ARTICLE 26
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The disdyakis triacontahedron is unique among the convex
polyhedra. Having more faces than any of the Platonic, Archimedean or other
Catalan solids, it is the 15th Catalan solid (including enantiomorphs), the 26th
member of the family of Archimedean solids and their duals (excluding
enantiomorphs) and the 31st polyhedron when Platonic solids included. It is
therefore prescribed by the Divine Names YAH (number value 15), YAHWEH (26) and
EL (31), as well as by the other Godnames. The musical counterpart of duality is
inversion symmetry of the intervals of the seven scales. This analogy implies
that the disdyakis triacontahedron and its dual, the truncated
icosidodecahedron, are the last and penultimate members of a sevenfold sequence
of polyhedra that are counterparts of the musical scales. The disdyakis
triacontahedron corresponds to the mathematically most harmonious Pythagorean
scale. Confirmation that the analogy is valid is the fact that the seven musical
scales show the same 7:11:15 pattern of Pythagorean tone ratios as that
displayed by the disdyakis triacontahedron in the sheets of its three types of
vertices. The 13 Archimedean solids and their duals correspond to the 13
different intervals and their inverses found in the seven scales. The shape of
the disdyakis triacontahedron is created by the 1680 vertices, edges and
triangles surrounding its axis of symmetry. They correspond to the 1680
circularly polarized oscillations shaping the helical whorl of the
E_{8}×E_{8} superstring described by Annie Besant and C.W.
Leadbeater. As confirmation, these geometrical elements group into two sets
whose numbers are the number values of ‘Cholem’ and ‘Yesodeth,’ the words making
up the Kabbalistic name for the Mundane Chakra of Malkuth. The disdyakis
triacontahedron is therefore the polyhedral representation of this superstring.
The number of yods surrounding its axis that lie on boundaries of the
tetractyses from which it can be constructed is that of 264 tetractyses. As the
inner Tree of Life contains 264 yods when built from tetractyses, this is
compelling evidence that the disdyakis triacontahedron is the polyhedral
counterpart of the former. The Platonic Lambda quantifies its archetypal
properties, which are prescribed by the Godnames of the ten Sephiroth of the
Tree of Life.

1
Table 1. Table of gematria number values of the ten Sephiroth in the four
Worlds.

SEPHIRAH 
GODNAME

ARCHANGEL

ORDER OF
ANGELS

MUNDANE
CHAKRA

1

Kether
(Crown)
620 
EHYEH
(I am)
21 
Metatron
(Angel of the Presence)
314 
Chaioth ha Qadesh
(Holy Living Creatures)
833

Rashith ha Gilgalim
First Swirlings
(Primum Mobile)
636 
2

Chokmah
(Wisdom)
73 
YAHWEH, YAH
(The Lord)
26,
15

Raziel
(Herald of the Deity)
248 
Auphanim
(Wheels)
187 
Masloth
(The Sphere of the Zodiac)
140 
3

Binah
(Understanding)
67 
ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation of God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317 

Daath
(Knowledge)
474 




4

Chesed
(Mercy)
72 
EL
(God)
31 
Tzadkiel
(Benevolence of God)
62 
Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness
(Jupiter)
194 
5

Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement Strength
(Mars)
95 
6

Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76 
Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light
(Sun)
640 
7

Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97 
Tarshishim or Elohim
1260

Nogah
Glittering Splendour
(Venus)
64 
8

Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light
(Mercury)
48 
9

Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living God)
49,
363

Gabriel
(Strong Man of God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame
(Moon)
87 
10

Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and King)
65,
155

Sandalphon
(Manifest Messiah)
280 
Ashim
(Souls of Fire)
351

Cholem Yesodeth
The Breaker of the Foundations
The Elements
(Earth)
168 
The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah
and Assiyah. Corresponding to them are the Godnames, Archangels, Order of
Angels and Mundane Chakras (their physical manifestation). This table gives
their number values obtained by the ancient practice of gematria, wherein a
number is assigned to each letter of the alphabet, thereby giving a number
value to a word that is the sum of the numbers of its letters.

2
1. Geometrical
composition of the disdyakis triacontahedron
The Tree of Life at the heart of Kabbalah represents Adam Kadmon, or ‘Heavenly Man.’ What this
means is that the Divine archetypes determining the nature of the manifestation in spacetime
of holistic systems, including humans, find symbolic expression in the sacred geometry of the
Tree of Life. Works on Kabbalah traditionally depict the Tree of Life in two dimensions as a
set of 16 triangles with their 10 vertices connected by 22 straight lines, with only brief
mention that it is actually 3dimensional. However, this is only the outer form of the Tree of Life. It has an inner
form as well (Fig. 1), consisting of two similar sets of seven regular polygons enfolded in
one another: triangle, square, pentagon, hexagon, octagon, decagon and dodecagon. One set is
the mirror image of the other, the mirror lying along the central Pillar of Equilibrium at
right angles to the plane containing the 14 polygons. Projected onto the latter, the three
Sephiroth on either side of the central pillar coincide with the uppermost and lowest
corners of the hexagons and the outer corners of the triangles (indicated by the red dots in
Fig. 1). The two endpoints of the ‘root edge’ shared by both sets of polygons
coincide with Daath and Tiphareth. This means that eight of the 70 corners of the polygons
are shared with the Tree of Life, leaving 62 unshared corners that
represent geometric degrees of freedom that are intrinsic to its inner form alone.
62 is the number value of Tzadkiel, the Archangel of Chesed (see
Table 1; numbers from this table will be written throughout the article in
boldface).
It was pointed out in Article 25 that these 62 degrees of
freedom correspond to the 62 vertices of the disdyakis triacontahedron — the
most visible sign that it is the 3dimensional realisation of the inner Tree of Life. The two
endpoints of the root edge correspond to any pair of diametrically opposite vertices and the 30
unshared corners associated with each half of the inner Tree of Life correspond to the 30
additional vertices forming each half of the polyhedron. The mirror symmetry displayed by the
two identical halves of the inner Tree of Life corresponds to the mirror symmetry of the
positions of all the vertices. The polyhedron has 62 vertices, 180 edges and
120 triangular faces — a total of 362 geometrical elements. Between any two opposite vertices
are 360 elements, where 360 = 36×10, showing how the Godname ELOHA assigned to
Geburah with number value 36 prescribes this polyhedron. Because of the
symmetry of this polyhedron with respect to inversion though its centre, every one of its
geometrical elements has
its inverted counterpart. It is therefore meaningful to
3
associate 31 vertices, 90 edges and 60 faces with one
‘half’ and their inverted mirror images with the other half, i.e., 181 geometric elements
constitute each half. This composition is expressed in the pair of dodecagons — the last of the
polygons forming the inner Tree of Life — as the 181 yods needed to construct each dodecagon
when its sectors are regarded as three tetractyses (Fig. 2). The 60 yods along the sides of the 12 sectors in each dodecagon —
that is, the yods defining their shapes — symbolize the 60 faces that shape each
half of the disdyakis triacontahedron. The 120 yods inside the 12 sectors of each dodecagon
denote the 120 vertices and edges making up each half of the polyhedron. The yods at the
centres of the two dodecagons denote any two diametrically opposite vertices.
As the disdyakis triacontahedron is the 3dimensional counterpart of the
inner Tree of Life, we should expect to see the number 362 embodied in the latter.
Indeed, this is so. Starting with the two pairs of seven enfolded polygons, the number of
yods lying
on the 175 edges of their 94 tetractyses is 362 (Fig. 3). The two endpoints of the root edge correspond to any two
diametrically opposite vertices of the polyhedron and the 180 yods lining the tetractyses
that are associated with each set of polygons correspond to the 180 vertices, edges
and triangles forming each half of the polyhedron. Each set of polygons has 47
tetractyses with 87 sides outside the root edge. 87 is the
number value of Levanah (“The Lunar Flame”), the Mundane Chakra of Yesod.
The 94 tetractyses have 80 corners, where
80 is the number value of Yesod.
4
2. The seven musical scales The ten Sephiroth
of the Tree of Life are traditionally depicted as spheres (circles in its 2dimensional
representation). Geometrically speaking, they are the 10 vertices of 16 triangles with 22
sides (Fig. 4). Just as the tetractys symbolizes the fourfold sequence of point,
line (its two endpoints), triangle (its three vertices) and tetrahedron (its four vertices),
so the ten Sephiroth mark the same sequence in 3dimensional space. This is the ‘trunk’ of
the Tree of Life. It comprises 10 vertices, 10 lines, 5 triangles and one tetrahedron
(Fig. 5), totalling 26 geometric elements. This
illustrates the archetypal quality of the Divine Name YAHWEH with number value
26: YHVH = 10 + 5 + 6 + 5 = 26. YAH (YH), its older
version with number value 15, prescribes the tetrahedron — the simplest
Platonic solid — because it is composed of 15 geometrical elements.
Inspection of Fig. 4 shows that there are 12 lines and 11 triangles outside the trunk,
i.e., there are (26+12+11=49) geometrical elements in the
Tree of Life.
This shows how the Godname EL CHAI assigned to Yesod with number value
49 determines the geometrical composition of the Tree of Life — both trunk and
branches.
The Pythagorean scale consists of five tone intervals T of 9/8 and two
leimmas L of 256/243 (corresponding to, but not identical with, the modern semitone):
T T L T T T L.
By starting successive sequences of intervals on the circumference of the
circle displayed in Fig. 6, it is seen that there can be only seven different octave species:
5
The eighth sequence of intervals is the same as mode 1. These are the eight
Church musical modes. The three ‘hypo’ versions of the Dorian, Phrygian and Lydian modes are
separated from the latter by a perfect fourth (Fig. 7). The early Catholic Church
added an extra mode called the "Hypomixolydian" (mode 8) to be the
counterpart of the Mixolydian. It is the same octave species as the Dorian but has a distinct
ethos because its finalis (ending note) and dominant (reciting note) are different.
Figure 8 shows the four Authentic Modes and the four Plagal Modes (the modern
notation S for the semitone is used here instead of the leimma L). The doubleheaded arrows
link an Authentic Mode and a Plagal Mode whose patterns of intervals are mirror images of
each other. The pattern of intervals of the Dorian mode:
T L T T T L T
6
is its own mirror image and so the addition of the Hypomixolydian mode is
needed to restore the leftright mirror symmetry between the three Authentic Modes and the
three Plagal Modes even though it is not distinct from the Dorian mode in terms of its pattern
of tone intervals, differing in the positions of the finalis and the dominant.
Table 2 displays the tone ratios of the seven types of musical scale. Cells
containing tone ratios that belong to the Pythagorean scale (C scale):

C

D

E

F

G

A

B

C'

Tone ratio = 
1

9/8

(9/8)^{2}

4/3

3/2

27/16

243/128

2

are coloured; white cells contain tone ratios that have nonPythagorean
values. The C scale is unique in having only Pythagorean notes (no white cells).
(Coloured cells indicate Pythagorean notes and white cells indicate
nonPythagorean notes.)
Table 3 shows the number of nonPythagorean notes in each scale.
Excluding the tonic with tone ratio 1 and the octave with tone ratio 2, the
seven musical scales have 26 Pythagorean tone ratios (coloured cells). Their
tonal spectrum is therefore prescribed by YAHWEH (YHVH), the values of the letters of the
Godname denoting the numbers of tone intervals of one or two types:
This prescription can be viewed in an alternative way. Table 2 indicates that the
7
distribution of Pythagorean tone ratios other than 1 in the seven scales
is:
Remarkably, the traditional Church numbering of the seven distinct musical
modes generates not only the correct letters values of the Hebrew letters of YHVH but also
the order in which they are pronounced and written! It is, indeed, the Dorian scale
(mode 1) — the only scale whose pattern of tone intervals is its own mirror image — that can be
considered the source of all the musical scales. This was confirmed in Article 19 by comparing
the eight church modes with the eight trigrams of the Chinese I Ching and by proving that there
is only one scheme of correspondence between them that permits mirror symmetry in
both.^{1} This is where the first trigram — the Heaven trigram —
corresponds to the Dorian mode and the eighth trigram — the Earth trigram — corresponds to
the Hypomixolydian mode — the eighth mode. Fittingly, in view of its mathematical
perfection, the seventh and last distinct scale generated from the Dorian scale is the
Pythagorean scale, or Hypolydian mode (see the bottom of p. 5).
The 49 notes of the seven musical scales below the octave
consist of 33 notes with tone ratios of the Pythagorean musical scale (coloured cells) and 16
notes with nonPythagorean ratios (white cells):
33 = 7(1) + 26 = 7 + 11 +
15,
where 7 = 7(1), 11 = 5(9/8) + 6(3/2) and
15 = 3(81/64) + 2(243/128) + 6(4/3) + 4(27/16). Alternatively, the 33
Pythagorean tone ratios may be thought of as made up of the seven tone ratios of the
Pythagorean scale (C scale) and their 26 repetitions in the six other
scales:
As before, the letter values of YHVH denote numbers of different notes in
various sets.
Comparing the geometrical composition of the Tree of Life discussed earlier
with the composition of notes in the seven musical scales, we find the following
correspondence:
8
The 26 notes of the seven scales other than their tonics
(or, alternatively, the 26 repetitions of the notes of the Pythagorean scale)
correspond to the 26 geometrical elements composing the trunk of the
Tree of Life. The 23 tonics or nonPythagorean notes (or, alternatively, the seven notes of the
Pythagorean scale and the 16 nonPythagorean notes) correspond to the 23 geometrical elements
making up the branches of the Tree of Life. This parallelism demonstrates the way in
which the seven musical scales conform to
the archetypal pattern of the Tree of Life.^{2} Music based upon these scales is intrinsically sacred. As the
Pythagoreans declared, “music is geometry.”
3. The 33 sheets of vertices of the
disdyakis triacontahedron
The 62 vertices of the disdyakis triacontahedron are made
up of the 32 vertices of a rhombic triacontahedron, whose 30 faces are Golden Rhombi,
9
and 30 raised centres of the latter. The long diagonals of each golden
rhombus are the edges of an icosahedron and its short diagonals are the edges of a
dodecahedron. Labelling the vertices of the 120 triangular faces A, B & C (Fig. 9), there are 30 A vertices. 12 B vertices (corners of the icosahedron)
& 20 C vertices (corners of the dodecahedron) arranged in seven sheets sandwiched
between diametrically opposite A vertices, in 11 sheets perpendicular to a BB axis and in
15 sheets perpendicular to a CC axis. The disdyakis triacontahedron has
therefore (7+11+15=33) generic sheets of vertices. There are more sheets
than this because each pair of opposite vertices defines its own set of sheets.
In Articles 2224,^{3} evidence was presented to indicate that the disdyakis
triacontahedron is the 3dimensional realisation of the inner form of the Tree of Life. The
33tree levels of 10 overlapping Trees of Life correspond to the 33 sheets of vertices
(Fig. 10). The 62 vertices of the disdyakis triacontahedron
correspond to the 62 of their 65 Sephirothic emanations
whose projections onto the plane containing the 14 polygons enfolded in each tree coincide
with corners of these polygons, that is, points shared between the inner and outer forms of
each tree. The lowest 26 tree levels, which are spanned by
50 Sephirothic emanations, may be physically interpreted^{4} as the 26 dimensions of spacetime predicted for
spinless strings by quantum mechanics (this demonstrates how ELOHIM, the Godname of Binah
with number value 50, prescribes the dimensionality of spacetime). They
correspond to the 26 sheets of vertices perpendicular to either a BB or a
CC axis. The highest seven tree levels of ten overlapping Trees of Life correspond to the
seven sheets of vertices perpendicular to an AA axis. The lowest 11 tree levels denote the
11 dimensions of spacetime predicted by supergravity theory; they correspond to the 11
sheets of vertices. The next 15 tree levels signify the 15
higher dimensions whose compactification generates superstrings and their unified forces
described by SO(32) and E_{8}×E_{8}; they correspond to the
15 sheets of vertices. The compactification of 11dimensional spacetime
into S_{1}×S_{6}×M_{4}, where S_{1} is a 1dimensional space
(line segment or circle), S_{6} is the 6dimensional, compactified space predicted
by superstring theory and M_{4} is Einsteinian, 4dimensional spacetime, conforms
to the letter values of V and H in the wellknown Godname, YHVH:
4. The
Archimedean & Catalan solids
Let us now compare the 33 generic sheets of vertices in the disdyakis triacontahedron with the
33 notes of the seven musical scales (seven tonics and 26 notes below the
octave) that have Pythagorean tone ratios. The following correspondences exist:
10
AA sheet of vertices tonic with tone ratio of 1;
BB sheet of vertices major 2nd (9/8) or perfect 5th (3/2);
CC sheet of vertices major 3rd (81/64), perfect 4th (4/3), major 6th (27/16) or
major7th (243/128).
If, as discussed earlier, the 26 notes are the repetitions
of the Pythagorean notes in the six other scales, an alternative set of correspondences is:
AA sheet of vertices Pythagorean note;
BB sheet of vertices tonic/perfect 5th;
CC sheet of vertices major 2nd/major 3rd/major 6th/major 7th.
If we follow the previously discussed scheme based upon the natural ordering
of successive sequences of intervals, starting with the Dorian, the correspondences are:
AA sheet of vertices tonic with tone ratio of 1;
BB sheet of vertices note of modes 6 and 7;
Whatever the scheme of correspondence, we see that the disdyakis
triacontahedron is the geometrical counterpart of the seven octave species. This polyhedron is
the spatial realisation of the Tree of Life and the seven species of octaves are its musical
manifestation.
Article 14^{5} showed that the seven types of musical scale conform to the archetypal
pattern of the Tree of Life blueprint. The fact that their notes group in the same
11:15 pattern that both the geometrical elements of the trunk of the Tree
of Life and the BB and CC sheets of vertices do is further, strong evidence that the
disdyakis triacontahedron is the 3dimensional counterpart of the Tree of Life.
26dimensional spacetime mapped by the lowest 26 tree
levels of the map of Reality called the ‘Cosmic Tree of Life,^{6} which consists of 91 overlapping Trees of Life.
The seven octave species that originate with the Dorian mode (D scale) and
realise their perfection in the Hypolydian mode (the Pythagorean, or C scale) are the musical
counterpart of seven polyhedra (five regular, two semiregular) that, starting from the
tetrahedron, and ordered according to increasing number of corners, completes their perfection
with the disdyakis triacontahedron:
tetrahedron is dual to tetrahedron 

Dorian mode is reverse of Hypomixolydian mode 
octahedron is dual to cube 

Mixolydian mode is reverse of Hypodorian mode 
icosahedron is dual to dodecahedron 

Lydian mode is reverse of Hypophrygian mode 
? is dual to ? 

Phrygian mode is reverse of Hypolydian mode 
Just as the tetrahedron is selfdual, so the Dorian mode has a pattern of
intervals that is the mirror image of itself. It is therefore the same as the pattern of the
Hypomixolydian mode, the eighth mode, which both completes and recommences the cyclic sequence
of intervals that started with the Dorian. Just as the octahedron and cube are dual to each
other, so the Mixolydian and Hypodorian modes possess patterns of intervals that are mirror
images of each other. Just as the icosahedron and dodecahedron are dual to each other, so the
Lydian and Hypophrygian modes are mirror images of each other. The Phrygian musical mode is the
reverse of the Hypolydian mode. What, however, is the pair of nonPlatonic polyhedra that are
dual to each other and are the counterpart of this fourth pair of modes? As the 32 corners of a
rhombic triacontahedron consist of the 12 corners of an icosahedron and the 20 corners of a
dodecagon, it might seem natural to think that the next member of the sevenfold sequence of
polyhedra after the five
11
Platonic solids is this one and that the disdyakis triacontahedron is
the seventh because it is just the rhombic triacontahedron with the centres of its faces
raised. However, it is not its dual. If the analogy between polyhedral duality and modal
symmetry is to be taken seriously so that the alternation between a polyhedron and its dual
persists beyond the five Platonic solids and applies as well to the last pair of
polyhedra making up the seven polyhedra, it means that either one or both of these
polyhedra cannot complete the sevenfold sequence of polyhedra that mirrors the seven musical
modes. It might be expected that the disdyakis triacontahedron should, indeed, be the
seventh polyhedron because of its status as the polyhedral form of the Tree of Life. But a
rigorous mathematical discussion cannot make this assumption, however, persuasively it is
suggested by the evidence presented in Articles 22–25. Instead, it must explain
why, in strictly mathematical terms, this polyhedron does possess a unique significance.
After all, the icosidodecahedron is an Archimedean solid with
only 32 faces. But it, too, acquires 120
triangular faces when the centres of these faces are raised above them
(Table 4). Moreover, the transformed polyhedron has 62
Archimedean solid

Types of faces

Number of triangular faces

cuboctahedron 
8 triangles + 6 squares

8×3 + 6×4 = 48 
icosidodecahedron 
20 triangles + 12 pentagons

20×3 + 12×5 = 120 
truncated tetrahedron 
4 triangles + 4 hexagons

4×3 + 4×6 = 36 
truncated cube 
8 triangles + 6 octagons 
8×3 +6×8 = 72 
truncated octahedron 
6 squares + 8 hexagons 
6×4 + 8×6 = 72 
truncated dodecahedron 
20 triangles +12 decagons 
20×3 + 12×10 = 180 
truncated icosahedron 
12 pentagons + 20 hexagons 
12×5 + 20×6 =180 
rhombicuboctahedron 
8 triangles + 18 squares 
8×3 + 18×4 = 96 
truncated cuboctahedron 
12 squares + 8 hexagons + 6 octagons 
12×4 + 8×6 + 6×8 = 144 
rhombicosidodecahedron 
20 triangles + 30 squares + 12 pentagons 
20×3 + 30×4 + 12×5 = 240 
truncated icosidodecahedron 
30 squares + 20 hexagons + 12 decagons

30×4 + 20×6 + 12×10 = 360 
snub cube 
32 triangles + 6 squares 
32×3 + 6×4 = 120 
snub dodecahedron 
80 triangles + 12 pentagons 
80×3 + 12×5 = 300 

vertices (30 old vertices and 32 new ones), which is the same number as the disdyakis
triacontahedron. It has 480 hexagonal yods^{7} — just as many as the disdyakis triacontahedron. It has (60 +
20×3 + 12×5 = 180) edges — the same number, which means that it, too, has
(62+180+120=362) vertices, edges and faces, a number that, as Section 1
revealed, is embodied in both the inner Tree of Life and in the last of its regular
polygons. Finally, it has (180×2 + 62 = 422) yods along its sides — the
same as the disdyakis triacontahedron. It thus shares some of the amazing properties of
this polyhedron and their correlations with the Tree of Life that were discussed in
Articles 22–25. The only other Archimedean solid with 62 corners and
120 faces when its faces are raised is the snub cube It, too, has 480 hexagonal yods,
362 vertices, edges and triangular faces and 422 yods along its sides. The lack of
uniqueness in such properties of the disdyakis triacontahedron means that they, alone,
cannot be regarded as evidence that makes it stand out from all other polyhedra. What
does pick it out is the following: there are 13 Archimedean solids (15,
including the two solids that possess enantiomorphic
12
Table 5. Properties of the Archimedean & Catalan solids.
(listed in order of increasing
number of vertices)


(listed in order of increasing
number of faces)

F

E

C

Archimedean solid

8

18

12

truncated tetrahedron 
14

24

12

cuboctahedron 
14

36

24

truncated cube 
14

36

24

truncated octahedron 
26

48

24

rhombicuboctahedron 
38

60

24

snub cube 
38

60

24

snub cube (chiral partner) 
32

60

30

icosidodecahedron 
26

72

48

truncated cuboctahedron 
32

90

60

truncated icosahedron 
32

90

60

truncated dodecahedron 
62

120

60

rhombicosidodecahedron 
92

150

60

snub dodecahedron 
92

150

60

snub dodecahedron
(chiral partner) 
62

180

120 
truncated icosidodecahedron 


Catalan solid

F

E

C

triakis tetrahedron 
12

18

8

rhombic dodecahedron 
12

24

14

triakis octahedron 
24

36

14

tetrakis hexahedron 
24 
36 
14

deltoidal icositetrahedron 
24

48

26

pentagonal icositetrahedron 
24

60

38

pentagonal icositetrahedron
(chiral partner) 
24

60

38

rhombic triacontahedron 
30

60

32

disdyakis dodecahedron 
48

72

26

triakis icosahedron 
60

90

32

pentakis dodecahedron 
60

90

32

deltoidal hexacontahedron 
60

120 
62

pentagonal hexacontahedron 
60

150

92

pentagonal hexacontahedron
(chiral partner) 
60

150

92

disdyakis triacontahedron 
120

180

62


C = number of vertices
E = number of edges
F = number of faces 
13
counterparts). This demonstrates how the Godname YAH of Chokmah with number
value 15 prescribes the family of solids that have at least two types of
regular polygons as their faces. The musical counterpart of this is the fact that the
15 notes of two octaves of the Pythagorean scale are needed to generate the
eight Church modes:
The duals of the Archimedean solids are the 15 Catalan
solids shown in Table 5. When listed in order of increasing number of corners, starting with
the truncated tetrahedron with 12 corners and eight faces, the Archimedean solid with most
corners is the truncated icosidodecahedron with 120 corners and 62 faces. When
listed in order of increasing number of faces, starting with the triakis tetrahedron with 12
faces and eight corners, the Catalan solid with the most faces is the disdyakis
triacontahedron with 62 corners and
120 faces. This is the disdyakis triacontahedron! There are 13 Catalan
solids apart from the enantiomorphic versions of the pentagonal icositetrahedron and the
pentagonal hexacontahedron. The disdyakis triacontahedron is therefore the
26th and last of the two families of Archimedean and Catalan solids.
The Divine Name YAHWEH with number value 26 prescribes these two families of
solids and determines the disdyakis triacontahedron as the solid with the most faces, whilst
its dual, the truncated icosidodecahedron, is the solid with the most corners. These most
complex of the Archimedean and Catalan solids are the sixth and seventh members of the
sevenfold sequence of polyhedra that mirror the seven musical scales:
tetrahedron is dual to tetrahedron 

Dorian mode is inverse of Hypomixolydian mode 
octahedron is dual to cube 

Mixolydian mode is inverse of Hypodorian mode 
icosahedron is dual to dodecahedron 

Lydian mode is inverse of Hypophrygian mode 
truncated

disdyakis Phrygian

Hypolydian

is dual to


is inverse of

icosidodecahedron

triacontahedron mode

mode

The disdyakis triacontahedron corresponds to the mathematically perfect
Pythagorean scale (Hypolydian mode). It completes the tetractys pattern representing the
generation from the mathematical point of the perfect solid (Fig. 11).
14
The centre and corners of a polyhedron with E edges define E internal
triangles. The (6+4×3=18) tetractyses inside and on the surface of a tetrahedron have 70
yods surrounding its centre.^{8} In other words, starting from this point, 70 yods are needed to
construct the tetrahedron from tetractyses. This is the meaning of the 70 yods in the Tree
of Life constructed from 16 tetractyses (Fig. 12). As the simplest Platonic solid, it is the first in the archetypal
sequence of seven solids leading to the 3dimensional realisation of the Tree of Life — the
disdyakis triacontahedron. As the ‘seed’ that grows into the flower of the latter, it must
possess these 70 degrees of freedom. 62 of the yods are hexagonal yods
(48 in the four faces, 14 internal). They symbolize in potentia
the 62 faces of the fullygrown, polyhedral Tree of Life. If, instead, the
internal triangles formed by the vertices and centre are divided into three tetractyses, the
tetrahedron has 124 surrounding its centre.^{9} In other words, 120 yods are necessary, starting from a
tetrahedron, to construct it from the 30 tetractyses. These symbolize in potentia
the 120 faces of the disdyakis triacontahedron.
As we saw in Section 1, the disdyakis triacontahedron has 362 geometrical
elements. Including its centre, there are 363 elements. This shows how SHADDAI
EL CHAI, the complete Godname of Yesod, prescribes this
polyhedron. With each internal triangle a single tetractys, there are 180
internal tetractyses with 62 internal sides ending on the vertices. The number
of geometrical
15
elements inside the polyhedron and on its faces = 362 + 180 +
62 + 1 = 605. Turning each internal triangle into three tetractyses generates
three sides, two triangles and one vertex for each one, creating in total (605 + 180×6 = 1685)
elements. Converting the triangular faces into three tetractyses adds (6×120=720) elements,
totalling 2405. Constructed both internally and externally from tetractyses, the polyhedron has
a central axis made up of its centre, two diametrically opposite vertices and the lines joining
them to this centre, that is, five geometrical elements. The disdyakis triacontahedron
therefore has 2400 (=240×10) elements surrounding its central axis, of which 1680
(=168×10) elements comprise it with single tetractyses for its 120 faces
(Fig. 13). This property is truly remarkable evidence (indeed, proof) that
the disdyakis triacontahedron does, indeed, represent the Tree of Life blueprint. This is
because the basic unit of matter described 110 years ago by the Theosophists Annie Besant
and C.W. Leadbeater, using a yogic siddhi, consists of ten closed curves, or “whorls,” each
a helix with 1680 turns (Fig. 14). This fundamental superstring structural parameter is the
number of geometrical elements in the disdyakis triacontahedron that surround any axis
joining two diametrically opposite vertices.
The division:
was encountered in many previous articles. Its significance in the context
of the superstring gauge symmetry group E_{8} is as follows: The roots of the
E_{8} algebra can be described in terms of eight orthonormal unit vectors {ui}. Eight
‘zero roots’ correspond to points at the centre of the root diagram and 240 ‘nonzero roots’
all have length √2. They are given by
±u_{i} ±u_{j}
(i, j = 1, 2,… 8)
and
½(±u_{1}, ±u_{2}, … ±u_{8})
(even number of +’s)
Their explicit forms as 8tuples and their numbers are listed below:
The 240 nonzero roots of E_{8} comprise 168 made
up of four sets of 28 and one set of 56, one set of 70 and two single ones. E_{6}, an
exceptional subgroup of E_{8}, has 72 nonzero roots. The 240 nonzero
roots therefore comprise 168 roots that do not belong to E_{6} and
72 roots that do. Each root defines its associated group generator and gauge
charge that couples to a 10dimensional gauge field. The 1680 geometrical elements of
16
the disdyakis triacontahedron surrounding its central axis correspond to the
1680 components of the 168 10dimensional gauge fields of the superstring
symmetry group E_{8} that are not also gauge fields of its subgroup E_{6}.
With its faces divided into three tetractyses, the surface of the disdyakis
triacontahedroh is made up of (62+120=182) vertices, (180 + 120×3 = 540) edges
and (120×3=360) triangles, that is, 1082 geometrical elements. Starting from a point denoting a
vertex, 1081 elements are needs to create the shape of the polyhedron.
1081 is the number value of Tiphareth, the Sephirah at the centre of the Tree
of Life. Tiphareth means “beauty.” The polyhedron embodies in its geometry the archetype of
Divine beauty.
Inside each triangular face divided into three tetractyses are 10 yods. The
number of yods other than vertices in the faces = 120×10 + 180×2 = 1560 = 156×10. The yods in
156 tetractyses are needed to construct the surface of the polyhedron. As 156 is the
155th integer after 1, this shows how ADONAI MELEKH, the complete Godname of
Malkuth with number value 155, prescribes the outer form of the disdyakis
triacontahedron, in keeping with the meaning of Malkuth as the outer physical form.
156 is also the sum of the number values of the four types of combinations
of the letters Y, H and V in YHVH, the Godname of Chokmah:
156 is the sum of the first 12 even integers that can be arranged in a
square:
The number of corners of the 360 tetractyses in the surface of the
polyhedron = 62 + 120 = 182. With internal triangles divided into three
tetractyses, the number of corners of the 540 internal tetractyses = 180 + 1 = 181. The
(360+540=900) tetractyses needed to build the disdyakis triacontahedron have
(182+181=363) corners. This is how the Godname SHADDAI EL CHAI with number
value 363 prescribes its tetractys composition (see Table 1).
There are seven yods on the boundaries of the three tetractyses inside each
of the 120 triangular faces. Each of the 180 sides of the polyhedron has two hexagonal yods
between their 62 ends. The number of yods on the boundaries of the (120×3=360)
tetractyses = 62 + 180×2 + 120×7 = 1262. There are 1260 such
yods between any two diametrically opposite vertices. 1260 is the number value
of Tarshishim, the Order of Angels assigned to Netzach. The number value of the name
of the angelic order assigned to the seventh Sephirah is the number of yods between
two diametrically opposite vertices that shape the 120 faces of the disdyakis triacontahedron,
the seventh polyhedron in the sequence matching the seven musical scales.
1260 is the number of yods in 126 tetractyses. It is
remarkable that the number 126 is the sum of the number values of the four types of
combinations of the letters A, H and
17
in AHIH, the Godname of Kether:
A = 1, H = 5, I = 10
1. A + H + I 
= 
16

2. AH + HI + AI + HH 
= 
42

3. AHI + HIH + AHH 
= 
47

4. AHIH 
= 
21

TOTAL

= 
126

This shows how EHYEH prescribes the population of yods between opposite
vertices that define the shapes of the 360 tetractyses in the faces of the disdyakis
triacontahedron.^{10}
The number 126 has the remarkable property that it is the arithmetic mean of
the first 26 triangular numbers, where 26 is the number value
of YAHWEH:
It is an example of the profound arithmetic connections between Godname
numbers. If the Pythagorean Tetrad 4 is assigned to the boundary yods of the disdyakis
triacontahedron, the sum generated by the 420 yods between diametrically opposite vertices
shaping the 120 faces is 1680. With the latter divided into three tetractyses, the sum
generated by the 1260 yods between diametrically opposite vertices shaping the
360 tetractyses is 4×1260 = 5040 = 3×1680. Turned into tetractyses, the 120 faces of the
disdyakis
triacontahedron embody the number of turns of each helical whorl of the
UPA/heterotic superstring (Fig. 15). Turned into three tetractyses, they embody the number of turns of
its three socalled ‘major whorls’ (the thicker whorls).
As
5040 = 71^{2} – 1 = 3 + 5 + 7 +… + 141,
and, as the Tree of Life turned into tetractyses contains 70 yods, the
number of helical turns in the three major whorls is simply the sum of the 70 odd integers
after 1 that can be assigned to the yods in the Tree of Life (Fig. 16). Its Lower Face (shown shaded) has 30 yods, the rest of the Tree of
Life having 40 yods. As
1680 = 41^{2} – 1 = 3 + 5 + 7 +… + 81,
18
the sum of the first 40 odd integers up to 81 outside the Lower
Face is 1680, and the sum of the next 30 odd integers composing the Lower Face is 2×1680. The
Tree of Life encodes arithmetically the number of turns in the three major whorls, which are
the string manifestation in 4dimensional spacetime of what Kabbalists call the ’Supernal
Triad,’ Christians call the ‘Holy Trinity’ and what is familiar to Hindus as the ‘Trimûrti’ of
Shiva, Vishnu and Brahma. Such is the close relationship between number and the
19
geometry of the Tree of Life that, when imagined as actually constructed
from the odd integers, starting with 3, its Lower Face reproduces the distinction between the
first member of the Supernal Triad and the two others in the context of their correspondence to
the three major whorls in the subquark state of the E_{8}×E_{8} heterotic
superstring.
The largest odd integer, which coincides with the position of Malkuth at the
base of the Tree of Life, is 141 = 76 + 65, i.e., the sum of
the numbers of YAHWEH ELOHIM and ADONAI. It is the 71st odd integer, where 71 =
21 + 50, i.e., the sum of the numbers of EHYEH and ELOHIM.
The integers are assigned in Fig. 16 sequentially from left to right.
Remarkably, the sum of the integers located at the positions of the 10
Sephiroth is:
3 70
15 21
70
70
59 49 83 = 700 =
70 70 70
97 107 125
141
70 70 70 70,
i.e., it is the sum of the number 10 (the Pythagorean Decad) assigned to
each of the 70 yods in the Tree of Life! It is an example of the way in which the beautiful,
mathematical properties of the Tree of Life clearly point to the designing
Intelligence behind it. The following, amazing property of the
20
disdyakis triacontahedron provides further proof that it is the polyhedral
version of the inner Tree of Life: when its 180 internal triangles are each divided into three
tetractyses, there are within each triangle seven yods lying on edges of the (3×180=540)
tetractyses. The line joining each polyhedral vertex to the centre of the polyhedron has two
hexagonal yods in it. Therefore, the number of yods lying on edges of the internal tetractyses
= 2×62 + 7×180 + 1 = 1385. We saw earlier that 1262 yods lie on the boundaries
of the 360 tetractyses forming the 120 faces of the polyhedron. The total number of boundary
yods inside and on the surface of the polyhedron = 1385 + 1262 = 2647. Four yods lie on the
common vertical edge of the 10 tetractyses symmetrically arranged around the axis that are part
of internal triangles having sides that converge on the uppermost vertex (Fig. 17). Actually, they are edges of the rhombic faces and raised halves of
their longer diagonals. Similarly, four yods lie on the common vertical edge of the 10
tetractyses belonging to internal triangles whose sides converge on the lowest vertex
diametrically opposite the highest one. This means that there are seven yods on the vertical
axis running through the highest and lowest vertices. The number of yods lying on the
boundaries of the (540+360=900) tetractyses that surround the vertical axis = 2647
– 7 = 2640 = 264×10. 264 is the number of yods in the seven enfolded polygons forming half
of the inner form of the Tree of Life (Fig. 18)! Assigning 10 to each such yod therefore generates as their sum the
number of yods lying on the boundaries of the 900 tetractyses that can be assembled into the
disdyakis triacontahedron As seen earlier, these tetractyses have 363
corners prescribed by SHADDAI EL CHAI, Godname of Yesod. Three of the corners lie on the
central axis (see Fig. 17). 360 corners surround the axis. This number is the sum of the
integers 10 that can be assigned to the 36 corners of the seven
enfolded polygons. There are (264–36=228) yods that are not corners. This
is the number of yods on the edges of the 900 tetractyses that are not corners. A
remarkable correspondence emerges between the inner Tree of Life and the disdyakis
triacontahedron, whose axis is equivalent to the root edge shared by both sets of polygons.
It exists because the disdyakis triacontahedron is its polyhedral form.
How do the properties of the dual of the disdyakis triacontahedron compare
with it? It is readily shown that the truncated icosidodecahedron has in its
62 faces the same numbers of yods, hexagonal yods, corners and boundary yods.
It also has the same number (363) of corners of tetractyses making up its
faces and interior. However, the latter differs in the numbers of its hexagonal and boundary
yods. The number of boundary yods inside and on the surface of the polyhedron is 2763, not the
number 2647 for the disdyakis triacontahedron that proved its correspondence with the seven
enfolded polygons. Its 900 tetractyses have 1563 corners and edges, a total of 2463 geometrical
elements. This compares with the 2405 elements for the disdyakis triacontahedron that implied
that 2400 elements surround its central axis. As expected, the properties of its dual fall
short of the beautiful correlations with the properties of the seven polygons, showing that the
disdyakis triacontahedron — not its dual — should be regarded as the seventh and last member of
the set of seven polyhedra because it represents exact correspondence with the Tree of
Life.
Of the 264 yods in the seven regular polygons forming each half of the inner
Tree of Life, 73 yods belong to the dodecagon. 69 of them lie outside the root
edge, leaving 195 yods in the first six polygons. Associating two yods in the shared edge with
each set of six polygons, there are 193 yods associated with each set. As the uppermost corner
of a hexagon coincides with the lowest corner of the hexagon enfolded in the next higher tree,
there are (192n+1) yods associated with the 6n polygons enfolded in n trees. The set of
polygons has 26 corners and 31 sides, each with two hexagonal
yods, where 26
21
is the number value of YAHWEH and 31 is the number value of
EL. Hence, there are (26 + 31×2 = 87) yods
along their boundaries, where 87 is the number value of Levanah, the
Mundane Chakra of Yesod. The number of boundary yods associated with each set of six polygons
is 85. There are (84n+1) boundary yods associated with the 6n polygons enfolded in n trees.
The Godname ADONAI with number value 65 prescribes the
lowest 10 trees with 65 Sephiroth. There are 1921 yods associated with the 60
polygons enfolded in them. Of these, 841 yods lie on their boundaries, inside which are 1080
yods. As the uppermost corner of the top hexagon coincides with the lowest corner of the
hexagon belonging to the adjacent tree, there are 840 yods intrinsic to each set of polygons,
i.e., there are (2×840=1680) boundary yods intrinsic to both sets, which have (2×1080=2160)
yods inside their
boundaries. We find that the disdyakis triacontahedron has as many
geometrical elements surrounding its axis as there are yods forming the shapes of the two sets
of the first six polygons enfolded in 10 Trees of Life. The yods symbolize bits of information
— geometrical elements in the
22
disdyakis triacontahedron and circularly polarised oscillations in each
whorl of the UPA/superstring. 216 is the number value of Geburah, the second
Sephirah of Construction. Its Godname ELOHA prescribes the 360 tetractyses in the 120 faces of
the disdyakis triacontahedron because its number value is 36.
If the number 10, which was regarded by the Pythagoreans as the perfect
number, is assigned to every yod in the 360 tetractyses, the sum of the integers assigned to
their 182 corners is 1820. This is the number of yods surrounding the centres of the five
Platonic solids when they are constructed from tetractyses.^{11} It is another remarkable property of the disdyakis
triacontahedron that illustrates its archetypal qualities. As 1820 = 70×26,
it is the sum of the number value 26 of YAHWEH assigned to each of the 70
yods that make up the Tree of Life when its 16 triangles are converted into tetractyses (see
Fig. 12). It indicates that the five regular solids constitute a
whole in themselves.
The sum of the integers 10 assigned to the 62 vertices of
the polyhedron is 620. This is the number value of Kether (“crown”), the first
Sephirah of the Tree of Life. It is also the number of corners intrinsic to the 70 polygons
enfolded on each side of 10 Trees of Life that are unshared with their Sephiroth (Fig. 20). A symbol of the number 10, the decagon divided into 2ndorder
tetractyses has 720 yods surrounding its centre. 620 of them are hexagonal
yods. The 62 vertices of the disdyakis triacontahedron correspond
to the 62
hexagonal yods per sector of the decagon.
As the dual of the disdyakis triacontahedron, the truncated
icosidodecahedron has the same number (1440) of hexagonal yods in its 62
faces.^{12} The five Platonic solids have (720=72×10) hexagonal
yods in their 50 faces.^{13} The seven solids have 3600 (=36×10×10) hexagonal
yods in their 232 faces, showing how the Godname ELOHA of Geburah with number value
36 prescribes the hexagonal yod population of this archetypal set of seven
solids. The first six solids have (720+1440=2160) hexagonal yods, which is the number of
yods in 216 tetractyses. The number value of Geburah is
216 and the number value of Chesed, the first Sephirah, is
72. It cannot be just coincidence that successive Sephirah have such
significant values.
Adding the five Platonic solids to the 26 Archimedean and
Catalan solids creates a new family of 31 solids. The Divine Name EL (“God”)
assigned to Chesed therefore
23
prescribes how many regular and quasiregular solids there are. Constructed
from tetractyses, the pentagon has 31 yods (Fig. 21). The yods on the sides of the tetractyses symbolize the Archimedean
and Catalan solids and the five yods at their centres represent the five Platonic
solids.
As
168 = 13^{2} – 1 = 3 + 5 + 7 + 9 + 11 + 13 +
15 + 17 + 19 + 21 + 23 + 25,
this number is the sum of the odd integers after 1 that can be assigned to
the 12 corners of the dodecagon, which is the seventh type of polygon appearing in the inner
Tree of Life and the tenth polygon per se (Fig. 22). The number 168 is also the number of yods
generated in the dodecagon when its sectors are each divided into three tetractyses
(Fig. 23). The total number of yods surrounding its centre is 180, which is
the number of sides of the disdyakis triacontahedron. This is how the last polygon in the
inner Tree of Life
embodies information about the properties of its polyhedral version.
168 is the number value of Cholem Yesodeth, the
Mundane Chakra of Malkuth. As the structural parameter of superstrings — the physical
manifestation or Malkuth aspect of the Tree of Life in the subatomic world — this number is
embodied in the disdyakis triacontahedron (apart from the Tree of Life factor of 10) as the
number of geometrical elements surrounding its axis. Just as the seven solids correspond to the
seven musical scales, so they correspond to the seven Sephiroth of Construction. Indeed, it is
the latter
that is responsible in the first place for all sevenfold patterns
in nature. Which solid is associated with which Sephirah? The solids comprise five Platonic
solids and
24
two Archimedean or Catalan solids, whilst the seven Sephiroth of
Construction consist of Chesed and Geburah, which are unshared by adjacent overlapping trees,
and the five Sephiroth that form the Lower Face of the Tree of Life and which are shared by
adjacent trees. In view of this, it is natural to make the following association:
Chesed 

disdyakis triacontahedron 
Geburah 

truncated icosidodecahedron 
Tiphareth 

dodecahedron 
Netzach 

icosahedron 
Hod 

cube 
Yesod 

octahedron 
Malkuth 

tetrahedron 
The ancient Greeks believed that the Platonic solids are the shapes of
particles of the five elements of Earth, Water, Air, Fire and Aether. The reasons they gave for
associating each element with its particular solid may now appear naïve. However, the
association of the elements with the five lowest Sephiroth of Construction certainly is not so.
Fig. 24 depicts the assignment of the seven solids to the Sephiroth
according to the ancient association of the Platonic solids with the Elements. It shows the
cube corresponding to Malkuth because it was regarded as the shape of particles of Earth,
the element corresponding to Malkuth. It is inconsistent with the tetractys array of solids
depicted in Fig. 11, which shows the disdyakis triacontahedron at the centre of
the tetractys — the yod that symbolizes the Malkuth level or aspect of any holistic system.
However, this discrepancy is not serious, because it merely reflects the fact that the
association between Elements and Platonic solids made by the ancient Greeks was based upon
wrong and naïve considerations, namely, the relation between the sensory experience of these
polyhedra and the sensory quality of water, fire, earth, etc. They were also based upon
incomplete considerations because in esoteric formulations of mystical traditions there are
seven Elements, not five, the latter forming part of their exoteric versions. The order of
the seven solids should be based on monotonic increase or decrease of the number of their
corners or faces, i.e., with the dimension of their symmetry groups (Table 6):
Table 6
Polyhedron

Dual Polyhedron

Number of Symmetries

Symmetry Group

tetrahedron 
tetrahedron 
24 (12)

T_{d} (T)

cube 
octahedron 
48 (24)

O_{h} (O)

octahedron 
cube 
icosahedron 
dodecahedron 
120 (60)

I_{h} (I)

dodecahedron 
icosahedron

truncated icosidodecahedron 
disdyakis triacontahedron 
120 (60)

I_{h} (I)

disdyakis triacontahedron 
truncated icosidodecahedron 
There are three polyhedra with up to 48 symmetries and four
polyhedra with 120 symmetries. This compares in the Tree of Life with the triad of Chesed,
Geburah & Tiphareth and the quaternary of Netzach, Hod, Yesod & Malkuth. As the
simplest regular polyhedron, the tetrahedron starts the mathematical sequence that ends with
the disdyakis triacontahedron. That makes it sensible to associate the simplest solid with
Chesed and the most complex solid with Malkuth. It also seems appropriate, given
25
that the number embodied in this solid is both the number value of the
Mundane Chakra of Malkuth and the structural parameter of the heterotic superstring — the
physical manifestation or Malkuth aspect of the Tree of Life. On the other hand, the analogy
between the 2:5 pattern displayed by the ChesedGeburah pair and the five lowest Sephiroth and
the 2:5 pattern of the Catalan solid and its dual and the five Platonic solids is a strong
argument for associating the disdyakis triacontahedron with Chesed rather than Malkuth. If
polyhedra are to be associated with Elements at all, then this solid and its dual
must be associated with, respectively, Chesed and Geburah and with their
corresponding sixth and seventh Elements. This argument has to be taken seriously because the
seven tone intervals in each of the seven musical scales are the musical manifestation of the
universal septenary principle, and the same 5:2 pattern exists in each of them as five tone
intervals and two leimmas.
The five Platonic solids correspond to the five whole tones and the
Archimedean and Catalan solids correspond to the two leimmas:
However, this association is only a formal one (although the sixth
polyhedron, the dual of the disdyakis triacontahedron is third in the row above). As
now explained, there is a more fundamental correspondence. The product of an interval of a
perfect fourth (4/3) and an interval of a perfect fifth (3/2) is the octave: 4/3×3/2 = 2. This
is the same note as the tonic but increased in pitch by a factor of 2. As intervals, the
perfect fourth and perfect fifth are in a sense inverses of each other. If a note is a perfect
fourth above a given note, it is also a perfect fifth below the same note in the next higher
octave. Replacing corners of a polyhedron by faces and then replacing faces by corners just
leaves the original polyhedron. Forming the dual of a polyhedron is like raising the pitch of a
note by a perfect fourth or fifth. An interval of a perfect fourth is ‘inverse’ to an interval
of a perfect fifth in a way that is analogous to how one polyhedron may be dual to another. The
octave can be divided into either four perfect fourths or four perfect fifths:
In fact, all eight notes of the octave can be generated by
continuous leaps of four perfects fourths and three perfect fifths, starting from the
tonic C. The seven intervals creating the eight notes consist of one upward jump of a fourth
(CF), three upward jumps of a fifth (DA, EB, FC') and three downward jumps of a fourth
(GD, AE, C'G). This 1:3:3 pattern is the same as for the seven musical scales, which consist
of three pairs of scales whose intervals are in reverse order, like a reflection in a mirror,
and the
26
Dorian, which is its own inverse or mirror image:^{14}
It is also the same for the seven polyhedra, which consist of three pairs of
duals and one (the tetrahedron), which is selfdual:
tetrahedron 

tetrahedron 
octahedron 

cube 
icosahedron 

dodecahedron 
icosidodecahedron 

disdyakis triacontahedron 
The pattern reflects the fact that the seven Sephiroth of Construction
consist of two triads: ChesedGeburahTiphareth and NetzachHodYesod, and one (Malkuth),
whilst the tetractys has seven hexagonal yods, six of which are arranged in two equilateral
triangles at the corners of a hexagon with the seventh at its centre
(Fig. 25). Indeed, it was proved in Article 14^{15} that the six notes above the tonic form only two chords of
three notes whose tone ratios are in the same relative proportions. These two natural
triads of notes are DFA and EGB. Corresponding members are separated by a
tone interval of 9/8 (Fig. 26). What this means is that the eight notes form four
pairs:
C C'
D E
F G
A B
27
The perfect fourth F corresponds to the icosahedron, the fourth Platonic
solid, the perfect fifth G corresponds to the dodecahedron, the perfect fifth, the major sixth
A corresponds to the icosidodecahedron and the major seventh B corresponds to the disdyakis
triacontahedron The tonic C and the octave C' correspond to the selfdual tetrahedron.
Using Table 3, the numbers of Pythagorean tone ratios in the seven scales can be
represented in Fig. 27 as converging to the number 7 of the C scale. It also depicts
how the numbers of faces in the seven solids increase to that of the disdyakis
triacontahedron.
Notice that dual polyhedra (indicated by the same coloured dots) lie on
opposite, sloping lines.
As pointed out earlier, the eight notes of the Pythagorean scale are
separated by four perfect fourths and four perfect fifths. The prime division of the octave
into the perfect fifth G, leaving an interval of a perfect fourth, corresponds to the
disdyakis
triacontahedron being dual to the truncated icosidodecahedron. The division
of the octave into a perfect fourth followed by a perfect fifth corresponds to the icosahedron
being dual to the dodecahedron. The perfect fifth between note D and note A followed by a drop
down by
28
a fourth corresponds to the duality of the octahedron and the cube. The
perfect fourth between D and G corresponds to the tetrahedron. The two possible divisions of
the complete octave — 3/2×4/3 and 4/3×3/2 — correspond to the disdyakis triacontahedron and its
dual and to the dodecahedron and its dual, whilst divisions within the octave correspond to the
tetrahedron, cube and octahedron (Fig. 28).
Inspection of Table 2 reveals that the seven octave species have notes with 13 different
tone ratios other than 1, which, together with their 13 inversions, totals
26. In ascending and descending magnitude of pitch, the two sets of
intervals are:
256/243 
9/8 
32/27 
81/64 
4/3 
1024/729 
729/512 
3/2 
128/81 
27/16 
16/9 
243/128 
2 
243/256 
8/9 
27/32 
64/81 
3/4 
729/1024 
512/729 
2/3 
81/128 
16/27 
9/16 
128/243 
1/2 
This shows how the Divine Name YAHWEH with number value 26
prescribes the number of different rising and falling intervals in the seven scales, i.e.,
their tonal range. These 13 different, rising intervals and their falling interval counterparts
correspond to the 13 Archimedean solid and their duals — the Catalan solids.
The numbers of these intervals vary from scale to scale, with most repeated
at least once. Let us consider the basic set of different intervals in each scale. Article 16
calculated that the seven basic sets have 53 Pythagorean intervals (including 1) and 37
nonPythagorean intervals.16 There are therefore a minimum of 90 intervals. The significance of
this number in the context of the disdyakis triacontahedron will appear shortly.
5. Plato’s Lambda & its connection to the disdyakis
triacontahedron In his Timaeus, Plato described how the Demiurge measured the World Soul, or
substance of the spiritual universe as a strip divided according to the simple proportions
of the first three squares of 2 and 3. This is traditionally represented by his ‘Lambda,’
socalled because of its resemblance to the Greek letter Λ (Fig. 29). These numbers line but two sides of a tetractys array of ten
numbers from whose relative proportions the scientists and musicians of ancient Greece
worked out the frequencies of the notes of the now defunct Pythagorean musical
scale.^{17} The numbers missing from the Lambda are shown in red in
Fig. 29. The sum of the 10 integers is 90 and the sum of the integers 1, 8
and 27 at the corners of the tetractys is 36. The seven integers at the centre and corners
of the hexagon shown in Fig. 29 with dashed edges add up to 54.
Comparing these properties of the Lambda with the number of intervals of the
seven musical scales given at the end of the last section, we find that it has the remarkable
property of defining not only the tone ratios of the Pythagorean scale as the ratios of its
numbers but also how many of these ratios are present as a basic set in the seven types of
musical scales! This is no coincidence, because the archetypal nature of the Lambda and its
connection with the Tree of Life,^{18} the I Ching table,^{19} superstrings,^{20} the musical scales^{21} and the nature of the spiritual cosmos^{22} has been demonstrated in previous articles. For example, in
conformity with the primacy of the number 4, the first four polygons in the inner
Tree of Life have 90 yods outside their shared edge (Fig. 30). This cannot be coincidence, because their individual yod
populations are also the sums of diagonal rows of integers in the Lambda. It should
therefore come as no surprise that the Lambda is relevant to the disdyakis triacontahedron,
as now explained. Constructed
29
from tetractyses, its 120 triangular faces divide into (120×3=360)
tetractyses. Each of its 180 edges is the side of an internal triangle formed by two vertices
and its centre. They are constructed from (180×3=540) tetractyses. The total number of
tetractyses forming the disdyakis triacontahedron = 360 + 540 = 900. The following
correspondences appear:
Lambda tetractys


disdyakis triacontahedron

Sum of 10 integers = 90


number of tetractys = 900 = 90×10; 
Sum of integers at corners = 36 

number of tetractyses in faces = 360 = 36×10; 
Sum of integers at centre & corners of hexagon = 54 

number of tetractyses inside solid = 540 = 54×10. 
The numbers 1^{3}, 2^{3} and 3^{3} at the
shapedefining corners of the Lambda tetractys add up to 36, which is the
number of tetractyses whose yod population is the number of tetractyses creating the shape of
the disdyakis triacontahedron. The seven numbers inside the Lambda tetractys add up to 54 — the
number of tetractyses whose yod population is the number of tetractyses inside
the disdyakis triacontahedron. Notice the similarity between the function and the character of
the two sets of integers.
Moreover, the largest of the seven integers in the hexagon is 18, whilst the
number of internal tetractyses creating the 180 edges is 180 = 18×10. Four arithmetic
properties of the Lambda correlate with the geometry of the disdyakis triacontahedron. This
cannot be coincidental.
The integers 1, 2, 3 and 4 symbolized by the four rows of dots in the
tetractys express the number of tetractyses generating the disdyakis triacontahedron. As:
30 = 1^{2} + 2^{2} + 3^{2} + 4^{2},
900 = 30^{2} =
(1^{2}+2^{2}+3^{2}+4^{2}).^{2}
Each of the 120 faces, when divided into three triangles, has three edges
meeting at a vertex inside it. The number of geometrical elements making up the insides of the
faces = 120×(1+3+3) = 120×7 = 840 = 84×10, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}.
The number of Golden Rhombicshaped sets of four faces in the disdyakis
triacontahedron = 30 = 1^{2} + 2^{2} + 3^{2}+ 4^{2}.
The number of triangular faces = 120 = 4×30 =
2^{2}(1^{2}+2^{2}+3^{2}+4^{2})
30
= 2^{2} + 4^{2} + 6^{2} + 8^{2}.
As 31 = 2^{0} + 2^{1} + 2^{2} +
2^{3} + 2^{4}, the number of vertices = 62 =
2×31
=
2(2^{0}+2^{1}+2^{2}+2^{3}+2^{4}) = 2^{1} +
2^{2} + 2^{3} + 2^{4} + 2^{5}.
The number of vertices surrounding the axis of the polyhedron =
62 – 2 = 60
= 2^{2} + 2^{3} + 2^{4} + 2^{5}.
We found earlier that the number of hexagonal yods needed to construct the
faces of this polyhedron = 1440 = 144×10, where
The minimum number of triangles needed to construct both the interior and
exterior = 120 + 180 = 300 =
(1^{2}+2^{2}+3^{2}+4^{2})(1+2+3+4). The number of triangles =
660, so that each half of the polyhedron has 330 triangles, where 330 =
(1!+2!+3!+4!)(1+2+3+4).
We also found that the number of geometrical elements surrounding the axis
of this polyhedron that make up the 900 tetractyses = 2400 =
(1^{3}+2^{3}+3^{3}+4^{3})1×2×3×4. All these expressions
illustrate how the Tetrad Principle^{23} determines properties of
archetypal objects that possess sacred geometry
31
32
6. The superstring structural parameter
1680 Surrounding the axis of the disdyakis triacontahedron are 1680
corners, edges and triangles created when its surface is formed from 180 internal
triangles with the centre of the solid as corners and which are then divided into three
tetractyses. This is easily seen from the table below:

Corners 

Edges


Triangles 
surface: 
62 

180


120 
interior: 
180 

62 + 3×180 = 602


3×180 = 540 
Total =

242 

782


660 
minus: 
–2 (two edges) 

–2 (two edges)


___ 
Total = 
240
+ 

780

+ 
660 = 1680 
There are 780 (=78×10) edges and 900 (=90×10) corners and faces. The
significance of this is as follows: the ten Sephiroth of the Tree of Life each manifest as
Godnames in Atziluth (the archetypal, or divine, world), Archangels in Beriah (World of
Creation), Angels (Yetzirah, or Formative World) and Mundane Chakra (Physical World). According
to Kabbalah, the Mundane Chakra of a Sephirah is assigned an astronomical object as its
material manifestation. The planet Earth is assigned to Malkuth. It symbolizes the physical
aspect of the Tree of Life. Its Mundane Chakra is Cholem Yesodeth, meaning
“breaker of the foundations." Through gematria, wherein the 22 letters of the Hebrew
alphabet are assigned integers, words acquire number values that
are equal to the sum of their letter values. The word ‘Cholem’ has the
number value 78 and the word ‘Yesodeth’ has the value 90, giving the Hebrew name of the Mundane
Chakra a value of 168 (Fig. 31). Remarkably, these numbers define (apart from the Pythagorean
factor of 10), respectively, the number of edges and the number of vertices and triangles of
the tetractyses needed to construct the disdyakis triacontahedron! Even if, implausibly, the
total number 1680 had happened by coincidence to match the number value of the Hebrew name
of the Mundane Chakra, it is improbable in the extreme that the geometrical composition of
the polyhedron could be quantified by three numbers that happen to be the values of the two
words in the name and their sum.
The numbers 90 and 78 appear in the first six polygons of the inner Tree of
Life, which themselves constitute a whole Tree of Life pattern, being prescribed by the Godname
numbers of all the Sephiroth.^{24} Both sets of polygons have 336 hexagonal yods, so that
168 such yods are associated with each set (Fig. 32). The triangle, pentagon and octagon have 78 yods and the square,
hexagon and decagon has 90 yods. Once again, it cannot, plausibly, be coincidental that some
combination of polygons contains 168 hexagonal yods. Even if this were so,
it would still be extremely unlikely that a subset of them would as well contain either 78
or 90 yods just by chance.
As
13^{2} – 1 = 168
33
34
and, as a tetractys array of ten tetractyses has 13 yods along each edge, a
parallelogram formed by two such arrays lying backtoback has 168 yods below
its apex. Ten parallelograms arranged as two pentagrams have 1680 yods surrounding their shared
apices (Fig. 33). As the number of yods in a tetractys array = 1 + 2 + 3 +…+ 13 =
91, there are 90 yods below its vertex, where 90 is the number value of Yesodeth. The
remainder of each arm of the tenpointed star has (91–13=78) yods, where 78 is the number
value of Cholem. Each arm is a geometrical representation of the number value of
Cholem Yesodeth. The inner half of the star has 900 yods, which is the number of
vertices and triangles in the interior and faces of the disdyakis triacontahedron
surrounding its axis of symmetry. The outer half has 780 yods, which is the number of edges
of the triangles. Each yod denotes one of the 1680 geometrical elements that surround any
axis of symmetry that joins two diametrically opposite vertices of the
polyhedron. With the 120 triangular
faces as single tetractyses, the number of yods in the 120 tetractyses on
the surface = 62 + 2×180 + 120 = 542. With the internal triangles as single
tetractyses, the number of internal yods = 2×62 + 180 + 1 = 305. The total
number of yods in its (120+180=300) tetractyses = 847. As there are seven yods along the axis,
the number of yods surrounding the centre of the polyhedron = 840. The inner and outer halves
of each whorl of the UPA/superstring each consist of 840 circularly polarised oscillations made
in 2½ revolutions around its spin axis (see Fig. 14). This superstring structural parameter is therefore embodied in the
disdyakis triacontahedron as the number of yods surrounding its axis of symmetry. The Tetrad
Principle determines this number because a tetractys array of 10 tetractyses (Fig. 34) has 85 yods, where
85 = 4^{0} + 4^{1} + 4^{2} + 4^{3},
so that 84 yods surround its centre, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}.
The Tetrad Principle also determines the structural parameter
168 (the number of oscillations in half a revolution of a whorl of the
UPA/superstring) because, starting from a square symbolizing the Tetrad, the fourth
stage of its construction from tetractyses requires 36 of them with
168 yods surrounding its centre (Fig. 35). Moreover, the fourth regular polygon is the hexagon. With
its sectors constructed from three tetractyses, it contains 91 yods, i.e., 85 yods other
than corners. The two joined hexagons in the inner form of the Tree of Life have
168 yods other than corners (Fig. 36). The superstring
35
36
structural parameter is the number of extra yods needed to fill up the
36 tetractyses in a pair of hexagons joined along one side. The Godname ELOHA
with number value 36 prescribes the structural parameter 168.
The number of hexagonal yods in two separate hexagons is 156, which is the
155th integer after 1. This shows how ADONAI MELKEH, the full Godname of
Malkuth with number value 155, prescribes the number characterizing the
oscillatory form of the E_{8}×E_{8} heterotic superstring.
7. The disdyakis triacontahedron as the polyhedral Tree of
Life The inner Tree of Life has 70 corners, eight of which coincide
with Daath and the seven Sephiroth other than Malkuth, Yesod and Kether (or, rather, the
projection of them onto the plane containing the polygons).
62 corners are intrinsic to the inner Tree of Life. These degrees
of freedom correspond to the 62 vertices of the disdyakis
triacontahedron.
The next level of differentiation of the outer Tree of Life is 10 Trees of
Life (Fig. 37). Each tree has its own inner form consisting of 14 polygons. The
140 polygons enfolded in 10 trees have (62×10=620) corners
unshared with their 62 Sephiroth. This is why the number value of Kether,
the first Sephirah, has to be 620. The 62 corners
that the 120 polygons enfolded in 10 trees share with the latter correspond to the
62 vertices of the disdyakis triacontahedron. There are 21
Sephiroth on each side pillar and 10 Tiphareths and 10 Yesods on the central pillar. The
62 Sephiroth therefore split into two sets of
31 associated with the active and passive pillars. They correspond to
the 31 vertices and their 31 inverted images. The
association of these two numbers is why the Godname EL of
Chesed has the number value 31 and the number value of its
Archangel Tzadkiel is 62.
The two sets of the first six polygons have 50 corners.
They are prescribed as a new Tree of Life pattern because 50 is the number
value of ELOHIM, the Godname of Binah.
37
38
There are 12n such polygons with (48n+2) corners enfolded
in n trees. The 120 polygons enfolded in 10 trees are the polygonal counterpart of the 120
faces of the disdyakis triacontahedron. They have 482 corners. Two of these — the uppermost
corners of the pair of hexagons — coincide with the lowest corners of the pair of hexagons
enfolded in the eleventh Tree of Life. 480 corners are intrinsic to the 120 polygons. 240 such
corners lie on each side of the central pillar. Fig. 38 shows that there are 240 hexagonal yods in 60 faces of the disdyakis
triacontahedron and 240 hexagonal yods in their 60 inverted images. These 480 degrees of
freedom generated by turning the 120 triangles into tetractyses are the counterpart of the
480 corners of the 120 enfolded polygons. They are also the counterpart of the (240+240=480)
hexagonal yods in the (7+7) separate polygons (Fig. 39) They symbolize the 480 nonzero roots of the
heterotic superstring gauge group E_{8}×E_{8}'. The
distinction between the 240
hexagonal yods and their inverted images shows itself in the direct product
of E_{8} and E_{8}' and in the two sets of polygons enfolded in the 10 trees,
which are mirror images of each other. Each hexagonal yod symbolizes a nonzero root of
E_{8} or E_{8}' and therefore a gauge boson transmitting the unified
superstring force. Hexagonal yods in a tetractys symbolize Sephiroth of Construction. The
variety of superstring forces is linked to the differentiation between these cosmic, spiritual
potencies.
840 yods lie on the 30 sides of the first six polygons on each side of the
central pillar (see Fig. 37). The 1680 boundary yods shaping the two sets of six polygons
correspond to the 1680 geometrical elements surrounding the axis of symmetry of the
disdyakis triacontahedron. Each yod symbolizes a bit of information needed to characterize
the form of the polyhedral version of the Tree of Life, just as the 1680 circularly
polarized oscillations of each whorl of the UPA/superstring create its helical shape.
Let us now examine the Tree of Life and the separate polygons. Divided into
three tetractyses, the 19 triangles of the lowest tree in any set of overlapping trees
acquire
39
240 yods^{25} (Fig. 40). These correspond to the 240 vertices of the 660 tetractyses making
up the disdyakis triacontahedron that surround its axis of symmetry. The seven separate
regular polygons have 48 corners. With their sectors constructed from three
tetractyses, they contain 720 yods surrounding their centres.^{26} These correspond to the 720 edges and triangles surrounding the
axis that make up the 330 tetractyses in one half of the polyhedron. The similar set of
seven polygons on the other side of the lowest tree also contains 720 such yods. They
correspond to the 720 edges and triangles in the inverted half of the polyhedron. This exact
parallelism between the disdyakis triacontahedron and the outer and inner Tree of Life is
powerful evidence that the former is their polyhedral form.
Between the tonic and the octave of a musical scale are six notes. The seven
scales have (6×7=42) notes between their tonic and octave, making 44 notes in total if the
latter are counted only once. They comprise 28 notes with Pythagorean tone ratios and 16 notes
with nonPythagorean tone ratios. Arranged on a halfcircle according to the magnitude of their
tone ratios (Fig. 41), the six notes between the tonic and octave of each scale occupy
positions that are diametrically opposite notes of the scale that is its mirror image, i.e.,
there are 21 notes and their mirror images. The counterpart of this
inversion symmetry in the disdyakis triacontahedron are the 210 (=21×10)
yods in one half lying along its edges between diametrically opposite vertices that have
their mirror images in the other half of the polyhedron. Actually, only the yods in
21 tetractyses are needed to create the shape of the polyhedron because the
rest of it is simply their mirror reflections. This shows how the Godname EHYEH (“I am”) of
Kether with number value 21 prescribes the disdyakis triacontahedron. Both
the seven musical scales and this polyhedron represent holistic systems, one of sound, the
other of geometry. The universal, divine archetypes find expression in many contexts, but
their mathematical character is always similar, despite the differences in their cultural
morphology.
The Godnames of the 10 Sephiroth prescribe the disdyakis triacontahedron in
the following ways:
Kether: EHYEH = 21. There are
420 (=42×10) yods on the edges of the polyhedron between diametrically opposite vertices. 210
(=21×10) yods belong to one half of the polyhedron and 210 yods belong to its
other, inverted half; EHYEH ASHER EHYEH = 543. 542 yods in faces surround the
centre of the polyhedron;
Chokmah: YAH = 15. The 30
golden rhombic faces of the rhombic triacontahedron underlie the disdyakis triacontahedron. 30
= 15th even integer. Including the chiral versions of the pentagonal
icositetrahedron and the pentagonal hexacontahedron, there are 15 Catalan
solids, among which the disdyakis triacontahedron has the most faces; YAHWEH =
26. The disdyakis triacontahedron is the 26th and last of the
two families of Archimedean and Catalan solids because it has the most faces;
Binah: ELOHIM = 50. There are
900 tetractyses when both faces and interior triangles are divided into three tetractyses. 90 =
2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} = sum of five
squares. 900 = 90×10 = sum of 50 squares;
Chesed: EL = 31. Of the
31 Archimedean, Catalan and Platonic solids, the disdyakis triacontahedron has
the most faces. It has 62 vertices, 31 in one half and
31 in its mirror image half;
Geburah: ELOHA: = 36. Number of
corners of 300 tetractyses surrounding an axis that
40
joins two diametrically opposite vertices = 360 = 36×10 =
number of vertices, edges and triangles in 120 faces surrounding this axis; 37 vertices in one
half of the polyhedron, where 37 is the 36th integer after 1.
Tiphareth: YAHWEH ELOHIM = 76.
The 31 Platonic, Archimedean and Catalan solids can be symbolized by the
31 yods of a pentagon with its sectors turned into tetractyses. When the
sectors are divided into three tetractyses, the pentagon contains 76 yods.
YAHWEH ELOHIM prescribes the pentagon determining the disdyakis triacontahedron as the
31st polyhedron;
Netzach: YAHWEH SABAOTH = 129.
Number of internal yods and yods on edges of the disdyakis triacontahedron = 727 =
129th prime number;
Hod: ELOHIM SABAOTH = 153.
Number of internal yods = 2×62 + 180×1 + 1 = 305 = 153rd odd
integer;
Yesod: SHADDAI EL CHAI = 363.
With faces and internal triangles divided into three tetractyses, the number of their vertices
= 62 + 120×1 + 180×1 + 1 = 363;
Malkuth: ADONAI MELEKH = 155.
1560 (=156×10) yods other than vertices in faces divided into three tetractyses. 156 =
155th integer after 1.
References
^{1} Phillips, Stephen M. Article 19: “I Ching and the Eightfold Way,”
(WEB, PDF), pp. 15, 17.
^{2} For further discussion of the Tree of Life character of the
musical modes, see: Article 14: “Why the Greek Musical Modes are Sacred,” by Stephen M.
Phillips, (WEB, PDF), pp. 17, 18.
^{3} Phillips, Stephen M. Article 22: “The Disdyakis Triacontahedron
the 3dimensional Counterpart of the Inner Tree of Life,” (WEB, PDF), Article 23: “The ‘Polyhedral Tree of Life,” (WEB, PDF) and Article 24: “More Evidence for the Disdyakis
Triacontahedron As the 3dimensional Realisation of the Inner Tree of Life and Its
Manifestation in the E_{8}×E_{8} Heterotic Superstring,” (WEB, PDF).
^{4} Phillips, Stephen M. Article 2: “The Physical Plane and Its
Relation to the UPA/Superstring and Spacetime, ” (WEB, PDF).
^{5} Ref. 2.
^{6} Phillips, Stephen M. Article 5: “The Superstring as Microcosm
of the Spiritual Macrocosm,” (WEB, PDF).
^{7} Divided into three tetractyses, each of the 20 triangular faces
has 9 hexagonal yods inside it. Divided into five tetractyses, each of the 12 pentagonal faces
has 15 internal, hexagonal yods. Total number of internal hexagonal yods = 20×9 + 12×15 = 360.
Each of the 60 sides has two hexagonal yods. Total number of hexagonal yods in the
icosidodecahedron = 360 + 60×2 = 480.
^{8} Proof: the tetrahedron has six edges, each with two hexagonal
yods. Number of yods on edges = 4 + 6×2 = 16. It has four faces, each with 10 yods inside its
edges. Number of yods on faces = 16 + 4×10 = 56. Each edge is the side of an internal
tetractys. Two hexagonal yods lie on the edge joining each corner to the centre of the
tetractys. One yod is at the centre of each tetractys. Number of yods inside tetrahedron = 4×2
+ 1 + 6 = 15. Total number of yods = 15 + 56 = 71. 70 yods surround the centre.
^{9} Proof: dividing each internal triangle into three tetractyses
instead of one tetractys adds 9 yods inside each one. Number of yods = 71 + 6×9 = 125. 120 yods
other than corners surround the centre of the tetrahedron.
^{10} The last two members of the seven enfolded, regular polygons
forming the inner Tree of Life — the decagon and dodecagon — have 126 yods outside their shared
edge. As all seven polygons have 264 yods, there are 138 yods in the first five polygons.
Excluding diametrically opposite vertices, there are 1260 (=126×10) yods on the boundaries of
the 360 tetractyses forming the faces of the disdyakis triacontahedron. Excluding yods along
the axis of symmetry joining this pair of vertices, there are 1380 (=138×10) yods on the
boundaries of the 540 tetractyses inside the disdyakis triacontahedron. This 5:2
41
differentiation of
polygons corresponds to the distinction between the yods on the sides of the internal
and external tetractyses making up the disdyakis triacontahedron. The musical pattern of five
tone intervals and two leimmas in each of the seven musical scales has its geometrical
counterpart in the first five polygons generating the boundary yods inside the disdyakis
triacontahedron and the last two polygons generating the boundary yods on its faces.
^{11} Phillips, Stephen M. Article 3: “The Sacred Geometry of the
Platonic Solids,” (WEB, PDF), pp. 11, 20.
^{12} Proof: The truncated icosidodecahedron has 30 square faces, 20
hexagonal faces and 12 decagonal faces. The square has 12 hexagonal yods inside it, the hexagon
has 18 internal hexagonal yods and the decagon has 30 internal hexagonal yods. The number of
internal hexagonal yods in the polyhedron = 30×12 + 20×18 + 12×30 = 1080. The polyhedron has
180 sides, each with two hexagonal yods. The total number of hexagonal yods = 1080 + 180×2 =
1440.
^{13} Ref. 11, pp. 10, 20.
^{14} Suppose that each rising interval is replaced by its falling
counterpart, i.e., TT^{–1} and LL^{–1}. An ascending scale, e.g., TLTTLTT, then
becomes
T^{–1}L^{–1}T^{–1}T^{–1}L^{–1}T^{–1}T^{–1},
which is the descending version of the ascending scale TTLTTLT. Inverting each interval of
a scale creates another scale whose pattern of intervals is its mirror image. Only the D
scale is its mirror image: TLTTTLT = TLTTTLT.
^{15} Ref. 2, pp. 4, 5.
^{16} Phillips, Stephen M. Article 16: “The Tone Intervals of the
Seven Octave Species and Their Correspondence with Octonion Algebra and Superstrings,”
(WEB, PDF), p. 11.
^{17} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its
Meaning, Generalisation and Connection to the Tree of Life,” (WEB, PDF).
^{18} Ibid, pp. 7, 16.
^{19} Phillips, Stephen M. Article 20: “Algebraic, Arithmetic and
Geometric Interpretations of the I Ching Table,” (WEB, PDF), pp. 18–22.
^{20} Ref. 17, p. 7.
^{21} Ref. 17, pp. 2–3, and Article 12, “New Pythagorean Aspects of
Music and Their Connection to Superstrings,” Stephen M. Phillips, (WEB, PDF), pp. 2–4.
^{22} Ref. 17, pp. 3–7.
^{23} Phillips, Stephen M. Article 1: “The Pythagorean Nature of
Superstring and Bosonic String Theories,” (WEB, PDF), p. 5.
^{24} Phillips, Stephen M. Article 4: “The Godnames Prescribe the
Inner Tree of Life,” (WEB, PDF), p. 4.
^{25} Proof: The lowest tree has 19 triangles with 11 vertices and
25 edges. Divided into three tetractyses, each triangle has 10 internal yods. Number of yods in
lowest tree = 11 +25×2 + 19×10 = 251. The 11 Sephiroth are at the vertices of the triangles.
Number of yods other than Sephiroth = 251 – 11 = 240.
^{26} Proof: Divided into three tetractyses, each sector of a
regular polygon has 10 internal yods. The seven polygons have 48 sectors. Each of the 48 sides
of the polygons has two hexagonal yods between its ends. Each side joining a corner to the
centre of a polygon has two hexagonal yods. Number of yods in seven separate polygons = 48×10 +
48×2 + 48×2 + 48 + 7 = 727. Number of yods surrounding the centres = 727 – 7 = 720.
"The Universe is a thought of the Deity. Since this ideal thoughtform has
overflowed into actuality, and the world born thereof has realized the plan of its creator, it
is the calling of all thinking beings to rediscover in this existent whole the original
design."
F. Schiller, Theosophie des Julius, The World of Thinking Beings.
42
