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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 = 15n + 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 31st 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 = 15 + 25 + 35 + 45, 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 E8×E8 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 polyogns 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 129th 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 31st 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 31st 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 E8 (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 153rd 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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