ARTICLE 36
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Viewed along an axis joining any pair of C vertices of the
disdyakis triacontahedron, its 60 vertices are shown to be arranged similarly,
although not identically, to the triangles in the Sri Yantra. The projection of
the ten triangles and five 6sided polygons formed by these vertices onto the
equatorial (XY) plane of the polyhedron generates 42 vertices and four pairs of
intersecting triangles, comparing with the 42 tips of the four pairs of
intersecting triangles that surrounding the central one in the Sri Yantra. There
are 46 vertices on either side of the equatorial plane that are arranged as
pairs, with one vertex directly above the other. The 23 pairs consist of a
diametrically opposite pair of C vertices and 22 other pairs. They correspond in
the polygonal Tree of Life to the 23 pairs of hexagonal yods on the 23 edges of
its pair of dodecagons. The 23 pairs of vertices or hexagonal yods symbolize the
23 pairs of chromosomes in the human cell. The Godname YAH prescribes the number
of polygons, the Godname EHYEH prescribes the 41 vertices of the 33 triangular
sectors of the eight types of polygons and the Godname ADONAI prescribes their
66 edges. The (240+8) roots of the superstring gauge symmetry group
E_{8} are encoded in the 15 polygons as the 240 geometrical elements
surrounding their centres, of which there are eight independent ones, the others
being their mirror images. The counterparts of these elements in the polygonal
form of the Tree of Life are the 240 hexagonal yods that surround the centres of
the seven separate, regular polygons. Constructed from tetractyses, the eight
types of polygons formed by the vertices of the disdyakis triacontahedron have
206 yods. They symbolize the 206 bones of the human skeleton, the 34 corners and
centres of the seven polygons above or below the XY plane denoting the 34 single
bones in the axial skeleton. The disdyakis triacontahedron embodies the number
137 associated with the finestructure constant as the number of vertices in the
polyhedron and 15 polygons when the 60 sectors of the latter are divided into
three triangles. When these triangles are turned into tetractyses, the 15
polygons have 840 yods other than polygonal corners. They symbolize the 840
circularly polarized oscillations in an inner or outer half of each of the ten
closed curves making up the E_{8}×E_{8} heterotic superstring.
The archetypal character of the polygons is confirmed by the equivalence of
their yod composition to Plato’s Lambda Tetractys, whose numbers have been known
for over 2000 years to have as their proportions the tone ratios of the notes of
the Pythagorean musical scale. This equivalence extends to the central hexagon
in the XY plane.

1
1. The 15 layers of
polygons in the disdyakis triacontahedron
The position vector of the vertex A (Fig. 1) at (0,0,2φ^{2}) is inclined at an angle α to the position
vector of the vertex C at (0,φ,φ^{3}), where tanα = φ/φ^{3} =
1/φ^{2}. As φ – 1 = 1/φ,
1/φ^{2} = 1 – 1/φ = 1 – (φ–1) = 2 – φ.
Therefore, tanα = 2 – φ
1 + tan^{2}α = 1/cos^{2}α = 1 + (2–φ)^{2} = 1 + 4
+φ^{2} – 4φ = 6 – 3φ = 3(2–φ) = 3/φ^{2} .
Therefore, cosα = φ/√3.
1 – cos^{2}α = sin^{2}α = 1 – φ^{2}/3 =
(3–φ^{2} )/3 = (3–(1+φ))/3 = (2–φ)/3 = 1/3φ^{2} .
Therefore, sinα = 1/φ√3.
An anticlockwise rotation of α° about the Xaxis of the Y and Zaxes makes
the positive Zaxis pass through vertex C at (0,φ,φ^{3}) instead of the vertex A at
(0,0,2φ^{2}). A point P has rectangular coordinates (x,y,z) and polar coordinates
(R,θ,ψ), where
y = rcosβ, z = rsinβ, and r^{2} = y^{2} + z^{2} =
R^{2}(cos^{2}ψ + sin^{2}ψsin^{2}θ).
and r is the projection of the position vector R of P onto
the YZ plane. Its coordinates (x’,y’,z’) with respect to the new coordinate system with a C
vertex on the Zaxis are given by:
x' = x,
y' = rcos(β–α) = rcosβcosα + rsinβsinα = (yφ+z/φ)/√3,
z' = rsin(β–α) = rsinβcosα – rcosβsinα = (zφ–y/φ)/√3,
where x, y & z are listed for each vertex of the disdyakis
triacontahedron in Table 1 of Article 25. The new coordinates are listed here in Table 1.
Fig. 2 shows that the rotation of the Zaxis creates above the XY plane three
upward pointing triangles (red, yellow & violet) and two downward pointing triangles
(orange & blue), whilst below the plane are two upward pointing triangles (orange &
blue) and three downward pointing triangles (red, yellow & violet). There are five
upward pointing triangles and five downward pointing triangles, whereas the Sri Yantra is
formed from five downward pointing triangles and four upward pointing ones. However, the two
pairs of triangles forming a Star of David with B (orange) vertices and C (yellow) vertices
at their tips are identical in size and shape. This means there are four distinct,
upward pointing triangles (red, yellow, violet & blue) and four distinct,
downward pointing ones (red, orange, violet & blue), thus matching the two sets of four
triangles of the Sri Yantra. Alternatively, the projection of the 10 triangles onto the XY
plane (the central
2
3
Table 2. 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, Orders 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. Numbers from the table are written
in boldface in the article. 
4
layer of six vertices) creates four different pairs of intersecting
triangles. There is nothing in the projection corresponding to the downward pointing central
triangle, although it may be argued that the C vertex formally plays its role, being the
starting point for the generation of the polyhedron, just as the triple Godhead symbolised by
the innermost triangle in the Sri Yantra is the source of all manifestation.
The eight distinct triangles have 24 vertices. There are five 6sided
polygons but only three differently shaped ones with 18 vertices. This because one of the
polygons with A vertices is the inversion of the other with C vertices on the same side
of
the XY plane, whilst the two others in the lower half of the polyhedron are
the inversions of these. When projected onto the XY plane, they become two polygons, one the
mirror image of the other. Hence, there are 42 vertices of distinct polygons. This is
the number of triangles in the four layers of the Sri Yantra. 36 vertices
originate in the layers above and below the XY plane. ELOHA, Godname of Geburah, prescribes
them because its gematria number value is 36 (Table 2). 21 vertices and their 21 mirror
images are in the XY plane. This shows how EHYEH, Godname of Kether with number value
21, prescribes the crosssection of the disdyakis triacontahedron viewed
along a CC axis.
Five vertices above the XY plane on either side of the Xaxis do not have
counterparts directly beneath them in the lower half. Starting from above the Xaxis in
Fig. 2, they are the B vertex on the Yaxis, the
three A vertices forming a triangle and the lowermost
A vertex. Similarly, five vertices below the XY plane have no
vertices above them. Starting from above the Xaxis, they are the A vertex, the three A vertices forming a triangle
and the B vertex. Of the remaining 52 vertices, six vertices
belong to the central hexagon, leaving 23 pairs of vertices, one of which is directly
above the other in the pair. One of
5
them is the pair of C vertices forming the apex and nadir of the disdyakis
triacontahedron. These (22+1) pairs of vertices can be regarded as reproducing the pattern of
the two sex chromosomes and the 22 pairs of chromosomes in the nucleus of the human cell. Given
that this polyhedron is the cosmic blueprint — the “image of God” according to which humans
were designed, according to Genesis 1:27, this pattern needs serious
consideration. This view is reinforced by the fact that the same division appears
in the polygonal form of the inner Tree of Life because there are 23 pairs of hexagonal yods on
the edges of the pair of dodecagons, with one
pair belonging to the shared root edge (Fig. 3). Hexagonal yods signify hidden degrees of freedom, and the 23 pairs
are the minimal number of degrees of freedom needed to delineate the boundary of the pair of
dodecagons — the Malkuth aspect of the Tree of Life corresponding to the human body. The
pairing of chromosomes is signalled not by the pairs of hexagonal yods on each edge in a
dodecagon but by the mirror image pair of dodecagons, so that there is in one dodecagon a
mirror image counterpart of each yod in the other dodecagon.
2. Geometrical composition of
the 15 polygons
Consider the 15 polygons divided into their sectors. Table 3 lists the numbers of vertices, edges & triangles in the
eight polygons in half the disdyakis triacontahedron.
Table 3. Number of vertices, edges & triangles in the eight polygons.

Number of
vertices

Number of
edges

Number of
triangles

Total


3 + 1 = 4

6

3

12 + 1 = 13


3 + 1 = 4

6

3

12 + 1 = 13


3 + 1 = 4

6

3

12 + 1 = 13


6 + 1 = 7

12

6

24 + 1 = 25


3 + 1 = 4

6

3

12 + 1 = 13


6 + 1 = 7

12

6

24 + 1 = 25


3 + 1 = 4

6

3

12 + 1 = 13

Subtotal =

27 + 7 = 34

54

27

108 + 7 = 115


6 + 1 = 7

12

6

24 + 1 = 25

Total =

33 + 8 = 41

66

33

132 + 8 = 140

6
The first seven polygons have 108 geometrical elements surrounding their
centres. The hexagon in the XY plane has 24 elements surrounding its centre. The eight polygons
have (108+24=132) elements surrounding their centres. They comprise 33 vertices and 33
triangles, where 33 (=1!+2!+3!+4!) is the number of layers of vertices that are orthogonal to
AA, BB and CC axes.^{1} The total number of geometrical elements is
140. This is the number value of Malachim, the Order of Angels
assigned to Tiphareth, and of Masloth, the Mundane Chakra of Chokmah (the Zodiac). It
includes 41 vertices, where 41 is the 21st odd integer. This shows how
EHYEH, the Godname of Kether with number value 21, prescribes the shapes of
all the polygons orthogonal to a CC axis. The eight polygons have 66 edges. As 66 is the
65th integer after 1, the Godname ADONAI with number value
65 prescribes the eight types of polygons.
The two sets of seven polygons have (2×108=216) elements
surrounding their centres, where 216 is the number value of Geburah, the sixth
Sephirah. All 15 polygons have 60 vertices, 120 edges & 60 triangles
surrounding their centres, i.e., 240 geometrical elements. This number is highly significant
visàvis the inner Tree of Life because it is the number of hexagonal yods in its seven
separate polygons (Fig. 4). Moreover, the first three of these polygons have 60 hexagonal yods,
comparing with the 60 vertices surrounding centres of the 15 polygons, the
next three polygons have 120 hexagonal yods, comparing with their 120 edges, and the last
polygon has 60 hexagonal yods, comparing with their 60 triangles. The correlation shows that
the 15 polygons constitute a Tree of Life pattern. The 3:3:1 division of
polygons generates the pattern of their vertices, edges and triangles. It reflects the fact
that the seven Sephiroth of Construction are grouped in the Tree of Life as two triads and a
single one — Malkuth.
The superstring gauge symmetry group E_{8} has 240 roots and eight
simple roots. The geometry of the 15 polygons embodies this group parameter
describing the unified symmetry of superstring interactions. The centres of the eight types of
polygons correspond to the eight simple roots. As each gauge field is 10dimensional, the
number of spacetime components of the gauge fields corresponding to the generators associated
with these roots is 2400. This was found in Article 27^{2} to be the number of vertices, edges and triangles in the faces
and interior of the disdyakis triacontahedron when the former are divided into three
triangles and when the interior triangles formed by edges are likewise divided. The
remarkable conclusion is that this polyhedron embodies the number of components of the 240
gauge fields of E_{8} associated with its roots, whilst the 15
polygons formed by its vertices embody this number of gauge fields.
Just as the 2dimensional form of the Sri Yantra was considered in Article
35, let us now imagine the 60 vertices in the 15 layers projected onto the XY
plane. The two pairs of intersecting triangles forming Stars of David become one pair, losing
eight vertices, 12 edges and six triangles. The four 6sided polygons become two such polygons,
losing 14 vertices, 24 edges and 12 triangles. Two A vertices and two B vertices of triangles
disappear. Also, two edges of triangles disappear. 18 triangles disappear altogether, leaving
(60–18=42) triangles. This is the number of triangles in
7
8
Table 4. Number of yods in the eight polygons (sectors as single
tetractyses).
the Sri Yantra that surround the central one. 26 vertices
disappear, leaving (75–26=49) vertices. 26
is the number value of YAHWEH, the Godname of Chokmah, and 49 is the number
value of EL CHAI, the Godname of Yesod.
3. Yod composition of
the 15 polygons
Now let us consider the triangular sectors of the polygons as tetractyses. Table 4 shows the numbers of yods in the eight polygons in one half of the
disdyakis triacontahedron. There are 41 yods at the corners of their 33 tetractyses, 165
hexagonal yods and 206 yods altogether. The latter is made up of 34 vertices of the 27
tetractyses in the first seven polygons and 172 yods that are either hexagonal yods (165) or
vertices of the tetractyses in the central hexagon (7). Compare this with the facts that the
axial skeleton of the human body has 34 single bones and that 172 bones in the axial and
appendicular skeletons exist in pairs (Fig. 5). The corners and centres of the seven polygons above the XY plane
symbolise the single bones of the axial skeleton. It is truly remarkable that the bone
composition of the body should be defined by the yods needed to construct the eight polygons
in one half of the disdyakis triacontahedron from tetractyses! It is evidence that the human
body is designed according to the blueprint of the Tree of Life, of which this Catalan solid
is the polyhedral realisation.
The 60 tetractyses in the 15 polygons have 60 vertices
other than centres of polygons and 300 hexagonal yods, totalling 360 (=36×10)
yods that surround their 15 centres. This shows how ELOHA, the Godname of
Geburah with number value 36, prescribes the yod population of the
15 layers of vertices. The central hexagon also has
36 yods surrounding its centre. There are 10 triangles and five 6sided
polygons. The numbers 10 and 5 are the values of, respectively, the letters yod
(י) and heh
(ה) making up
the Divine Name YAH (יה) with number value 15, demonstrating
how it prescribes the layers of vertices in the disdyakis triacontahedron.
The first six polygons have 142 yods on the edges of their 27 tetractyses.
Both sets of six polygons have 284 boundary yods. The central hexagon has 31
yods on the edges
9
10
Table 5. Geometrical composition of the eight polygons (sectors as three
triangles).
of its six tetractyses, showing how EL, the Godname of Chesed, prescribes
the shape of the polygon in the middle of the 15 layers of vertices. The
15 polygons have (284+31=315) boundary yods. Of these, 75 are
vertices, leaving 240 hexagonal yods on the edges of tetractyses. We found earlier that the
15 polygons have 240 geometrical elements surrounding their centres. The two
numbers agree because each sector of a polygon adds four geometrical elements, whilst,
considered as a tetractys, it adds four hexagonal yods on the two edges associated with it.
Both properties demonstrate that the number 240 characterises the shape of the
15 polygons. Many previous articles have discussed it in the context of the
E_{8}×E_{8} heterotic superstring, whose 16800 circularly polarized
oscillations about its spin axis are the spacetime manifestation of the 240 gauge charges
associated with the nonzero roots of E_{8}. Likewise, there are 240 geometrical
elements surrounding the centres of the 15 polygons. The disdyakis
triacontahedron embodies the structure of the superstring.
Now consider the sectors of the polygons divided into three triangles.
Table 5 lists their geometrical composition. The first seven polygons have 61
vertices of their 81 triangles. Both sets of seven polygons have 122 vertices. The hexagon
in the XY plane has 13 vertices. The 15 polygons have (122+13=135)
vertices. Including the two C vertices, the disdyakis triacontahedron has 137 vertices. This
is the 33rd prime number, where 33 is the number of corners of the eight polygons in
onehalf of the polyhedron. Its reciprocal is approximately the finestructure constant α ≡
e^{2}/ħc ≈ 1/137 that measures the coupling of the electric charge of an electron to
the electromagnetic field. One of the most important numbers in physics is encoded in the
geometry of the 15 layers of vertices of this polyhedron! It is encoded in
the inner Tree of Life as the 1370 yods in its 14 polygons when their 94 sectors are each
constructed from three tetractyses (Fig. 6).
The first seven polygons have 270 geometrical elements surrounding their
centres. The central hexagon has 60 elements surrounding its centre. Surrounding the centres of
the 15 polygons are (270+60+270=600) elements made up of 120 vertices, 300
edges and 180 triangles. Constructed from triangles, the disdyakis triacontahedron has 600
geometrical elements (60 vertices, 240 edges and 300 triangles) surrounding any axis
11
that joins two diametrically opposite vertices.^{3} It is remarkable that the same number of geometrical
elements should be displayed by both the polyhedron and its 15 layers.
Table 6. Number of yods in the eight polygons (sectors as three
tetractyses).
Table 5 also indicates that the eight polygons have 165 edges & 99
triangles, that is, 264 edges & triangles. This demonstrates that they constitute a Tree
of Life pattern described by the structural parameters of the inner Tree of Life, whose
seven enfolded polygons contain 264 yods when constructed from tetractyses (Fig. 7).
Now suppose that each triangle is a tetractys. Table 6 lists the yod populations of the eight polygons. The 99 sectors of
the eight polygons in onehalf of the polyhedron have 74 vertices. Including the two C
vertices, there are 76 vertices. This
shows how YAHWEH
ELOHIM, the Godname of Tiphareth with number value 76, prescribes the
polyhedron through its 15 layers of vertices. The two 6sided polygons
above the XY plane have 26 vertices, leaving 50 vertices
(two C vertices and 48 vertices of sectors). 26 is the
number value of YAHWEH and 50 is the number value of ELOHIM.
Surrounding their centres are 66 yods at vertices of tetractyses. 66 is the
65th integer after 1, showing how ADONAI, the Godname of Malkuth with number
value 65, prescribes the shapes of the polygons through the vertices of their
99 tetractyses. The first seven polygons have 351 hexagonal yods.
351 is the number value of Ashim, the
12
Order of Angels assigned to Malkuth. It is also the sum of the first
26 integers:
351 = 1 + 2 + 3 + …. + 26.
This shows how YAHWEH, the Godname of Chokmah, determines the population of
hexagonal yods in the seven polygons above or below the XY plane. There are 385 yods other than
polygonal corners in the 81 tetractyses making up the first seven polygons, where
This demonstrates the mathematical beauty intrinsic to the disdyakis
triacontahedron. The Decad determines the number of extra yods needed to construct the polygons
in either half of the polyhedron above or below its central plane.
All eight polygons have 495 yods surrounding their centres. As 495 = 9×55 =
3^{2}×(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}) =
3^{2} + 6^{2} + 9^{2} + 12^{2} +
15^{2}, we see how the Godname YAH with number value
15 prescribes the yod population of these polygons. It also prescribes the 470
yods on the edges of their 99 tetractyses, for 47 is the 15th prime number.
All 15 polygons have 135 vertices, 780 hexagonal yods (that is, ten times the
number of hexagonal yods in the central hexagon) and 915 yods in all. 900 yods surround the
15 centres, where 900 = 30^{2} =
(1^{2}+2^{2}+3^{2}+4^{2})^{2} =
3^{2}(1+2+3+4)^{2} = (3+6+9+12)^{2}. This illustrates the Tetrad
Principle^{4} at work in expressing the mathematical
properties of holistic systems. When the 180 interior triangles formed by
the edges of the disdyakis triacontahedron are each divided into three tetractyses, they form
540 tetractyses. When its 120 faces are each divided into three triangles, they form 360
tetractyses. Therefore, the construction of the polyhedron requires (540+360=900) tetractyses.
Remarkably, the same number of tetractyses creates the polyhedron as there are yods surrounding
the centres of the polygons in its 15 layers! Earlier, we found that it also
has the same number (600) of geometrical elements as the 15 polygons.
There are 85 yods in the central hexagon (Fig. 8) other than its six corners, where
85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}.
84 yods other than these corners surround its centre, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}.
These are more examples of how the Tetrad Principle determines properties of
holistic systems, e.g., the disdyakis triacontahedron. The seven polygons above the XY
plane
13
14
have 378 yods other than corners that surround their centres. The
15 polygons have (378+84+378=840) such yods (Fig. 9). Remarkably, they have ten times the corresponding number in the
central hexagon. This number is a structural parameter of the heterotic superstring, being
the number of circularly polarized oscillations in an outer or inner half of each of the ten
closed curves that comprise it.
The arithmetic reason for this very important encoding of superstring
structure in the 15 layers of vertices is simple. The number of yods in
an nsided polygon whose sectors are each divided into three tetractyses is
15n + 1. The number of yods other than corners of the polygon that surround
its centre is 14n. The number of such yods in the 15 polygons =
14Σn = 14×60 = 840. The five 6sided polygons have 30 corners
and (14×30=420) yods other than corners, as do the ten triangles with 30 corners. This
equal division of 420 yods is prescribed by the Godname YAH (YH) because the letter value
5 of H (heh) specifies the number of 6sided polygons and the letter value 10 of Y (yod)
specifies the number of triangles.
It was shown in Article 27 that, when constructed from tetractyses, the
disdyakis triacontahedron has 840 yods surrounding any axis joining diametrically opposite
vertices.^{5} Once again, it is remarkable that the same yod population
appears in both contexts. The 840 yods in the 15 polygons other than
their 60 corners consist of 60 vertices of tetractyses and 780 hexagonal yods.
Similarly, the 840 yods surrounding the axis of the disdyakis
triacontahedron comprise 60 polyhedral vertices and 780 hexagonal yods. The same 60/780
composition of yods exists in each context as well.
A 120:720 division in the number 840 occurs as well. The
15 polygons have (60×2=120) hexagonal yods along their 60 edges and 720
yods inside them. The latter comprise 360 hexagonal yods along the three edges inside each
sector and 360 yods made up of 120 hexagonal yods along edges of sectors, 60 vertices and 180
hexagonal yods at centres of tetractyses. The disdyakis triacontahedron has 120 hexagonal yods
at centres of its faces and 720 yods that comprise 360 hexagonal yods along its 180 edges and
360 yods made up of 120 hexagonal yods on 60 internal edges of triangles, 60 vertices and 180
hexagonal yods at centres of 180 internal tetractyses.
4. The Lambda Tetractys pattern
of the 15 polygons
The total number of yods surrounding the centres of the 15 polygons =
15Σn = 15×60 = 900, that is, the number of
yods in 90 tetractyses. They consist of the 60 vertices of the disdyakis triacontahedron
(the yods of six tetractyses) and the 840 remaining yods surrounding the centres of these
tetractyses (the yods of 84 tetractyses). This 6:84 division appears in the Lambda
Tetractys (Fig. 10). This is the interpolated form of
15
Plato’s Lambda, the ratios of whose numbers have been known for over 2,000
years to define the tone ratios of the notes of the Pythagorean musical scale.^{6} If we consider its integers as denoting numbers of tetractyses,
then the integer 6 denotes six tetractyses whose 60 yods correspond to the 60 vertices of
the disdyakis triacontahedron, whilst the sum of its nine other integers is 84, and 84
tetractyses have 840 yods, which is the number of yods other than these vertices. This is
intuitively appealing because, in terms of the equivalence between the Tree of Life and the
tetractys, the central yod symbolizes Malkuth, so that the 60 vertices in the
15 polygons constitute the Malkuth aspect of the disdyakis triacontahedron
as the polyhedral Tree of Life as they create its form, whilst the other yods constitute its
higher aspect.
The Lambda Tetractys was encountered in a previous article^{7} in the context of the disdyakis triacontahedron as the 900
tetractyses needed to build its faces and interior triangles when each one is divided into
three tetractyses. The 120 faces comprise (120×3=360) tetractyses and the 180 interior
triangles formed by its 180 edges comprise (180×3=540) tetractyses. The former correspond to
the sum (36) of the integers at the corners of the Lambda Tetractys and the latter
correspond to the sum (54) of the integers at the centre and corners of a hexagon. Both the
disdyakis triacontahedron and its 15 layers of polygons conform to the archetypal
pattern of the Lambda Tetractys.
Another way of comparing the 15 polygons to the Lambda Tetractys is
as follows: the ten triangles and the central hexagon have 36 sectors with
(36×15=540) yods surrounding their 11 centres. This is the
number of yods in 54 tetractyses. The four 6sided polygons have 24 sectors with
(24×15=360) yods surrounding their 4 centres. This is the number of yods in
36 tetractyses. The sum of the ten integers is 90, which is the number of
tetractyses whose yod population equals that of the 15 polygons. The four
6sided polygons with
360 yods correspond to the integers at the corners of the Lambda, whose sum
is 36, whilst the ten triangles and the central hexagon correspond to the seven
remaining integers, whose sum is 54.
The central hexagon itself displays the 36:54 division in the Lambda
Tetractys. Regarded as a single tetractys, each sector contributes six yods, so that 36
yods surround its centre. Regarded as three tetractyses, a sector contributes nine more yods
(all hexagonal), so that 54 more yods are added to the hexagon. However, unlike the Lambda
Tetractys with the number 6 at its centre, a group of six yods cannot be distinguished in this
addition. The only way the comparison may be extended is to consider the six hexagonal yods
lying on edges of tetractyses inside each sector as contributing to the number 36 and
the remaining nine yods as contributing to the number 54. In this way, the central number 6
denotes the number of corners of the
16
hexagon (Fig. 11), as one would expect, intuitively, as the corners are the
shapedefining yods, a quality consistent with the meaning of Malkuth symbolized by
the yod at the centre of a tetractys and therefore with the number 6 that occupies this
position in the Lambda Tetractys. The hexagon in the XY plane with A vertices at its corners
is therefore analogous to the complete set of 15 polygons. As found several
times earlier, the properties of the latter are exactly ten times the corresponding ones of
the former. The whole is contained in another whole. The simple, arithmetic reason
for this is that the central hexagon has six sectors, whilst the 15
polygons have 60 sectors, that is, ten times as many, so that any property of the sectors of
the hexagon become multiplied by 10 for the corresponding properties of the complete set of
polygons.
References
^{1} Phillips, Stephen M. Article 25: “The 33 Vertex Sheets of the
Disdyakis Triacontahedron Signify the 33 Tree Levels of Ten Overlapping Trees of Life,”
(WEB, PDF).
^{2} Phillips, Stephen M. Article 27: “How the Disdyakis
Triacontahedron Embodies the Structural Parameter 1680 of the E_{8}×E_{8}
Heterotic Superstring,” (WEB, PDF), pp. 4–5.
^{3} Ibid., p. 20, Table 4.
^{4} Phillips, Stephen M. Article 1: The Pythagorean Nature of
Superstring and Bosonic String Theories,” (WEB, PDF), p. 5.
^{5} Ref. 2, p. 19.
^{6} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its Meaning,
Generalisation and Connection to the Tree of Life,” (WEB, PDF), and Article12: “New Pythagorean Aspects of Music and
Their Connection to Superstrings,” (WEB, PDF).
^{7} Phillips, Stephen M. Article 26: “How the Seven Musical Scales
Relate to the Disdyakis Triacontahedron,” (WEB, PDF), p. 29.
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