geometry of the dodecahedron.

There are 350 sides & triangles in the dodecahedron. This is a parameter of holistic
systems, being, for example, the sum of the 20 integers in the Tetrahedral Lambda (see here), the 200 sides of triangles in the dodecahedron corresponding to the sum
of the 12 integers on the six edges of the Tetrahedral Lambda and the 150 triangles corresponding to the sum of
the integers at its vertices and centres of its faces. The number 350 is also the number of hexagonal yods in
the (7+7) enfolded Type A polygons of the inner Tree of Life (see Figure 7 here). The triangles in the 12 faces of the dodecahedron have 90 sides,
leaving 260 (=**26**×10) triangles & internal sides. This 90:260 division in the number 350 is
characteristic of holistic systems (see here). The 90 sides correspond to the tetractys of integers in the first
face of the Tetrahedral Lambda because their sum is 90, whilst the 260 remaining triangles & internal sides
correspond to the sum of the ten integers in the three other faces. The embodiment in the dodecahedron of the
numbers 90, 260 & 350 is a clear indication of its holistic nature, as confirmed by its embodiment
(discussed earlier) of the superstring structural parameter 840 and (as we found above and shall confirm below)
the number 550 — a basic parameter of the Cosmic Tree of Life (see here).

The dodecahedron has (63+150=213) corners & triangles, so that 210 (=**21**×10)
corners & triangles surround an axis passing through two opposite vertices and the centre of the
dodecahedron. This shows how EHYEH with number value **21** prescribes its geometry.

The yod composition of the dodecahedron with Type A faces and internal triangles is:

Faces: [32, 180, 60];

Interior: [

31, 220, 90];Total = [63, 400, 150].

Each pentagonal face has **31** yods and the 60 tetractyses in its 12 faces
have 32 corners, where 32 is the **31**st integer after 1. Its interior has
**31** corners and 310 (=**31**×10) hexagonal yods. These properties show how EL,
the Godname of Chesed with number value **31**, prescribes the construction of the dodecahedron
from tetractyses. **62** corners of 150 tetractyses surround its centre.
**62** is the **31**st even integer and the number value of *Tzadkiel*,
the Archangel of Chesed. Here is another example of the conjunction of two numbers referring to the
*same* Sephirah.

The dodecahedron has (400+150=550) hexagonal yods. It was shown earlier that each half of this
Platonic solid comprises 550 lines & triangles. We now see that it contains the same number of hexagonal
yods! *They correspond to the 550 SLs of the Cosmic Tree of Life*. This is another way in which the
dodecahedron is a polyhedral map of the spiritual cosmos. The 240 hexagonal yods in its faces symbolize the 240
roots of the superstring gauge symmetry group E_{8} (see also here). 60 hexagonal yods line the 30 edges of the dodecahedron, leaving 490
(=**49**×10) hexagonal yods in its faces and interior, where **49** is the number
value of Yesod. Its faces have **272** yods, where **272** is the number
value of *Cherubim*, the Order of Angels assigned to Yesod. Here is another example of the conjunction in
an object possessing sacred geometry of two numbers that refer to the same Sephirah. In fact, the conjunction is
between *three* numbers, for — as was pointed out earlier — **80** yods line the
30 edges of the dodecahedron, where **80** is the number value of Yesod!

The dodecahedron is composed of (63+400+150=613) yods. 612 yods surround its centre, i.e., 306
yods in each half. 306 is the **153**rd even integer, showing how ELOHIM SABAOTH, the Godname of
Hod with number value **153**, prescribes the dodecahedron. (63+400=463) yods line the 200 sides of
the 150 tetractyses. 231 such yods in each half of the polyhedron surround its centre, showing again how it is
prescribed by EHYEH, because 231 = 1 + 2 + 3 +... + **21**.

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