ARTICLE 55
by
Stephen M. Phillips
Flat 4, Oakwood House, 117-119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
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Construction of the faces of the tetrahedron, octahedron, cube & icosahedron requires 248 points & straight lines. Believed by the ancient Greeks to be the shapes of the particles of the four Elements Earth, Water, Air & Fire, these Platonic solids, therefore, embody E8, the rank-8, exceptional Lie group with dimension 248 at the heart of superstring theory that describes the symmetry of the unified force between superstrings — the basic particles of matter. It is implausible that this is coincidence, because the numbers of points & lines in various subsets of the set of 248 points & lines are the numbers of roots in the four exceptional subgroups of E8. 2480 geometrical elements surround the axes of the five Platonic solids when their faces and interiors are constructed from triangles. This is the number of space-time components of the 248 10-dimensional vector gauge fields that transmit superstring forces. This cannot be due to chance because numbers that quantify the geometrical compositions of various subsets of elements are (apart from the Pythagorean factor of 10) equal to the numbers of roots of the exceptional subgroups of E8. On average, 496 geometrical elements other than vertices surround the axes of the five Platonic solids, 248 elements making up each half. This is the regular polyhedral counterpart of E8×E8, the symmetry group with dimension 496 which is associated with one of the two types of heterotic superstrings. 1680 geometrical elements (including their vertices) surround the axes of the first four Platonic solids. This number is the number of circular turns in each of the ten helices making up the basic constituents of matter, as described over a century ago by C.W. Leadbeater with the aid of a yogic siddhi. Many earlier articles have identified these particles as the subquark state of the E8×E8 heterotic superstring. That this is not coincidental is indicated by how the gematria number values of the Hebrew words making up the Kabbalistic name of the Mundane Chakra of Malkuth refer naturally to the sides, corners & triangles making up these Platonic solids. It is confirmed by the fact that the same numbers manifest in other sacred geometries, such as the outer & inner Trees of Life and the disdyakis triacontahedron. This is because, as sacred geometry, the Platonic solids embody the same mathematical archetypes as those found in these geometries. It is also confirmed by the fact that 168 yods on average in the faces and axes of the five Platonic solids surround their centres when they are constructed from tetractyses. |
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Table 1. Gematria number values of the ten Sephiroth in the four Worlds.
ANGELS
CHAKRA1 Kether
(Crown)
620EHYEH
(I am)
21Metatron
(Angel of the
Presence)
314Chaioth ha Qadesh
(Holy Living
Creatures)
833Rashith ha Gilgalim
First Swirlings.
(Primum Mobile)
6362 Chokmah
(Wisdom)
73YAHWEH, YAH
(The Lord)
26, 15Raziel
(Herald of the
Deity)
248Auphanim
(Wheels)
187Masloth
(The Sphere of
the Zodiac)
1403 Binah
(Understanding)
67ELOHIM
(God in multiplicity)
50Tzaphkiel
(Contemplation
of God)
311Aralim
(Thrones)
282Shabathai
Rest.
(Saturn)
317Daath
(Knowledge)
4744 Chesed
(Mercy)
72EL
(God)
31Tzadkiel
(Benevolence
of God)
62Chasmalim
(Shining Ones)
428Tzadekh
Righteousness.
(Jupiter)
1945 Geburah
(Severity)
216ELOHA
(The Almighty)
36Samael
(Severity of God)
131Seraphim
(Fiery Serpents)
630Madim
Vehement Strength.
(Mars)
956 Tiphareth
(Beauty)
1081YAHWEH ELOHIM
(God the Creator)
76Michael
(Like unto God)
101Malachim
(Kings)
140Shemesh
The Solar Light.
(Sun)
6407 Netzach
(Victory)
148YAHWEH SABAOTH
(Lord of Hosts)
129Haniel
(Grace of God)
97Tarshishim or
Elohim
1260Nogah
Glittering Splendour.
(Venus)
648 Hod
(Glory)
15ELOHIM SABAOTH
(God of Hosts)
153Raphael
(Divine Physician)
311Beni Elohim
(Sons of God)
112Kokab
The Stellar Light.
(Mercury)
489 Yesod
(Foundation)
80SHADDAI EL CHAI
(Almighty Living
God)
49, 363Gabriel
(Strong Man of
God)
246Cherubim
(The Strong)
272Levanah
The Lunar Flame.
(Moon)
8710 Malkuth
(Kingdom)
496ADONAI MELEKH
(The Lord and King)
65, 155Sandalphon
(Manifest Messiah)
280Ashim
(Souls of Fire)
351Cholem Yesodoth
The Breaker of the
Foundations.
The Elements.
(Earth)
168
(Numbers in this table referred to in the article will be written in boldface).
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1. The faces of the first four Platonic solids embody the dimension 248 of E8
The five Platonic solids are known to mathematicians as the five regular, convex, 3-dimensional polyhedra. They are named after Plato, who propounded in his Timaeus the Pythagorean doctrine that particles of the four Elements Earth, Water, Air & Fire had the shapes of the cube, icosahedron, octahedron & tetrahedron.
Figure 1. The five Platonic solids have 50 faces with 180 sectors.
The Pythagoreans associated the fifth regular polyhedron, the dodecahedron, with the celestial sphere. It came, later, to be identified with Aether, the fifth Element. The faces of the five Platonic solids are either equilateral triangles (tetrahedron, octahedron & icosahedron), squares (cube) or pentagons (dodecahedron). They have 50 vertices, 50 faces & 90 edges (Fig. 1). The number 50 is the number value of ELOHIM, the Godname of Binah (see Table 1), showing how this Godname prescribes their shapes. On average, the five solids have 10 vertices, 10 faces & 18 edges, i.e., 28 vertices & edges, where
28 = 1 + 2 + 4 + 7 + 14.
1, 2, 4, 7 & 14 are the factors of 28, so that it is the second perfect number [1]. Later, we shall reveal how the five Platonic solids embody the third perfect number 496.
Table 2. Geometrical composition of the faces of the five Platonic solids.
Polyhedron
V
E
F
m
C
e
T
Total = C + e + T
Tetrahedron
4
6
4
3
8
18
12
38
Octahedron
6
12
8
3
14
36
24
74
Cube
8
12
6
4
14
36
24
74
Icosahedron
12
30
20
3
32
90
60
182
Subtotal
30
60
38
–
68
180
120
368
Dodecahedron
20
30
12
5
32
90
60
182
Total =
50
90
50
–
100
270
180
550
(V = number of vertices, E = number of edges, F = number of faces, m = number of sectors in a face, C = V + F = number of corners of sectors, e = E + mF = number of sides, T = mF = number of triangles in F faces).
Division of their faces into their sectors generates 180 triangles with (50+50=100) corners and (90+180=270) sides. Table 2 lists the number of corners, sides and triangles in the faces of each solid. The faces of the five Platonic solids are composed of 550 points, lines & triangles, where
55
55 55
550 = 55 55 55
55 55 55 55
and
1
2 3
55 = 4 5 6
7 8 9 10 .
As 55 is the tenth triangular number, the Pythagorean Decad (10) determines their geometrical composition.
[1] Perfect numbers are numbers that are the sum of their factors. The first three perfect numbers are 6, 28 & 496.
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This beautiful property reveals the archetypal character of the polyhedral sacred geometry of the five Platonic solids. ELOHIM with number value 50 prescribes them because their 50 faces have 500 (=50×10) geometrical elements other than their 50 vertices.
Table 3 shows the numbers of polyhedral vertices (C) and non-polyhedral (V) corners, edges (E) and internal sides (e) of the triangular sectors of the faces of the first four Platonic solids:
Table 3. Geometrical composition of the faces of the first four Platonic solids.
Polyhedron
C
V
E
e
Total
Tetrahedron
4
4
6
12
26
Octahedron
6
8
12
24
50
Cube
8
6
12
24
50
Icosahedron
12
20
30
60
122
Total =
30
38
60
120
248
Notice how the Godname YAHWEH with number value 26 prescribes the tetrahedron (the simplest Platonic solid) with 26 corners & sides of the 12 triangles in its four faces. Notice also that the octahedron and cube each has 26 corners & polyhedral edges and 50 corners & sides, where 50 is the number value of ELOHIM, the Godname of Binah. The 24 triangles in the faces of each of these two polyhedra have 36 sides, where 36 is the number value of ELOHA, the Godname of Geburah, which is the Sephirah next below Binah on the Pillar of Judgement of the Tree of Life. The 120 triangular sectors of the 38 faces of the first four Platonic solids have 68 corners and 180 sides, i.e., 248 corners & sides. This shows how they embody the dimension 248 of E8, the rank-8, exceptional Lie group. The ancient Greeks believed that the tetrahedron, octahedron, cube & icosahedron were the shapes of the particles of, respectively, the elements Fire, Air, Earth & Water. We now see that these Platonic solids, indeed, do represent the properties of physical matter because their geometrical composition is quantified by the very number that, according to E8×E8 heterotic superstring theory, is the number of particles transmitting the unified force between superstrings of ordinary matter.
The dimension of a Lie group is the number of its so-called “roots,” the definition of which need not detain us (see here). This set of roots contains a subset of “simple roots” (the number of which defines the rank of the group). The exceptional Lie group with the largest rank (8) is E8. It has eight simple roots and 240 roots. The four vertices and the four face-centres in the tetrahedron constitute eight points, so it seems, intuitively speaking, more natural to associate them with the eight simple roots of E8, rather than the eight centres of the faces of the octahedron or the eight vertices of the cube. The remaining 240 points and straight lines making up the first four Platonic solids correspond to its 240 roots. The 248 corners & sides comprise 128 corners & edges and 120 internal sides of sectors.
There are (6+12=18) lines in the tetrahedron and (6+8+8+6+12+20=60) points in the octahedron, cube & icosahedron (see Table 3). Hence, they form a set of 78 points & lines. (30+60=90) lines are in the faces of the icosahedron. There are (78+90=168) points & lines other than the (12+24+12+24=72) lines in the octahedron & cube. 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, 78 being the number of Cholem and 90 being the number of Yesodoth (Fig. 2).
Figure 2. The gematria number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, is 168.
The number 72 could be the sum of other combinations of numbers in Table 3. However, it is readily verified that the simplest and most natural combination is the two pairs of numbers 12 & 24. Article 53 proves that sacred geometries embody the number 240 and that it displays the factorization 10×24. This means that we should expect the number 72 appearing in the geometrical composition of the faces of the first four Platonic solids to display the factorization 3×24 if it refers to the correct combinations. This, indeed, is the case for the 72 lines in the octahedron and cube, which have 24 edges, each polyhedron having 24 more sides of its 24 sectors. Notice also that, as the tetrahedron has 8 points in its faces, the number 80 (=8+72 ) arises naturally in the set of the first four Platonic solids. The division:
4
248 = 80 + 168
is characteristic of holistic systems. An example is the set of 248 yods up to Chesed of the fifth Tree (the 31st SL) when the triangles in the lowest five Trees are tetractyses, the lowest Tree having 80 yods (see here).
The fact that the first four Platonic solids embody the holistic parameter 248 in their geometry illustrates the Tetrad Principle formulated in Article 1. It states that the first four members of a class of mathematical object or, alternatively, the fourth member of that class, always express parameters of holistic systems.
2. Counterparts of exceptional subgroups of E8
The sceptic would find it difficult to defend his suggestion that the number 248 arises in the first four Platonic solids simply by chance if it were pointed out that the numbers of roots in the four exceptional subgroups of E8:
G2 F4 E6 E7 Number of roots: 12 48 72 126 also manifest in the points & lines making up the faces of the first four Platonic solids, as is now shown. Ignoring the eight points in the tetrahedron (its four vertices & four face-centres) that correspond to the eight simple roots, Table 4 indicates that the octahedron, cube & icosahedron have (26+34=60) points and 54 edges, whilst the tetrahedron has 12 internal lines. They are indicated by the numbers in the yellow cells, which sum to 126. They correspond to the 126 roots of E7, the largest exceptional subgroup of E8.
Table 4. The 126:114 division of the 240 points & lines in the first four Platonic solids.
Polyhedron
C
V
E
e
Total
Tetrahedron
–
–
6
12
18
Octahedron
6
8
12
24
50
Cube
8
6
12
24
50
Icosahedron
12
20
30
60
122
Total =
26
34
60
120
240
The numbers in the remaining orange cells add up to 114. As G2≤F4≤E6≤E7≤E8, the geometrical counterpart of the roots of an exceptional group must contain all the geometrical elements corresponding to a smaller, exceptional subgroup. The choice of numbers adding to 240 is consistent with this requirement because:
E7: the 240 points & lines contain a subset of 126 points & lines:
126 = 26 + 34 + 12 + 12 + 30 + 12;
E6: the subset of 126 points & lines contains a subset of 72 points & lines:
72 = 26 + 34 + 12;
F4: the subset of 72 points & lines contains a subset of 48 points & lines:
48 = 6 + 8 + 8 + 6 + 20
or
48 = 8 + 12 + 8 + 20;
G2: the subset of 48 points & lines contains a subset of 12 points & lines:
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12 = 6 + 6
or, alternatively, the 12 vertices of the icosahedron. The ambiguity of choice is, of course, irrelevant to the point demonstrated here, namely, that the original selection of 240 points & lines contain subsets that correspond to the roots of the exceptional subgroups of E8.
Suppose that the 50 faces of the five Platonic solids are divided into their sectors and that their 50 vertices and 50 face-centres are joined to their centres (Fig. 3). Sides of sectors are sides of triangles inside the polyhedra with vertices and their centres at their corners. For consistency with the division of all F polygonal faces of a Platonic solid with C vertices and E edges into their nF sectors (n = 3, 4 or 5), these internal triangles must be divided likewise, i.e., regarded as Type A triangles. Tabulated below are the numbers of non-vertex corners, sides & triangles in their faces and interiors that surround their axes, each containing two vertices and two sides of internal triangles:
Table 5. Formulae for non-vertex corners, sides & triangles in a Platonic solid.
Corners
Sides
Triangles
Total
Faces:
F
E + nF
nF
E + F + 2nF
Interior:
E + nF
C − 2 + F + 3E + 3nF = 4E + 3nF
3E + 3nF
8E + 7nF
Total:
E + F + nF
5E + 4nF
3E + 4nF
9E + F + 9nF
(Euler's formula for a polyhedron: C − E + F = 2 has been used to simplify expressions). Tabulated below are the numbers of:
Table 6. Geometrical composition of the faces & interiors of the five Platonic solids surrounding their axes.
Platonic solid
Faces
Interior
V
S
T
Subtotal
c
s
t
Subtotal
c′
s′
t′
Subtotal
Total
Tetrahedron (n = 3)
4
18
12
34
6
24
18
48
12
36
36
84
166
Octahedron (n = 3)
8
36
24
68
12
48
36
96
24
72
72
168
332
Cube (n = 4)
6
36
24
66
12
48
36
96
24
72
72
168
330
Icosahedron (n = 3)
20
90
60
170
30
120
90
240
60
180
180
420
830
Subtotal =
38
180
120
338
60
240
180
480
120
360
360
840
1658
Dodecahedron (n = 5)
12
90
60
162
30
120
90
240
60
180
180
420
822
Total =
50
270
180
500
90
360
270
720
180
540
540
1260
2480
We find that 2480 points, lines & triangles are needed to construct the five Platonic solids around their axes, given their 50 vertices. This is the number of geometrical elements that surround their axes, each of which contains three points and two lines. Amazingly, as the unified superstring force with E8 symmetry is transmitted by 248 vector gauge fields, each with 10 space-time components, these fields have 2480 vector components. To every such component, there is a corresponding geometrical element in the five Platonic solids. 248 is the number value of Raziel, the Archangel of Chokmah. YAHWEH, the Godname of this Sephirah, prescribes the dimension of E8 because it is the mean of the squares of all the integers after 1 up to 26, which is the number value of YAHWEH:
248 = (22 + 32 + 42 + ... + 262)/25.
The following evidence indicates that this result is not due to coincidence. The five exceptional groups are G2 (dimension = 12+2), F4 (dimension = 48+4), E6 (dimension = 72+6), E7 (dimension = 126+7) & E8 (dimension = 240+8), where "+n" means that the group has n simple roots. As G2≤F4≤E6≤E7≤E8, the geometrical counterpart of an exceptional group must contain all the geometrical elements corresponding to any smaller, exceptional subgroup of this group. Table 6 shows that inside the five Platonic solids and surrounding their axes are 1260 geometrical elements other than vertices that belong to triangles with sides of face sectors (green triangles of the type indicated in Fig. 3). 1260 is the number value of Tarshishim, the Order
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of Angels assigned to Netzach. They include (180+540=720) points & lines. Of these, the table indicates that (120+360=480) belong to the first four Platonic solids, which contain 120 points. Compare the correspondence between the roots of the five exceptional groups and these sets of elements (in the case of E8, its eight simple roots are included):
G2: 12 roots ↔ 120 internal points in first four Platonic solids;
F4: 48 roots ↔ 480 internal points & lines in first four Platonic solids;
E6: 72 roots ↔ 720 internal points & lines in five Platonic solids;
E7: 126 roots ↔ 1260 internal points, lines & triangles in five Platonic solids;
E8: 248 roots ↔ 2480 points, lines & triangles in five Platonic solids.
We see that not only do the five Platonic solids have 2480 geometrical elements other than vertices surrounding their axes but they also are composed of sets of elements that are ten times the numbers of roots (excluding simple roots) in the four exceptional subgroups of E8. This cannot be as well the result of chance! Rather, the more sensible conclusion to be drawn is that an isomorphism exists between the geometrical composition of the five Platonic solids and the root composition of E8 and its exceptional subgroups. Given that these famous polyhedra have been shown in previous articles to constitute a holistic system with features analogous to other sacred geometries, it should not seem surprising that the number 2480 manifests in their geometrical composition. This remarkable analogy demonstrates that superstring theory is part of the holistic pattern embodied in these geometries.
The icosahedron is the fourth Platonic solid. According to Table 2 & Table 6, it has (20+60=80) points & triangles in its faces. They also have 80 points in its faces and in internal triangles formed by sides of their sectors. Intuitively speaking, the latter seems the more likely counterpart of the eight simple roots of E8 because they are all points, whereas the former mixes points with triangles. It is readily verified by combining the various numbers in the table that no other Platonic solid has a combination of 80 geometrical elements of the same kind. The 60 corners of the internal triangles of the icosahedron formed by the sides of the sectors in its faces are the counterparts of the six simple roots of E6, whilst the centres of its 20 faces are the counterparts of the two extra simple roots of E8. The remaining 2400 geometrical elements consist of the following:
Faces
Interior
Interior
Corners
Sides
Triangles
Subtotal
Corners
Sides
Triangles
Subtotal
Corners
Sides
Triangles
Subtotal
Total
30
270
180
480
90
360
270
720
120
540
540
1200
2400
As pointed out above, the 720 geometrical elements in the blue interior triangles generated by the edges correspond to the 72 roots of E6. The (480+1200=1680) geometrical elements in the faces and in the interior green triangles generated by the sides of sectors of faces correspond to the remaining 168 roots of E8. Previous articles have given many examples of the 72:168 division of the parameter 240 embodied in sacred geometries. It re-appears naturally in the geometrical composition of the five Platonic solids as the 720 internal geometrical elements generated by edges that surround the axes of the five Platonic solids and as the remaining 1680 elements other than the 80 corners in either the faces of the icosahedron or the internal triangles formed by sides of their sectors.
Referring to Table 6, surrounding the axis of a Platonic solid are, on average, (2480/5=496) geometrical elements other than vertices. 248 geometrical elements other than vertices on average surround its axis in each half. This is the regular polyhedral basis of the E8×E8 heterotic superstring, the counterpart of the direct product being the fact that each geometrical element in one-half of the “average” Platonic solid has a corresponding mirror-image element in its other half. 320 points other than vertices surround the axes of the five Platonic solids, the average number of points being 320/5 = 64, which is the number of Nogah, the Mundane Chakra of Netzach. Including their centres, the average number of points other than vertices is 65. This is the number value of ADONAI, the Godname of Malkuth. Including the 40 vertices surrounding their axes, the average number of points surrounding the axis of a Platonic solid = 65 + 40/5 = 73. This is the number value of Chokmah, the second Sephirah of the Tree of Life. According to Table 6, the number of sides surrounding the axes of the five Platonic solids = 270 + 360 + 540 = 1170. The number of triangles = 180 + 270 + 540 = 990. Therefore, (1170+990=2160=216×10) sides & triangles surround their axes, where 216 is both the number of Geburah, the fifth Sephirah, and the average number of sides & triangles in each half of a Platonic solid that surround its axis.
3. First four Platonic solids embody the superstring structural parameter 1680
Including their 30 vertices, there are (30−8=22) vertices surrounding the axes of the first four Platonic solids. The table above shows that 1658 other geometrical elements surround their axes. The total number of such elements = 1658 + 22 = 1680 = 168×10. This is another fundamental parameter of superstring physics that string theorists have yet to discover, being the number of circular turns that C.W. Leadbeater counted in each helical whorl of the UPA/subquark superstring (Fig. 4). Confirmation that this number does not appear by chance in the context of the first four Platonic solids is provided by their composition when their vertices are included:
Corners
Sides
Triangles
Total
22 + 38 + 60 + 120 = 240
180 + 240 + 360 = 780
120 + 180 + 360 = 660
1680
780 (=78×10) sides and (240+660=900=90×10) corners & triangles surround their axes. The number 168 is the gematria number value of Cholem Yesodoth
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the Mundane Chakra of Malkuth (see Table 1). As indicated in Figure 2, the number value of Cholem is 78 and the number value of Yesodoth is 90.
We find that the first four Platonic solids, thought by the ancient Greeks to be the shapes of the particles of the Elements Earth, Water, Air & Fire, not only embody the superstring structural parameter 1680 but also reproduce in their geometrical make-up the number values of the two Hebrew words in the Kabbalistic name of the Mundane Chakra of Malkuth! It is improbable in the extreme that both numbers, as well as their sum, should appear by chance in the geometry of these Platonic solids in such a natural way. Instead, we are witnessing here the manifestation of the mathematical archetypes embodied in the Kabbalistic system of Godnames, Archangelic Names, Orders of Angels and Mundane Chakras (see Table 1). They determine the properties of all sacred geometries because the latter are isomorphic, so that they possess properties that are quantified by the same numbers — namely, the gematria number values of the Godnames, etc.
Figure 4. The first four Platonic solids embody superstring structural parameters.
As further confirmation that this amazing property of the first four Platonic solids does not arise by chance, consider the geometrical composition of each Platonic solid when its vertices are included:
Table 7. Geometrical composition (including vertices) of the first four Platonic solids.
Platonic solid
Faces
Interior Interior
Total
Tetrahedron 36 48 84 168 Octahedron 72 96 168 336 Cube 72 96 168 336 Icosahedron 180 240 420 840 Total = 360 480 840 1680
We find that surrounding its axis:
8
The superstring structural parameter 168 (the number of turns in a half-revolution of a whorl of the UPA/subquark superstring) appears explicitly in the tetrahedron, octahedron & cube, whilst its structural parameter 840 (the number of turns in the 2½ revolutions of the outer or inner half of a whorl) appears explicitly in the icosahedron. Such repeated appearances of the number value 168 of Cholem Yesodoth, as well as the numbers 840 & 1680, cannot be due to chance! Rather, it demonstrates the truly amazing fact that the sacred geometry of the first four Platonic solids — both individually and collectively — embed a number (in fact, several numbers) characterizing the structure of superstrings that was paranormally obtained over 100 years ago by the remote viewing of atoms. This is because both superstrings and these Platonic solids are holistic systems that conform to the archetypal pattern described in The holistic pattern.
Figure 5. Correspondence between the geometrical composition of the first four Platonic solids, the disdyakis triacontahedron and the outer & inner Trees of Life.
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When the 180 internal triangles formed by joining adjacent vertices of the disdyakis triacontahedron to its centre are divided into their sectors, there are 1680 corners, sides & triangles surrounding any axis that passes through two diametrically opposite vertices (Fig. 5). They consist of 240 corners, 780 sides & 660 triangles. This is remarkable, for the table above indicates that the same numbers of corners, sides & triangles surround the axes of the first four Platonic solids. The reason for this is simple: both holistic systems are isomorphic representations of the same archetypal pattern.
The 1680 geometrical elements surrounding the axes of the first four Platonic solids consist (referring to the last table) of 240 points, (180+240+120+180=720) sides & triangles either in the faces or in their interior formed by edges and (360+360=720) internal sides & triangles formed by sides of sectors. This 240:720:720 division has its parallel in the 1-tree, which has 240 yods other than Sephiroth, and the two sets of seven separate Type B polygons making up the inner Tree of Life, each set containing 720 yods that surround their centres (Fig. 5).
4. Yod composition of the Platonic solids
According to Table 5, a regular polyhedron with F faces, E edges & n sectors in each face has F corners other than vertices, (E+nF) sides & nF triangles in its faces that surround its axis passing through two opposite vertices. If each triangle becomes a tetractys, the number of yods other than vertices in its faces that surround its axis = F + 2(E+nF) + nF = F + 2E + 3nF. Including the (C–2) vertices surrounding its axis, the total number of yods = C + F – 2 + 2E + 3nF = 3E + 3nF, where Euler’s formula for a polyhedron:
C – E + F = 2
has been used to replace C by E and F. The number of yods surrounding the axes of the five Platonic solids = 3∑ (E+nF) = 3(90+180) = 810. Their axes each consist of three yods on either side of its centre. Therefore, (810 + 5×6 = 840) yods in their faces and axes surround the centres of the five Platonic solids, where
840 = 292 – 1 = 3 + 5 + 7 + … + 57
is the sum of the first 28 odd integers after 1. This demonstrates how the second perfect number 28, which we found in Section 1 to be the average number of vertices & edges in the five Platonic solids, determines how many yods surround their centres. The number 840 is a superstring structural parameter, being the number of circular turns in an outer or inner half of a whorl of the UPA/subquark superstring (see Fig. 4). It is expressed by the Tetrad Principle as
840 = 10×84 = (1+2+3+4)(12 + 32 + 52 + 72).
Figure 6. When nested, the five Platonic solids have 841 yods in their faces & axes. This is the number of yods in the 1-tree and the two sets of seven separate polygons of the inner Tree of Life.
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The centre of a Platonic solid on average is surrounded by (840/5 = 168) yods in its faces & axes. This demonstrates how the superstring structural parameter 168 is embodied in the five Platonic solids.
Suppose that the five Platonic solids are nested in one another, occupying the same centre but with their axes pointing in different directions, each axis having three yods on either side of its centre, and with no vertices coinciding. The number of yods that surround the centre of a Platonic solid in its faces and on its axis = 3E + 3nF + 6. The five nested Platonic solids contain 841 yods (Fig. 6). The 1-tree with Type A triangles contains 251 yods and the seven separate Type A polygons contain 295 yods, so that the number of yods in the 1-tree and both sets of seven separate Type A polygons = 251 + 295 + 295 = 841. This demonstrates the Tree of Life nature of the Platonic solids. As two hexagonal yods lie on each edge, there are (90×2 + 50 = 230) yods on the 90 edges of the five Platonic solids. There are (5×4 + 1 = 21) yods other than vertices on their axes when they are nested. Therefore, (230+21=251) yods line their axes and edges, which leave 590 yods in their faces, 295 such yods being in their upper halves and 295 yods being in their lower halves. This is the precise counterpart of the 251 yods in the 1-tree and the 295 yods in each set of seven separate polygons. It is convincing evidence that it is not by chance that the five nested Platonic solids contain the same number of yods as the 1-tree and the two sets of seven separate polygons! The inner form of the 1-tree manifests in the five nested Platonic solids as their faces, whilst its outer form manifests as their edges and axes — those geometrical components that determine their forms.
Surrounding their axes are 40 vertices, 50 centres of faces, 90 edges lined by 180 hexagonal yods and 180 hexagonal yods at centres of the 180 tetractyses in their faces, i.e., (40+50+180+180=450) yods. There are also 360 hexagonal yods lining the 180 sides of these tetractyses that meet at face-centres, whilst lining the axes are 30 yods surrounding centres of Platonic solids. The 840 yods surrounding the latter are made up of (30+360=390) yods lining either axes or interior sides and 450 yods either lining edges or at centres of tetractys. On average, (390/5=78) yods are of the former type and (450/5=90) yods are of the latter type. As Figure 2 indicates, the average yod population of the axes and faces of the five Platonic solids reproduce the number value 78 of Cholem and the value 90 of Yesodoth. Once again, this is evidence that the number 168 does not appear by chance in this context.
Figure 7. The five Platonic solids embody the number value 73 of Chokmah ("Wisdom").
Referring again to Table 5, surrounding the axis of a regular polyhedron are (E+F+nF) corners other than vertices, (5E+4nF) sides & (3E+4nF) triangles. Therefore, the total number of yods other than vertices surrounding the axis = E + F + nF + 2(5E+4nF) + 3E + 4nF = 14E + F + 13nF. For the five Platonic solids, the number of such yods = 14×90 + 50 + 13×180 = 3650 = 365×10. On average, a Platonic solid has (3650/5=730=73×10) yods other than vertices surrounding its axis (Fig. 7). 73 is the gematria number value of Chokmah ("Wisdom"), the second Sephirah of the Tree of Life. Remarkably, it is the number of yods up to Chokmah in the lowest of any set of overlapping Trees of Life when their triangles are tetractyses (see here). According to Table 5, the interior of a Platonic solid has (E+nF) corners, (4E+3nF) sides & (3E+3nF) triangles surrounding its axis. The number of internal yods surrounding its axis = E + nF + 2(4E+3nF) + 3E + 3nF = 12E + 10nF. The number of internal yods in the five Platonic solids = 12×90 + 10×180 = 2880 = (1+2+3+4)(11+22+33+44). On average, the axis of a Platonic solid is surrounded by (2880/5=576) internal yods. This, too, is expressed by the integers 1, 2, 3 & 4 because 576 = 242 = 12×22×32×42. It is a beautiful illustration of how these integers symbolized by the tetractys express properties of sacred geometries.
5. Conclusion
The five Platonic solids are an example of sacred geometry not because a famous philosopher wrote about them over 2000 years ago or because their symmetries and shapes are ubiquitous in nature — as though regularly used by a designing God — but because they are the regular polyhedral realisation of the divine mathematical archetypes, embodying the same patterns as those found in the sacred geometries of various religions. Their geometrical and yod compositions are quantified by the respective dimensions 248 & 496 of E8 and E8×E8 and by the structural parameters 168, 840 & 1680 of the UPA described by C.W. Leadbeater over a century ago. This is exemplified by:
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Suggested reading
"The mathematical connection between religion and science," by Stephen M. Phillips, Antony Rowe, Publishing. England (2009).
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