
triangle

square

pentagon

octagon

decagon

dodecagon

Total

Number of vertices = 
3

4

5

8

10

12

42

Number of edges = 
6

8

10

16

20

24

84

Number of triangles = 
3

4

5

8

10

12

42

Total = 
12

16

20

32

40

48

168

The 168 elements in each set of polygons other than the
hexagon can be sorted into the following groups:

168 = 84 + 84.
84 vertices & triangles and 84 edges;

168 = 56 + 56 + 56.
Number of geometrical elements in square & decagon: 16 + 40 = 56;
Number of vertices & triangles in triangle, pentagon, octagon & dodecagon =
(3+5+8+12) + (3+5+8+12) = 28 + 28 = 56;
Number of edges in triangle, pentagon, octagon & dodecagon = 6 + 10 + 16 + 24 =
56;

168 = 7×24.
The 168 geometrical elements can be grouped into seven sets of 24
elements (colours denote sets):
The counterparts of these three types of factorisations in the I Ching
table, the Sri Yantra and the disdyakis triacontahedron will be discussed in later sections.
Their significance visàvis the Klein Quartic and the mapping of its 168
automorphisms on the 3torus will be discussed in Section 6.
The 24 geometrical elements surrounding the centre of the Type A hexagon
consist of three consecutive vertices, three edges, three internal edges, three triangles and
their diametrically opposite counterparts. In other words, they comprise four triplets of
elements and their diametric opposites. The seven separate polygons therefore consist of eight
triplets of elements and three sets of 56 elements. The latter can be regrouped as 56 triplets
of elements, one taken from the square & decagon, one (vertex or triangle) from the
triangle, pentagon, octagon & dodecagon and one (edge) from this set. This means that the
geometrical composition of the inner Tree of Life divides naturally into
(8+56=64) triplets of geometrical elements. The number 64 is
the gematria number value of Nogah, the Mundane Chakra of Netzach. It characterizes
the DNA molecule
in a way that will be discussed in Section 7.
6
Type B polygons
The seven separate polygons comprise 240 vertices & triangles
surrounding their seven centres. Including the root edge separating the two sets of seven
separate polygons, the ends of which may be formally counted as vertices, there are
(247+2+247=496) vertices & triangles (Fig. 5). 248 vertices & triangles are associated with
each set. They comprise an endpoint of the root edge, seven centres and 240 vertices &
triangles. This is the geometrical counterpart of the eight simple roots and the 240 roots
of the superstring gauge symmetry group E_{8}. The inner form of the Tree of Life,
when viewed as 14 separate regular polygons divided by their root edge, has
(248+248=496) vertices & triangles
that are the counterparts of the (248+248) roots of
E_{8}×E_{8}. That this is no coincidence is demonstrated by the fact that
the first six polygons have 72 vertices surrounding their six centres,
leaving 168 vertices & triangles. This compares with the six simple
roots and 72 roots of E_{6}, an exceptional subgroup of
E_{8}, and the 168 roots of E_{8} that are not roots of
E_{6}. As bits of information defining the geometry of the inner Tree of Life, the
248 vertices and triangles are the counterparts of the 248
roots of E_{8}. As was shown in Article 4,^{3} the Godnames assigned to the ten Sephiroth of the Tree of Life
prescribe both the inner Tree of Life and its largest subset of polygons — the first six
polygons and the first (6+6) polygons. For example, YAHWEH with gematria number value
26 and ELOHIM with number value 50 prescribe this subset
because the first six enfolded polygons have 26 corners and the first (6+6)
enfolded polygons have 50 corners, whilst ELOHA with number value
36 prescribes the six separate polygons because they have
36 corners.
The seven separate Type B polygons have 480 geometrical elements surrounding
their centres that are made up of 240 vertices & triangles and 240 edges. 240 geometrical
elements surround the centres of the hexagon, octagon & decagon; similarly, 240 geometrical
elements surround the centres of the triangle, square, pentagon & dodecagon. The
counterparts of this 240:240 division in the Platonic solids, thought by the ancient Greeks to
be the shapes of the particles of Fire, Air, Earth & Water, are the 240 hexagonal yods in
the 18 faces of the tetrahedron, octahedron & cube and the 240 hexagonal yods in the 20
faces of the icosahedron when they are constructed from tetractyses.^{4} Their 38 faces comprise 120 triangles with 248
vertices & edges,^{5} where
120 = 2^{2} + 4^{2} + 6^{2} + 8^{2}.
They are the counterparts of the 248 vertices &
triangles associated with the seven separate Type B polygons. This is how the first four
Platonic solids embody the dimension 248 of the symmetry group E_{8}.
Surrounding the centres of the seven polygons are 336 vertices & edges.
They comprise 168 vertices & edges in the triangle, square, pentagon &
dodecagon (let us call this set ‘S’) and 168 vertices & edges in the
hexagon, octagon & decagon (let us call this set ‘S'’). Notice that S' determines the
division of both 480 into two 240s and 336 into two 168s. According to
Table 1, S' determines also the division of the 384 edges & triangles in
the seven polygons into two equal groups — the 192 edges & triangles in S and the 192
edges & triangles in S'. In fact, this equal division is true for any
combination of geometrical elements because the hexagon, octagon & decagon have 24
corners — the same as the number of corners of the triangle, square, pentagon &
dodecagon — so that the formulae listed above for each combination imply the same number of
elements. The 168:168 division manifests in the ten closed
curves of the E_{8}×E_{8} heterotic superstring as the 168
circularly polarized oscillations in a halfrevolution of each curve. It appears in the Sri
Yantra as the 168 yods on the sides of the 21 triangles in
each half when they become tetractyses. It manifests in the disdyakis triacontahedron as the
168 hexagonal yods on its 84 edges on each side of its central plane. The
24:24 division of corners displayed by these two sets of polygons is characteristic of
holistic systems.^{6 }It reappears in the first (6+6) enfolded polygons as the 24
corners associated with each set of six polygons. It appears in a single polygon as the 24
edges and 24 vertices & triangles in the Type A dodecagon, which embodies holistic
properties analogous to all seven polygons. Its analogue in the seven separate Type B
polygons are the 24 sides of the triangle, square, pentagon & dodecagon and the 24 sides
of the hexagon, octagon & decagon (note: sides, not corners, because they
belong to the set of 192 edges & triangles in each set).
The edge/triangle composition of S and S' is shown below (‘+n’ denotes the n
sides of a polygon):
Each set has 96 internal edges and 72 triangles, i.e.,
168 edges & triangles. The triangle and pentagon
7
have 32 internal edges and 24 triangles; the square and dodecagon have
64 internal edges and 48 triangles, that is, twice as many.
Similarly, the octagon has 32 edges and 24 triangles and the hexagon and decagon have
64 internal edges and 48 triangles, that is, twice as many.
For S and S', the number 168 has the factorisation 7×24, where ‘7’ is the
number of internal edges & triangles per sector of a polygon and ‘24’ is the number of
sectors in S or S'. Let us label the 32 internal edges & 24 triangles in the triangle &
pentagon as A, the 32 internal edges & 24 triangles in half the square & dodecagon as
B, the 32 internal edges & 24 triangles in their other half as C, the 32 internal edges
& 24 triangles in the octagon as A', a similar set in half the hexagon & decagon as B'
and a similar set in their other half as C'. Then S = {A,B,C,24} and S' = {A',B',C',24}, where
A, B, C, etc has 56 edges & triangles and ‘24’ denotes the 24 sides of the polygons in S
& S'. Each number 168 therefore has the factorisation 56×3. It also has
the factorisation 84×2, for the dodecagon has 84 edges & triangles and the triangle, square
& pentagon have 84 edges & triangles, whilst the two halves of the polygons in S' each
has 84 edges & triangles.
The 24 sides of polygons belonging to S comprise the three sides of the
triangle, four sets of three consecutive sides of the dodecagon and three sets of three sides
of the square and pentagon, that is, eight triplets of edges. The 24 sides of polygons
belonging to S' comprise two sets of three consecutive sides of the hexagon and six sets of
three sides of the octagon and decagon, that is, eight triplets of edges. Therefore, S and S'
each consist of 64 triplets of edges & triangles, eight of them being the
sides of their polygons. Remarkably, the same 64×3 factorisation
is
displayed in the seven Type A polygons as in the seven Type B polygons. The
number 384 characterizing holistic systems manifests in both sets of
8
polygons because each is a whole, which means that they embody the
same characteristics of the Tree of Life by manifesting in their own way the same
parameters, such as 480, 384, 336 & 168. For example, the 14 Type A
polygons have 384 geometrical elements, whilst the seven Type B polygons have 384 edges &
triangles. As previous articles have proven, all holistic systems exhibit the same patterns
defined by the same set of parameters. This amazing property enables physical objects like the
superstring and the DNA molecule to be recognized as such systems. This was demonstrated in
earlier articles for the former. It will be proven for the latter in Section 8.
2. The 64×3 pattern in the I Ching
table
The ancient Chinese oracular system is known in the West as I Ching (易經), or
‘Book of Changes.’ Pronounced “Ee Jing”, I Ching is about 5,000 years old, making it one of the
oldest available sources of spiritual wisdom. It will not be necessary to explain how it is
used to provide advice about the future. According to Taoist philosophy, yang and yin,
respectively, the positive, or active, and negative, or reactive, phases of nature’s great
cycles, emerged from Tai Yi, the “Great Unity,” symbolized by a circle. As opposite states
represented by the positive (+) and negative (–) signs, they can be paired in four ways: + +, –
+, + – & – –. (Fig. 6). Adding a plus or minus sign at the end creates eight possible
combinations:
As eight stages of transformation of pure yang (Heaven trigram) into pure
yin (Earth trigram), they can be represented in Figure 7 by an eightcornered flow chart. The Tai Chi Tu, the ancient symbol
of the polar opposites of yin and yang, is its equivalent.
The eight trigrams can be formally assigned to Daath and the seven Sephiroth
of Construction of the Tree of Life (Fig. 8), with the unbroken line (symbol of yang) assigned to Chokmah at the
head of the Pillar of Mercy, which embodies the generative/active/male principle in Nature,
and with the broken line (symbol of yin) assigned to Binah at the head of the Pillar of
Judgement embodying the formative/reactive/female principle. The trigrams can be paired in
(8^{2}=64) ways. The resulting table of 64 pairs
of trigrams (hexagrams) shown in Figure 9 was given algebraic, arithmetic & geometric interpretations in
Articles 20 & 21.^{7} Its diagonal consists of the eight trigrams paired with
themselves, i.e., the 48 yin/yang lines of the eight hexagrams comprise 24
yin lines and 24 yang lines. The 28 hexagrams in the upper half of the table consist of 56
trigrams with (56×3=168) yin/yang lines. Similarly, the 28 hexagrams in the
lower half, which are the inversions of those in the upper half, comprise
168 yin/yang lines. Hence, the 384 yin/yang lines in the
64 hexagrams consist of
9
192 yin lines and 192 yang lines. Each diagonal half of the table comprises
96 yin lines and 96 yang lines. Of these, 12 yin lines and 12 yang lines make up the eight
trigrams in the diagonal that are associated with each half and 84 yin lines and 84 yang lines
make up the 56 offdiagonal trigrams. Each of the eight trigrams is repeated seven times in the
28 offdiagonal hexagrams, so that the number of yin/yang lines is 7×8×3 = 7×24, i.e., there
are seven copies of the eight trigrams composed of 24 yin/yang lines.
We see that the I Ching table displays the following divisions and
factorisations:
192 = 24 + 168 =
64×3;
24 = 8×3;
168 = 56×3 = 84×2 = 7×24.
Comparing them with the geometrical composition of the inner Tree of Life,
we see that the 384 yin/yang lines correspond to the 384 geometrical elements of the two sets
of seven separate Type A polygons. The 192 lines in each half of the table correspond to the
192 geometrical elements in each set. The 24 yin/yang lines in the eight diagonal trigrams of
each half correspond to the 24 geometrical elements in the hexagon in each set, the 84 vertices
& triangles and the 84 edges in the remaining polygons corresponding to the 84 yang and 84
yin lines in the 56 offdiagonal trigrams in each half of the table. These 56 triplets of
yin/yang lines are the counterparts of the three sets of 56 elements in the polygons other than
the hexagon. The seven copies of the eight trigrams correspond to the seven sets of 24 elements
making up these polygons.
When the polygons are Type B, the 384 yin/yang lines correspond to their 384
edges & triangles. The 24 yin/yang lines in a set of trigrams making up the upper or lower
half of the diagonal correspond to the 24 outer edges of polygons in, respectively, S or S'.
The 168 offdiagonal yin/yang lines in each half correspond to the
168 edges & triangles in S or S'. Their division into 56 trigrams is the
counterpart of the 56 groups of three edges & triangles into which S and S' can be divided.
Their division into 84 yin lines and 84 yang lines corresponds to the 84:84 division of edges
& triangles in S or S'. Their seven copies of 24 yin/yang lines in a set of eight trigrams
correspond to the seven edges & triangles per sector in S and S', each of whose three
polygons has 24 sectors.
The 384 yin/yang lines in the 64 hexagrams of the I Ching
table therefore symbolize the 384 geometrical elements of the two sets of Type A polygons
making up the inner Tree of Life and the 384 edges & triangles in each set of Type B
polygons. Each is an example of a holistic system characterized by the same set of
parameters and displaying the same patterns of division and factorisation. It is
highly implausible that this could be coincidental. This is confirmed by the emergence of
analogous patterns in the seven musical scales, as next explained.
3. The 64×3 pattern in the intervals
of the seven musical scales
The octave of the modern equaltempered musical scale is divided into 12
equal semitones. Its progenitor was the Pythagorean scale, an eightnote scale whose notes have
the tone ratios:
1

9/8

81/64

4/3

3/2

27/16

243/128

2

T

T

L

T

T

T

L


and are separated by five tone intervals T = 9/8 and two leimmas L =
256/143. Taking successive notes as the starting note, or tonic, generates seven possible
musical scales. They are the four authentic modes: Dorian (D scale), Phrygian (E scale), Lydian
(F scale) & Mixolydian (G scale); and the three plagal modes: Hypodorian (A scale),
Hypophrygian (B scale) & Hypolydian mode (C scale). The fourth plagal mode (Hypomixolydian)
is identical to the Mixolydian in terms of its sequence of intervals. Table 3 lists the tone ratios of the notes of the seven scales (red fractions
denote notes not belonging to the Pythagorean scale (C scale).
Table 3. The tone ratios of the seven musical scales.
Musical scale 
Tone ratio

B scale (Hypophrygian mode) 
1

256/243

32/27

4/3

1024/729

128/81

16/9

2

A scale (Hypodorian) 
1

9/8

32/27

4/3

3/2

128/81

16/9

2

G scale (Mixolydian) 
1

9/8

81/64

4/3

3/2

27/16

16/9

2

F scale (Lydian) 
1

9/8

81/64

729/512

3/2

27/16

243/128

2

E scale (Phrygian) 
1

256/243

32/27

4/3

3/2

128/81

16//9

2

D scale (Dorian mode) 
1

9/8

32/27

4/3

3/2

27/16

16//9

2

C scale (Hypolydian mode) 
1

9/8

81/64

4/3

3/2

27/16

243/128

2

The 42 notes between the tonic and octave comprise 26
Pythagorean notes and 16 nonPythagorean
10
notes. The Godname YAHWEH with number value 26 prescribes
the Pythagorean character of the seven scales. The inversion of an interval n is the interval
2/n between it and the octave. For example, the perfect fifth with tone ratio 3/2 is the
inversion of the perfect fourth with tone ratio 4/3 because 2/(4/3) = 3/2. Only the Dorian mode
is made up of notes and their inversions. Table 3 indicates that the seven musical scales are composed of 14 types of
notes:
1 
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 
(the vertical line separates notes from their inversions). They consist of
seven pairs of notes and their inversions, each pair of intervals spanning an octave. A note
and its inversion are equidistant from the ends of the scale. In order of increasing magnitude,
they are:
1. 

1

2

T^{5}L^{2} 
1×2 = 2

2. 
L 
256/243

243/128

T^{5}L 
256/243×243/128 = 2

3. 
T 
9/8

16/9

T^{4}L^{2} 
9/8×16/9 = 2

4. 
TL 
32/27

27/16

T^{4}L 
32/27×27/16 = 2

5. 
T^{2} 
81/64

128/81

T^{3}L^{2} 
81/64×128/81 = 2

6. 
T^{2}L 
4/3

3/2

T^{3}L 
4/3×3/2 = 2

7. 
T^{2}L^{2} 
1024/729

729/512

T^{3} 
1024/729×729/512 = 2

The 14 notes consist of two pairs of Pythagorean notes, four pairs of a
Pythagorean and a nonPythagorean note and one pair of nonPythagorean notes, i.e., eight
Pythagorean and six nonPythagorean notes. Twelve types of notes make up the 42 notes between
the tonic and octave of the seven scales. They comprise six Pythagorean notes and six
nonPythagorean notes. The 13 rising intervals above the tonic are mirrored by 13 corresponding
falling intervals with tone ratios that are the reciprocal of those of the former. The number
value 26 of YAHWEH is the number of different rising and falling tone
intervals in the 14 types of notes. Consistent with its meaning of prescribing the domain of a
holistic system, YAHWEH prescribes the intervallic range of the notes of the seven scales. The
same 13:13 division is found in the semiregular convex polyhedra, which consist of 13
Archimedean solids and their duals, the 13 Catalan solids.^{8} The duality of these polyhedra corresponds to the rising and
falling distinction between intervals separating the types of notes that belong to the seven
musical scales.
According to Table 3, the 42 notes between the tonic and octave divide equally into two
sets of 21 notes and their 21 inversions:
Notice that the number of times the inversion of a note occurs is the same
as the number of times the note does, although it does not always appear in the same scale —
only the Dorian mode (D scale) has this property because it is composed solely of notes and
their inversions. The number of notes for which their inversions do occur in the same
scale is 15, showing how the Godnames EHYEH with number value
21 and YAH with number value 15 prescribe the notes in the
seven scales. Notice also that the perfect fourths and perfect fifths occur most frequently.
The letter value 1 of A (aleph) in the Hebrew Godname AHIH denotes the single pair of notes
(1024/729, 729/512), the letter value 5 of H (heh) denotes the
two pairs (256/243, 243/128) & the
three pairs (81/64, 128/81), the letter
value 10 of I (yod) denotes the four pairs (32/27, 27/16) & the six pairs (4/3, 3/2) and the letter value 5 of the
second H denotes the five pairs (9/8, 16/9). No other assignment is possible.
There are 28 intervals between the eight notes of a musical scale, i.e., 27
intervals below the octave. The seven scales have (7×27=189) such rising intervals and 189
falling intervals, that is, 378 rising & falling intervals. The number 378 is the
26th triangular number after 1:
1 + 2 + 3 + … + 27 = 378.
It shows how YAHWEH with number value 26 prescribes the
number of rising and falling intervals between the notes of the seven scales, as well as the
number of their Pythagorean notes. The 189 intervals below the octave are therefore made up of
21 notes and 168 intervals that comprise their
21 inversions and 147 intervals between the seven notes above the tonic in the
seven scales. Figure 10
11
displays the composition of the 189 intervals.^{9} They comprise 123 Pythagorean intervals and 66 nonPythagorean
intervals. The 125 intervals up to 1024/729 comprise 83 Pythagorean and 42 nonPythagorean intervals. They
have 64 inversions made up of 40 Pythagorean and 24 nonPythagorean
intervals. There are 63 pairs of intervals and their inversions and 63 unpaired intervals,
that is, 63 triplets of intervals, each triplet comprising a pair of intervals and an
unpaired interval. Of the 63 pairs, 21 are pairs of notes and
their inversions and 42 are pairs of intervals other than notes or their inversions.
Altogether, the 189 intervals comprise 21 notes that are paired with their
inversions, 84 paired intervals, 21 inversions of the notes and 63 unpaired
intervals, i.e.,
84 intervals that are either inversions of notes or unpaired intervals. The
168 intervals other than the 21 notes up to
1024/729 therefore divide into two
sets of 84 intervals. Their composition is:

interval

number

interval

number

84: 





256/243

2 
243/128

2 

9/8

5 
16/9

5 

32/27

8 
27/16

8 

81/64

6 
128/81

6 

4/3

18 
3/2

18 

1024/729

3 
729/512

3 

Total =

42 
Total =

42 
84: 





256/243

10 
243/128

2 

9/8

25 
16/9

5 

32/27

12 
27/16

4 

81/64

9 
128/81

3 

4/3

6 
3/2

6 

1024/729

0 
729/512

2 

Total =

62 
Total =

22 
The 168 intervals trivially divide into seven groups of 24
intervals because there are 24 intervals in each scale other than the 21 notes
tabulated earlier that have 21 inversions.
As the scales contain the same tonic and octave, as well as the 63 triplets
of intervals, there is a final triplet consisting of two notes (the tonic and its inversion,
the octave) and the generic interval of the octave. Hence, all possible rising intervals
between the notes of the seven scales consist of 64 triplets. Formally
speaking, it is meaningful to distinguish the rising and falling intervals between the tonic
and itself even though they are both 1 in magnitude. The falling interval between the tonic and
octave is ½, which is also the generic falling octave interval. All possible falling intervals
consist of 64 triplets. Hence all possible rising and falling intervals
between the notes of the seven scales comprise 64 triplets and, so to speak,
their 64 ‘inverses.’
This double 64×3 pattern was found in the discussion of
both the inner Tree of Life and the I Ching table, The 192 yin/yang lines in the upper,
diagonal half of the latter signify the 192 rising intervals between the notes of the seven
scales, with the three lines of the upper Heaven trigram in the diagonal symbolizing
12
the rising interval of 1 between the tonic and itself, the octave and the
rising octave interval of 2. The 192 yin/yang lines in the lower, diagonal half of the table
denote the 192 falling intervals, with the three lines of the lower Heaven trigram in the
diagonal symbolizing the falling interval 1 between the tonic and itself, the falling octave
interval of ½ and the generic falling octave interval of ½. The 21 yin/yang
lines in the seven remaining diagonal trigrams in the upper half of the table signify the
21 notes, whilst the 21 lines in their counterparts in the
lower half denote the 21 inversions of these notes. The 168
offdiagonal yin/yang lines in each half denote the 168 rising intervals other
than these notes and their 168 falling counterparts. Their sevenfold
repetition corresponds to the seven types of musical scales.
4. The 64×3 pattern in the Sri
Yantra
Mandalas have been used by Hindus and Buddhists as objects of meditation and
as spiritual teaching tools. Yantras are the yogic equivalent of these plans or charts that
symbolize the cosmos — both physical and spiritual. The Sri Yantra is the most revered as the
mother of all Yantra. Mapping man’s inward journey from physical existence to spiritual
enlightenment, its system of interlocking triangles is surrounded by a lotus of eight petals, a lotus of sixteen petals, and an
earth square resembling a temple with four doors. The Sri Yantra (Fig. 11) is generated from nine primary triangles. Five downwardpointing
triangles symbolizing the feminine, creative energy of the Goddess Shakti intersect four
upwardpointing triangles symbolizing the masculine, creative energy, popularly conceived in
India as the God Shiva. Their overlapping generates 43 triangles. 42 triangles arranged in
four layers of eight, ten, ten and 14 triangles surround a downward pointing triangle whose
corners denote the triple Godhead, or Hindu Trimûrti, of Shiva, Brahma and Vishnu. At its
centre is a point, or ‘bindu,’ representing the Absolute, or transcendental unity. Including
its square enclosure and two sets of circumscribing lotus petals, the Sri Yantra has nine
levels.
There are two ways of conceiving the set of 43 triangles. The more common
one is to regard them as lying in the same plane, so that outwardpointing vertices of
triangles touch either the bases or vertices of triangles in the adjacent, surrounding set.
When their geometrical composition is analysed, this means that the vertices of some triangles
meet at a single point in this plane, whilst other vertices lie on edges of triangles. The
other way is to regard the Sri Yantra as a threedimensional stack of four layers of triangles,
so that only triangles in the same layer touch one another. Both cases were analyzed in Article
35,^{10} where the equivalence of the Sri Yantra and the inner form of
the Tree of Life was established. In this article, we shall focus on only one of the
features of the threedimensional Sri Yantra.
As representations of the Divine Whole, the Tree of Life is equivalent to
the tetractys (Fig. 12). This is a triangular array of ten points that had a special role
in the number mysticism of the ancient Pythagoreans. Because the yod (י), the tenth letter
of the Hebrew alphabet, looks somewhat like a dot, these points will be called yods. The
Supernal Triad of Kether, Chokmah & Binah correspond to the three yods at the corners of
the tetractys. The seven Sephiroth of Construction correspond to the seven ‘hexagonal yods,’
socalled because they are located at the corners and at the centre of a hexagon. Malkuth
corresponds to the central hexagonal yod and the two triads of Sephiroth of Construction:
ChesedGeburahTiphareth & NetzachHodYesod correspond to the hexagonal yods at the
corners of two equilateral triangles intersecting to form a Star of David (known in Hinduism
as the ‘Sign of Vishnu’). When it is constructed out of the template of the tetractys, an
object possessing sacred geometry is turned into a set of numbers, namely, the numbers of
various classes of yods that mark out its structure in space. As many previous articles have
demonstrated, these numbers characterize holistic systems like the superstring and the seven
musical scales.
When the 42 triangles surrounding the central triangle in the Sri Yantra are
turned into tetractyses, 378 yods are generated (Fig. 13). 42 hexagonal yods are at the centres of the 42 tetractyses, which
have 84 yods at corners (42 in each half of the Sri Yantra) and 252 hexagonal yods lie on
their 126 edges. Hence, there are 189 yods in each half that are made up of
21 hexagonal yods at centres of tetractyses and
(42+126=168) yods lining their edges.
13
14
It might seem natural to assign: 1. the 21 note intervals
and their falling counterparts in the seven musical scales to the hexagonal yods at the centres
of the set of 21 tetractyses in each half of the Sri Yantra, and 2. the
remaining 168 rising intervals and their falling counterparts to the
168 yods on their edges. However, the analogous
21:168 division of the I Ching table tells us that
21 yods in the Sri Yantra must consist of seven triplets corresponding to the
seven upper trigrams in the diagonal hexagrams other than the Heaven trigram. The 63 triplets
of yods in each half of the Sri Yantra consist of 42 triplets of hexagonal yods at corners of
two intersecting triangles within each triangle and 21 triplets of yods at the
corners and centres of triangles. The latter divide up uniquely into 14 triplets in
the first three layers of triangles and seven triplets in the fourth layer:
The seven upper trigrams correspond to the seven triplets of corners &
centres of the triangles in one half of the fourth layer, the seven lower trigrams
corresponding to the corners and centres of the seven triangles in its other half. Can their
counterpart 21 notes actually be grouped into seven sets of three, as
required? Yes, uniquely, according to Fig. 10:
The three sets of seven notes can be thought of as seven sets of three, one
being drawn from each set of seven (similarly for their inversions). The composition of the 189
rising intervals was found in the previous section to be:
21 notes + 42 intervals & their 42 inversions +
21 inversions of notes + 63 unpaired intervals.
The 42 intervals and their 42 inversions are assigned to the 14 pairs of
intersecting triplets of hexagonal yods in the 14 tetractyses of the first three layers in half
of the Sri Yantra, whilst the 21 inversions of notes and the 63 unpaired
intervals are assigned, respectively, to the 21 hexagonal yods in seven
triplets of the seven triangles in the fourth layer and to the 63 yods that are either at
corners & centres of the 14 tetractyses in the first three layers or in the seven other
triplets of hexagonal yods in the seven tetractyses of the fourth layer. The 189 falling
intervals are similarly assigned to their diametrical opposites in the other half of the Sri
Yantra. The (42+42=84) hexagonal yods forming triplets in the first three layers denote the 42
intervals and their inversions. The 168 yods other than the
21 corners & centres of the seven tetractyses in the fourth layer in each
half of the Sri Yantra symbolize the 168 rising intervals and the
168 falling intervals other than the primary 21 notes up to
that with tone ratio 1024/729. Just as
the 168 intervals are divided into two sets of 84, so the 168
yods split into two sets of 84, as shown in the diagram above.
Viewed as a tetractys, the central triangle has six hexagonal yods on its
edges arranged at the corners of a Star of David. The musical counterparts of the two
intersecting triangles are: 1. the tonic, octave & rising octave interval, and 2. the unit
interval, the octave ½ and the falling octave interval of ½. As the source of the holistic
system that is the seven musical scales, these intervals express in potentia the
musical domain of rising and falling intervals represented by the 378 yods of the 42
tetractyses that surround the central one. Hence, the 378 yods in the 42 tetractyses, together
with the two triplets of hexagonal yods in the central triangle, are composed of
64 triplets and their 64 inverses, reflected across its
centre. This double 64×3 pattern is exactly the same as that shown by the
64 trigrams belonging to each diagonal half
15
of the I Ching table (see Fig. 13). The correspondence between the Sri Yantra and I Ching table is
shown below:

I Ching table

Sri Yantra

1. 
two Heaven trigrams of hexagram in top lefthand corner 
→ two triplets of hexagonal yods in central triangle; 
2. 
21 yin/yang lines in 7 other diagonal trigrams in each half 
→ 21 corners & centres of triangles in the 7 triangles; 
3. 
168 yin/yang lines in 28 offdiagonal hexagrams in each half 
→ 168 yods other than 21 corners &
centres; 
4. 
64 trigrams in each half 
→ 64 triplets of yods in each half; 
5. 
192 yin/yang lines in each half 
→ 192 yods in each half; 
6. 
384 yin/yang lines in 64 hexagrams 
→ 384 yods in 64 pairs of triplets. 
The counterparts in the inner Tree of Life of the 192 yods making up each
half of the Sri Yantra are the 192 vertices, edges & triangles surrounding the centres of
the two sets of seven Type A polygons discussed in Section 1, which we saw can be grouped into
64 sets of three (p. 6). The 24 yin/yang lines in the eight upper or lower
trigrams in the diagonal of the I Ching table correspond to the 24 geometrical elements
surrounding the centre of the hexagon in each set. The 168 offdiagonal
yin/yang lines in each half of the table correspond to the 168
geometrical elements


surrounding the centres of the six remaining regular polygons in each set.
They are present in 56 trigrams, which correspond to the 56 triplets of elements in these
polygons. The 48 geometrical elements in the pair of hexagons are the
counterparts of the 48 yin/yang lines in the eight diagonal hexagrams and of
the 48 yods either at centres or corners of tetractyses in the fourth layer of
the Sri Yantra or on edges of the central tetractys. As we shall further illustrate, this
48:336 division is characteristic of the 384 degrees of freedom displayed by
holistic systems. So also is their 2×64×3 factorisation. The factor of 2
arises because they consist of two halves that are mirror images or inverses of each other —
the two sets of seven polygons, the two halves of the I Ching table and the Sri Yantra and the
rising and falling aspects of every interval.
The factor of 3 stems from the way in which the 192 geometrical elements
surrounding the centres of the seven polygons can be naturally grouped into three sets, from
the trigrams, from the grouping of the musical intervals into three sets (paired
notes/intervals and unpaired intervals) and from the grouping into triplets of, firstly, the
two corners per tetractys and its centre and, secondly, the Star of David formed by hexagonal
yods on edges of tetractyses.
16
5. The 64×3 pattern in the disdyakis
triacontahedron
There are 13 convex, semiregular polyhedra. Called the ‘Archimedean
solids,’ their counterparts generated by replacing vertices by faces and vice versa are called
the ‘Catalan solids.’ The Catalan solid with the most number of vertices is the disdyakis
triacontahedron (Fig. 14). It has 62 vertices, 180 edges & 120 triangular faces. 12
vertices (called ‘B’) are the vertices of a icosahedron and 20 vertices (called ‘C’) are
vertices of a dodecahedron. The 32 B & C vertices belong to the rhombic triacontahedron
(Fig. 15), which has 30 Golden Rhombic faces, the ratios of whose diagonals
(edges of the icosahedron & dodecahedron) is the Golden Ratio φ, considered for hundred
of years by artists and architects as a proportion most pleasing to the human eye. The
apices of the 30 pyramids having these faces as their bases are called ‘A’ vertices.
It was proved in previous articles^{11,12,13} that the disdyakis triacontahedron is the polyhedral version of the
Tree of Life. More generally, it is the polyhedral manifestation of the archetypal pattern
intrinsic to all holistic systems. The surface of this Catalan solid has 362 geometric
elements. This attribute is encoded in the pair of Type B dodecagons — the last of the
regular polygons making up the inner Tree of Life — because the division of their 24 sectors
into three tetractyses produces 362 yods (Fig. 16). Each yod denotes a geometric element. The two centres denote any
pair of diametrically opposite A vertices and the 360 yods surrounding them denote the 60
vertices, 180 edges and 120 triangles between these opposite poles.
The 181 yods in each dodecagon denote the 181 geometrical elements in each
half of the disdyakis triacontahedron. The archetypal ‘form’ quality of the Godname ELOHIM
assigned to Binah in the Tree of Life is illustrated by the fact that its number value
50 is the number of corners of the 72 tetractyses in the pair
of dodecagons. The same applies to the Godname ELOHA assigned to Geburah below Binah because
its number value is 36, which is the number of tetractyses in a dodecagon.
The Decad (10) determines geometrical properties of the disdyakis
triacontahedron because it has: 1. 120 triangles, where 120 is the sum of the first 10 odd
integers after 1 that can be assigned to the corners of the decagon:
2. 60 vertices surrounding an axis of the polyhedron passing through two
diametrically opposite vertices,
17
where 60 is the number of hexagonal yods in a Type A dodecagon, the
tenth regular polygon, and 3. 180 edges, where 180 is the number of yods surrounding
the centre of a Type B dodecagon (Fig. 16).
Suppose that we choose as the Zaxis one of the 15 possible lines joining
two opposite A vertices. Figure 17a shows the projection onto the XY plane of the 62
vertices. They are located on seven sheets. Three sheets are above or below the central one
in the XY plane. The 12 vertices and 12 edges lying in the latter form a 12sided polygon
(Fig. 17b). They comprise four A vertices, four B vertices, four C vertices,
four AB edges, four BC edges & four AC edges. In other words, there are three types of
vertices, each repeated four times, and three types of edges, each repeated four times.
These are indicated in Fig. 17b by the three black lines of different thicknesses, their
diametrically opposite counterparts being yellow lines. There are 84 polyhedral edges either
above or below the central polygon with 12 edges, which are also polyhedral edges. The 12
sectors of the polygon have 12 internal edges (coloured purple in three thicknesses, the
lines opposite them being light purple, with pairs of diametrically opposite sectors
coloured in light and dark shades of red, blue & violet). If O is the centre of the
polygon, there are four OA edges, four OB edges and four OC edges. Hence, there are 192
edges, 24 belonging to the central polygon and grouped into eight triplets of edges and
168 belonging solely to the polyhedron. The 180 edges comprise 60 AB edges,
60 BC edges and 60 AC edges. This means that the 168 edges comprise
(60–4=56) edges of these three types. They are divided into 28 AB, 28 BC & 28 AC edges
above the XY plane and their counterparts below it. The 192 edges are grouped into
64 edges, each of three types, eight belonging to the central polygon and
56 belonging solely to the polyhedron.
Each edge can be regarded as a doubleheaded arrow, i.e., as arrows that
point in opposite directions. If we write the arrow pointing from A to B as E_{A}, the
arrow pointing from B to C as E_{B} and the arrow pointing from C to A as
E_{C}, the arrow pointing from B to A is –E_{A}, etc. Similarly, the arrow AO
pointing from A to O is e_{A} and the arrow OA pointing from O to A is –e_{A}
(similarly for vertices B & C). Using primes to indicate the repeated set of vertices &
edges, there are four triplets of arrows:
AB = E_{A} 
AO = e_{A} 
A'B' = E_{A'} 
A'O = e_{A'} 
BC = E_{B} 
BO = e_{B} 
B'C' = E_{B'} 
B'O = e_{B'} 
CA = E_{C} 
CO = e_{C} 
C'A' = E_{C'} 
C'O = e_{C'} 
and four triplets of their counterparts pointing in the opposite direction.
There are 384 arrows comprising 168 arrows either above or below the central
polygon, each made up of two sets of 28 groups of three arrows, and 48 arrows
belonging to the latter, which are further divided into two sets of 24, each set consisting of
eight triplets. 192 arrows point along the edges of each half of the polyhedron and sectors of
its central polygon.
This pattern of two sets of 64 triplets of arrows is
identical to the 64 pairs of trigrams in the I Ching table. The two possible
opposite directions of each arrow are the counterparts of the yin and yang polarities of each
line in a trigram. The pairing of similar trigrams in the diagonal of the table represents the
fact that every set of three arrows belonging to one half of the central polygon has its
inverted counterpart in its other half. In the case of the 384 geometrical elements of the two
sets of seven polygons constituting the inner Tree of Life, we found that the 24 geometrical
elements in one hexagon have their mirror image counterparts in the hexagon belonging to the
other set. The inversion of pairs of trigrams in opposite diagonal halves of the table
corresponds to the counterparts of their arrows being spatially inverted in the other half of
the polyhedron. What makes the disdyakis triacontahedron holistic is that it is the polyhedral
representation of all 64 possible pairs of eight trigrams when yin and yang
lines are identified as the orientated edges of its faces and sectors of its central
polygon.
The 168 edges of the disdyakis triacontahedron above and
below the central polygon divide into 56 groups of three because all edges are of three types.
They divide into 84 edges above the polygon and 84 edges below it, that is, into two sets of
seven groups of 12 edges, a group consisting of four triplets of AB, BC & AC. The
168 arrows either above or below the XY plane consist of seven groups of 24
arrows, a group comprising eight triplets of arrows. This is the same as the seven sets of
trigrams either above or below the diagonal of the I Ching table. The disdyakis triacontahedron
exhibits the same 56×3, 84×2 & 7×24 as found earlier for the inner Tree of Life, the seven
musical scales and the I Ching table.
6. The 64×3 pattern in the {3,7}
tessellation of the 3torus
The Klein quartic
Felix Klein (1849–1925) is wellknown to mathematicians as one of the nineteenth century’s
great geometers. The onesided, closed surface called the “Klein bottle” is named after him. In
1878, he discovered that the curve:
X^{3}Y + Y^{3}Z + Z^{3}X = 0,
18
where X, Y & Z are complex numbers, has the 336fold symmetry of the
group SL(2,7), the special, linear group of 2×2 real matrices with determinant 1 over a field
of order 7. This curve is known as the “Klein quartic.” The “Klein surface” is the Riemann
surface of the Klein quartic. Klein showed that it is mapped onto itself (hence
“automorphisms”) by 168 analytic transformations. Hurwitz showed that a
Riemann surface of genus g has at most 84(g–1) automorphisms and the same number of
antiautomorphisms. A Riemann surface with the maximum number of automorphisms is called a
“Hurwitz curve” of genus g (g≥3). A Hurwitz curve of genus 3 has 84(3–1) = 84×2 =
168 automorphisms. The Klein curve is the Hurwitz curve with the smallest
genus g = 3.
A regular tessellation, or tiling, is a covering of the plane by regular
polygons so that the same number of polygons meets at each vertex. The Schläfli symbol {n,k} is
the tessellation by polygons with n vertices, k polygons meeting at each vertex. If 1/n + 1/k
< 0, {n,k} is a tessellation of the hyperbolic plane with negative curvature. If 1/n + 1/k =
0, {n,k} is a tessellation of the Euclidean plane with zero curvature. If 1/n + 1/k > 0,
{n,k} is a tessellation of the elliptic plane with positive curvature.
The regular tessellations of the Euclidean plane are wellknown. They are:
{3,6} in which equilateral triangles meet six at each vertex; {4,4} in which squares meet four
at each vertex; and {6,3} in which hexagons meet three at each vertex. The mathematician
Poincaré described ways in which points in the hyperbolic plane can be represented conformally
in the Euclidean plane. One of them is to represent the hyperbolic plane by points within the
socalled ‘Poincaré disk’. {n,k} is the Poincaré dual of {k,n} in which vertices and polygons
are interchanged.
The {3,7} tessellation of the 3torus
SL(2,7) is the doublecover of the Klein quartic symmetry group PSL(2,7). Its order 336 is the
number of automorphisms and antiautomorphisms of the Klein quartic. The 168
elements of PSL(2,7) are represented either by the {7,3} tiling of the Poincaré disk with 24
heptagons having 168 vertices or by its {3,7} tiling with 56 triangles having
168 vertices (Fig. 18). The Riemann surface of the Klein quartic is a
168sheeted covering of the sphere, branched over three points of the
sphere:

Above one of these points, the 168 sheets join in
groups of seven to give 24 points of the surface. These are the points of inflection of
the curve described by the Klein quartic. They are also what are called the “Weirstrass
points.” The 24 orbits of subgroups of order 7 correspond to these points.

Above another branch point, there are 84 points of the surface,
where the sheets join in twos. These are the sextactic points through which pass a
conic section that has sixfold contact with the curve. The 84 orbits of subgroups of
order 2 correspond to them.

Above the third branch point, the sheets join in threes to give 56
points of the surface. These 56 points are the points of contact of the Klein quartic
with the 28 bitangents, or lines that are tangent to the curve at two points. The 56
orbits of subgroups of order 3 correspond to them.
Having genus 3, the hyperbolic, Riemann surface of the socalled ‘Klein
Configuration’ can be embedded
19
on the surface of a 3torus (Fig. 19a), which has the same genus because it is topologically equivalent
to a sphere with three handles attached to it. The 3torus is also topologically equivalent
to six square antiprisms with their 12 square faces stuck to the 12 square faces of four
triangular prisms in a tetrahedral formation, each antiprism being given a twist^{14} (Fig. 19b). This leaves 56 triangular faces, which, as separate triangles,
have 168 vertices (eight faces from the triangular prisms with 24 vertices,
48 faces from the square antiprisms with 144 vertices). Each triangular
prism has nine edges and each square antiprism has 16 edges, of which eight edges coincide
with edges of the former when they are stuck together, resulting in (4×9 + 6×8 = 84) edges
of triangles. Figure 19b shows the {3,7} tessellation onto the 3torus of the 56 orbits of
order 3 corresponding to the 56 bitangent points of the Klein quartic. The
168 antiautomorphisms of the Klein quartic have a {3,7} tiling on a
3torus that is turned insideout. As discussed in Article 43,^{15} the 24 vertices and 84 edges of the 56 triangles in the
tetrahedraldeformed 3torus and the 24 vertices and 84 edges in its turned insideout
version are the respective counterparts of the 24 vertices and 84 edges above the central
plane of the disdyakis triacontahedron that surround an axis joining two opposite A vertices
and the mirror images of these vertices and edges below this plane. Indeed, a onetoone
correspondence was shown to exist between the geometries of the polyhedron and this pair of
3tori. As the former is the polyhedral form of the inner Tree of Life, the {3,7}
tessellation of the 168 automorphisms of the Klein quartic onto the 3torus
constitutes a holistic pattern that is isomorphic to all sacred geometries. The aspect of
this equivalence that is relevant to this article will be discussed next.
{3,7} tessellation of 3torus with tetractyses
Consider the triangular faces of the triangular prisms and the square antiprisms making up the
3torus, firstly, turned into tetractyses and, secondly, divided into their three sectors and
the latter then turned into tetractyses (i.e., Type A triangles with tetractyses as their
sectors). Their square faces must be ignored because they are not part of the surface
of the 3torus. We shall calculate the numbers of yods in each case, noting that the 12 squares
of the prisms become stuck to the 12 squares of the antiprisms, so that they play no part in
the properties of the surface of the 3torus.
The geometrical elements making up the two types of prisms are listed
below:
A triangular prism has 6 vertices, 9 edges, 2 triangles & 3 squares.
A square antiprism has 8 vertices, 16 edges (8 unshared), 8 triangles & 2 squares.
Table 4. Geometrical & yod composition of the 56 triangles in the {3,7}
tessellation of the 3torus.

Tetractys

Type A triangle


triangular
prism

square
antiprism

Total

triangular
prism

square
antiprism

Total

Number of vertices 
4×6=24

—

24

24+8×1=32

48×1=48

80

Number of edges 
4×9=36

6×8=48

84

36+8×3=60

48+48×3=192

252

Number of triangles 
4×2=8

6×8=48

56

8×3=24

48×3=144

168

Number of geometrical
elements 
68

96

164

116

384

500

Number of hexagonal yods
on edges of tetractyses 
2×36=72

2×48=96

168

2×60=120

2×192=384

504

Number of hexagonal yods 
8+72=80

48+96=144

224

24+120=144

144+384=528

672

Number of yods on edges
of tetractyses 
24+72=96

96

192

32+120=152

96+384=432

584

Number of yods 
8+96=104

48+96=144

248

24+152=176

144+432=576

752

20
The four triangular prisms have 24 vertices and the six square antiprisms
have 48 vertices.
Let us now consider them joined together at their square faces to form the
3torus. Let us also regard them as tetractyses. Table 4 displays their geometrical and yod composition. With single
tetractyses as their faces, the four triangular prisms and the six square antiprisms are
composed of 56 tetractyses with 248 yods (see pink cell). This is very
remarkable, for 248 is the number of E_{8} gauge bosons that
transmit the unified force between superstrings of ordinary matter! Furthermore, the
248 yods comprise eight yods at centres of the faces of the four triangular
prisms, leaving 240 yods made up of 24 yods at vertices, 48 hexagonal yods
at centres of tetractyses (i.e., 72 yods that are either vertices or
centres) and 168 hexagonal yods on their 84 edges (see green cell). The
division:
248 = 8 + 240
corresponds to the eight simple roots of E_{8} and its 240 roots.
The division:
240 = 72 + 168
corresponds to the 72 roots of E_{6}, an
exceptional subgroup of E_{8}, and the 168 remaining roots of
E_{8}. The root structure of the superstring gauge symmetry group E_{8} is
embodied in the yod composition of the tetractyses tessellating the 3torus! As was so often
found in previous articles, the number 168 defines the shape of the
holistic system. Here — in the case of faces as single tetractyses — it is the number of
hexagonal yods on the boundary of the tetractyses that compose it. In the case of Type
A faces, it is the number of triangles. The number 72 merely defines the
corners and centres of the 56 tetractyses.
According to Table 4, 192 yods lie on the 84 edges of the 56 tetractyses. 24 yods are
vertices of the four triangular prisms. The same applies for the 3torus when it is turned
insideout. This 24:168 division corresponds in the inner form of the Tree
of Life to the 24 geometrical elements that surround the centre of the hexagon and to the
168 geometrical elements surrounding the centres of the six other regular
polygons (Fig. 20). It corresponds in the I Ching table to the 24 yin/yang lines of
the eight upper trigrams in its diagonal and to the 168 yin/yang lines in
the 56 trigrams above the diagonal. Its counterparts in the Sri Yantra are the 24 yods in
each half that are either at centres of tetractyses or hexagonal yods in the central
tetractys and the 168 yods lying on edges of tetractyses. The turnedinside
out 3torus, whose
21
{3,7} tessellation into 56 triangles map the 168
antiautomorphisms of the Klein quartic, is the counterpart of the mirrorimage set of seven
polygons in the inner Tree of Life and the lower diagonal half of the I Ching table. This
implies that the 384 symmetries of the Klein quartic constitute a whole. Furthermore,
(248+248=496) yods are needed to turn the
(56+56=112) triangles of the two 3tori into tetractyses. This demonstrates a
connection with the (248+248=496) gauge
bosons transmitting the forces between E_{8}×E_{8} heterotic superstrings. Each
yod symbolizes one of these messenger particles. Clearly, the doublecover nature of the group
SL(2,7) of automorphisms and antiautomorphisms of the Klein quartic is related to the direct
product nature of the heterotic superstring gauge symmetry group E_{8}×E_{8}.
The question is: why?
The eight triangular faces of the four triangular prisms consist of eight
sets of three vertices. They correspond to the eight upper trigrams in the diagonal of the I
Ching table (like the latter, they are divided into two sets of four faces, each with three
vertices). The 168 hexagonal yods on the edges of the ten prisms and
antiprisms and belonging to 56 tetractyses are arranged as 56 sets of two interlaced triangular
arrays of three yods shared between adjoining tetractyses, that is, as 28 independent sets of
two arrays. They correspond to the 28 hexagrams with 168 yang/yin lines above
the diagonal. Similarly, the 168 hexagonal yods lining edges of the 56
tetractyses in the turned insideout 3torus correspond to the 28 hexagrams with
168 yin/yang lines below the diagonal. The 84×2 factorisation of the
168 hexagonal yods arises from the fact that a pair of hexagonal yods lies on
each of the 84 edges. They belong to oppositely interlaced triangles. This is the counterpart
of the yin/yang duality of lines and broken lines in the trigrams. In the case of the 24
vertices of the triangular prisms, vertices belonging to their opposite triangular
faces correspond to yin and yang lines. The 192 yods on edges of tetractyses in each 3torus
consist of (8+56=64) triplets. The eight triplets of vertices correspond to
the 24 geometrical elements surrounding the centre of the hexagon in the inner Tree of Life.
The counterpart here of the two sets of four triplets of vertices is the two sets of three
successive sectors, each with four geometrical elements, which regroup into two sets of four
triplets of elements.
The nine edges of the triangular prism consist of the three edges of each
opposite triangular face and three edges that join vertices of these faces. Hence, they consist
of three types of edges, each occurring three times. The four triangular prisms have three
types of edges, each occurring 12 times. Two hexagonal yods lie on each edge. Therefore, there
are 24 hexagonal yods on each of three types of edges. The square faces of the square antiprism
are rotated relative to each other by 45°. Each vertex of one square face is joined to two
vertices of the other square (Fig. 21). This means that the eight edges of the square antiprism that are
not edges of squares divide into four primary sets of two edges. The six square antiprisms
have 48 edges that group into four primary sets of 12 edges. As each edge
has two hexagonal yods, the 96 hexagonal yods group into four sets of 24 hexagonal yods.
Hence, the 168 hexagonal yods on the 84 edges of the prisms and antiprisms
consist of 24 hexagonal yods on each of the three types of edges of the triangular prisms
and on each of the four types of edges of the square antiprisms. In other words, there are
seven classes of 24 hexagonal yods. This 7×24 division has its counterpart in the I Ching
table as the seven copies of each trigram either above or below the diagonal, i.e., as the
seven copies of the 24 yin/yang lines of the eight types of trigrams. Their counterparts in
the
seven musical scales are the 24 inversions and intervals between notes in
each scale. Their counterparts in the inner Tree of Life are the seven groups of 24 geometrical
elements surrounding the centres of the polygons other than the hexagon (p. 6).
The 48 triangular faces of the six square antiprisms
display the same 24:24 division as the 48 yin/yang lines of the eight
hexagrams in the diagonal of the I Ching table. This is because the faces are of two
22
types — a downwardpointing face and an upwardpointing one. The 24 faces of
the former type share edges with upper square faces like the light grey one shown in
Fig. 21. The 24 faces of the latter type share edges with lower square
faces. Upward and downwardpointing faces (depicted in Fig. 21 by two different shades of grey) embody the polarities of yin and
yang. The six sets of (4+4) faces are the counterparts of the six yin/yang lines in each of
the two sets of four diagonal hexagrams. The reason for this correspondence is that
Table 4 shows that, when these faces are Type A triangles, there are 384
hexagonal yods lying on edges of tetractyses, 192 being on edges of downwardpointing
tetractyses and 192 being on edges of upwardpointing tetractyses. They are the counterparts
of the 192 yin lines and 192 yang lines in the table. This 192:192 division, which
characterises holistic systems and which is present in the yods lining the edges of the
tetractyses making up the 3torus and its turned insideout version, is also present in
each 3torus when the triangles are Type A. Indeed, this division manifests in the
geometrical composition of the square antiprisms as well, for they have 384
geometrical elements, of which 48 are vertices (24 in upwardpointing
triangles and 24 in downwardpointing ones) and 336 are edges & triangles
(168 in upwardpointing triangles and 168 in
downwardpointing ones). In other words, they comprise 192 ‘yin’ geometrical elements and
192 ‘yang’ geometrical elements. Alternatively, according to Table 4, whereas there are 384 yods lining the edges of the triangles in both
3tori when they are tetractyses, there are 384 hexagonal yods lining the edges of
tetractyses in the square antiprisms forming the ‘arms’ of either 3torus. As we found in
Section 1 for the Type A and Type B polygons of the inner Tree of Life, holistic parameters
like 384 characterize not only the whole system at one level of differentiation but each
half of it at another level. The whole system is both sets of seven polygons and both
3tori. Holistic systems are binary because they have a cyclic quality that is divided into
an outward, expansive (yang) phase and a returning, contractive (yin) phase that in some way
is the mirror image, or reverse, of the former. One example is the 192 rising intervals and
the 192 falling intervals between the notes of the seven musical scales; the 192
doublyorientated edges of the disdyakis triacontahedron and the sectors of its central,
12sided polygon are another example, one half being the mirror image of the other half in a
spatial balance of yang and yin.
We have established that the tetractys transformation of the 56 hyperbolic
triangles on the 3torus reproduces the same 84×2, 56×3 & 24×7 divisions of yods as found
in the 168 automorphisms of the Klein quartic, which form 84 subgroups of
order 2 (sextactic points of its Riemann surface), 56 subgroups of order 3 (points of contact
of bitangents) and 24 subgroups of order 7 (points of inflection). The symmetries of the
equation contain the characteristic patterns of a holistic system as represented by sacred
geometries and the seven musical scales.
7. The 64 codons
DNA is made of chemical building blocks called nucleotides. These building
blocks are made of three parts: a phosphate group, a sugar group (2deoxyribose, which is a
pentose (fivecarbon) sugar) and one of four types of nitrogen bases. To form a strand of DNA,
nucleotides are linked into chains, with the phosphate and sugar groups alternating. The four
types of nitrogen bases found in nucleotides are: adenine (A), thymine (T), guanine (G) and
cytosine (C) (Fig. 22). The order, or sequence, of these bases determines what biological
instructions are contained in a strand of DNA. For example, the sequence ATCGTT might
instruct for blue eyes, while ATCGCT might instruct for brown. Each DNA sequence that
23
contains instructions to make a protein is known as a gene. The size of a
gene may vary greatly, ranging from about 1,000 bases to 1 million bases in humans. The
complete DNA instruction book, or genome, for a human contains about 3 billion bases and about
20,000 genes on 23 pairs of chromosomes. The largest human chromosome contains approximately
220 million base pairs.
To understand DNA’s double helix from a chemical standpoint, picture the
sides of the ladder as strands of alternating sugar and phosphate groups — strands that run in
opposite directions. Each “rung” of the ladder is made up of two nitrogen bases, paired
together by hydrogen bonds (Fig. 23). Because of the highly specific nature of this type of chemical
pairing, base A always pairs with base T, and likewise C with G. These rules of
complementary base pairing are strictly followed because the four different sorts of
nucleotides are different shapes, so in order for the strands to close up properly, an A
nucleotide must go opposite a T nucleotide, and a G opposite a C.
DNA's unique structure enables the molecule to copy itself during cell
division. When a cell prepares to divide, the DNA helix splits down the middle and becomes two
single strands. These single strands serve as templates for building two new, doublestranded
DNA molecules — each a replica of the original DNA molecule. In this process, an A base is
added wherever there is a T, a C where there is a G, and so on until all of the bases once
again have partners.
In addition, when proteins are being made, the double helix unwinds to allow
a single strand of DNA to serve as a template. This template strand is then transcribed into
messenger RNA, which is a molecule that conveys vital instructions to the cell's proteinmaking
machinery.
The genetic code consists of (4×4×4=64) triplets of
nucleotides called codons.With three exceptions, each codon encodes for one of the 20 standard
amino acids used in the synthesis of proteins (humans can produce 10 of the 20 amino
acids.^{16} The others must be supplied in the food). That produces some
redundancy in the code: most of the amino acids are encoded by more than one codon.
The genetic code can be expressed as either RNA codons or DNA codons. RNA
codons occur in messenger RNA (mRNA) and are the codons that are actually “read” during the
synthesis of polypeptides (the process called ‘translation’). But each mRNA molecule acquires
its sequence of nucleotides by ‘transcription’ from the corresponding gene. Figure 24 shows a typical short section of an mRNA.
RNA is a singlestranded molecule with a much shorter chain of nucleotides
than DNA. Whereas DNA contains deoxyribose, RNA contains ribose, whose OH group makes it less
stable than DNA. The complementary base to adenine is not thymine but uracil (U), an
unmethylated form of thymine (compare their molecules in Fig. 22).
There are three ‘stop’ or ‘nonsense’ codons signifying the end of the coding
region; these are the UAA, UAG, and UGA codons. Sixtyone codons code solely for amino acids.
One codon (AUG) in mRNA serves two related functions:

it signals the start of translation. Hence, it is called the start
codon;

it codes for the incorporation of the amino acid methionine into the
growing polypeptide chain.
An anticodon is a sequence of three adjacent nucleotides in transfer RNA
(tRNA) that correspond to the
24
three bases of the codon on the mRNA. An anticodon is complementary to the
codon in messenger RNA that binds to it and designates a specific amino acid during protein
synthesis at a ribosome. Each tRNA contains a specific anticodon triplet sequence that can
basepair to one or more codons for an amino acid.
In transcription, RNA is synthesized from the DNA template by an enzyme —
RNA polymerase. The enzyme binds to a promoter sequence in the DNA, causing the double helix to
unwind. The enzyme progresses along the template strand, synthesizing a complementary RNA
molecule. In socalled ‘translation,’ messenger RNA (mRNA) carries information about a protein
sequence to the ribosomes, which synthesize protein in the cell. Ribosomes are complexes of RNA
and protein. Transfer RNA is a small RNA chain of about 80 nucleotides that
transfers a specific amino acid to
a growing polypeptide chain at the ribosomal site of protein synthesis
during translation. It has sites for amino acid attachment and an anticodon region that can
base pair to the corresponding threebase codon region on mRNA through hydrogen binding. tRNA
reads the sequence of the messenger RNA and is attached to the amino acids. Each type of tRNA
can be attached to only one type of amino acid, but because the genetic code contains multiple
codons that specify the same amino acid, tRNA molecules bearing different anticodons may also
carry the same amino acid.
Table 5 lists the 64 codons in mRNA and the amino acids for
which they code.
Table 5. The 64 codons (MRNA base triplets) and the amino
acid for which each codon codes.
Amino acid 



U

C

A

G

1st base 
U

UUU Phenylalamine
UUC Phenylalamine
UUA Leucine
UUG Leucine 
UCU Serine
UCC Serine
UCA Serine
UCG Serine 
UAU Tyrosine
UAC Tyrosine
UAA Ochre (Stop)
UAG Amber (Stop) 
UGU Cysteine
UGC Cysteine
UGA Opal (Stop)
UGG Tryptophan 
C

CUU Leucine
CUC Leucine
CUA Leucine
CUG Leucine 
CCU Proline
CCC Proline
CCA Proline
CCG Proline 
CAU Histidine
CAC Histidine
CAA Glutamine
CAG Glutamine 
CGU Arginine
CGC Arginine
CGA Arginine
CGG Arginine 
A

AUU Isoleucine
AUC Isoleucine
AUA Isoleucine
AUG Methionine 
ACU Threonine
ACC Threonine
ACA Threonine
ACG Threonine 
AAU Asparagine
AAC Asparagine
AAA Lysine
AAG Lysine 
AGU Serine
AGC Serine
AGA Arginine
AGG Arginine 
G

GUU Valine
GUC Valine
GUA Valine
GUG Valine 
GCU Alanine
GCC Alanine
GCA Alanine
GCG Alanine 
GAU Aspartic acid
GAC Aspartic acid
GAA Glutamic acid
GAG Glutamic acid 
GGU Glycine
GGC Glycine
GGA Glycine
GGG Glycine 
Notice that each amino acid is encoded by several codons. In the case of
leucine, there are many as six codons. Tryptophan is encoded by only one codon, as is
methionine, whose codon is the start codon.
Table 6 lists the 64 base triplets in DNA, the
64 mRNA codons and the 64 tRNA anticodons.
25
26
Table 6. Table of DNA base triplets, mRNA codons & tRNA anticodons.
(AT or U, CG)
Amino
acid

DNA Base
Triplets

mRNA Base
Codons

tRNA Base
Anticodons

alanine 
CGA, CGG, CGT, CGC

GCU, GCC, GCA, GCG

CGA, CGG, CGU, CGC

arginine 
GCA, GCG, GCT, GCC,
TCT, TCC

CGU, CGC, CGA, CGG,
AGA, AGG

GCA, GCG, GCU, GCC,
UCU, UCC

asparagine 
TTA, TTG

AAU, AAC

UUA, UUG

aspartate 
CTA, CTG

GAU, GAC

CUA, CUG

cysteine 
ACA, ACG

UGA, UGC

ACA, ACG

glutamate 
CTT, CTC

GAA, GAG

CUU, CUC

glutamine 
GTT, GTC

CAA, CAG

GUU, GUC

glycine 
CCA, CCG, CCT, CCC

GGU, GGC, GGA, GGG

CCA, CCG, CCU, CCC

histidine 
GTA, GTG

CAU, CAC

GUA, GUG

isoleucine 
TAA, TAG, TAT

AUU, AUC, AUA

UAA, UAG, UAU

leucine 
AAT, AAC, GAA, GAG,
GAT, GAC

UUA, UUG, CUU, CUC,
CUA, CUG

AAU, AAC, GAA, GAG,
GAU, GAC

lysine 
TTT, TTC

AAA, AAG

UUU, UUC

methionine 
TAC

AUG

UAC

phenylalanine 
AAA, AAG

UUU, UUC

AAA, AAG

proline 
GGA, GGG, GGT, GGC

CCU, CCC, CCA, CCG

GGA, GGG, GGU, GGC

serine 
AGA, AGG, AGT, AGC,
TCA, TCG

UCU, UCC, UCA, UCG,
AGU, AGC

AGA, AGG, AGU, AGC,
UCA, UCG

stop 
ATG, ATT, ACT

UAA, UAG, UGA

AUU, AUC, ACU

threonine 
TGA, TGG, TGT, TGC

ACU, ACC, ACA, ACG

UGA, UGG, UGU, UGC

tryptophan 
ACC

UGG

ACC

tyrosine 
ATA, ATG

UAU, UAC

AUA, AUG

valine 
CAA, CAG, CAT, CAC

GUU, GUC, GUA, GUG

CAA, CAG, CAU, CAC

The four bases appear (64×3=192) times in each set of
64 codons. Degeneracy results because a triplet code designates 20 amino acids
and a stop codon. Each base appears 48 times. The
factorisation of 192 as 48×4 manifests in the inner Tree of Life as the four
geometrical elements (corner, outer edge, inner edge & triangle) per sector surrounding the
centres of the 48 sectors of each set of seven separate Type A polygons
(Fig. 25). Each of four types of geometrical elements appears
48 times in a set and manifests in RNA as one of the four bases C, G, A
& U. The elements pair into two boundary elements (corner & edge) and two
internal elements (edge & triangle) per sector of a polygon. These two classes
correspond to the nonbinding pairs (A,C) and (U,G):
One half of the inner Tree of Life expresses the codons in mRNA and the
other half expresses the anticodons in tRNA. The boundary/interior distinction between their
geometrical elements corresponds to the complementary nature of the four bases. (For the
purpose of illustration, base A has been associated with a corner and base C has been
associated with an outer edge. Which is the correct association is an issue that need not
detain us here). We showed earlier that the same, 2×64×3 pattern exists in
other sacred geometries. For example, the 64 trigrams in one diagonal half of
the I Ching table correspond to the 64 mRNA codons and the 64
trigrams in its other half correspond to the 64 tRNA anticodons (Fig. 26). The yin/yang lines denote bases. The yin/yang duality of opposites
manifests in the codons as the pair of complementary bases A and U and as the pair of
complementary bases C and G. The 64 triplets of yods in each half of the
Sri Yantra also represent the 64 codons in RNA and the 64
anticodons in tRNA.
8. The sacred geometry of the
codons
Because there are four bases, triplet codons are required to produce at
least 21 different codes, that is, codons for the 20 amino acids and at least
one stop codon). Doublet codons would produce only (4^{2}=16) different codes. We see
that EHYEH, the Godname of Kether with number value 21, determines the
threefold nature of codons. EL, the Godname of Chesed with number value
31, prescribes the 61 codons coding only for the 20 amino acids because 61 is
the 31st odd integer. The Decad (10), the
27
Pythagorean measure of divine wholeness, determines this important number
because the decagon, its geometrical symbol, has 61 yods when its sectors are tetractyses
(Fig. 27). Amazingly, the 61st note in the Pythagorean scale, the perfect
fifth of the ninth octave, is the 31st harmonic. Moreover, it has a tone
ratio of 384 (Fig. 28), which is the very number of times the four nitrogen bases A, C, G
& U appear in the 64 mRNA base codons and in the 64
tRNA anticodons! This is not coincidental, for holistic systems exhibit the same set of
defining parameters and are
prescribed by Godnames. 64 is the 32nd even integer, where
32 is the 31st integer after 1, showing how EL prescribes the 64 codons. There are 128 possible codons in
an mRNA and a tRNA molecule, where 128 is the 127th integer after 1 and 127 is the
31st prime number. EL prescribes the complete codon domain available for
life — the number of codons, the number of codons encoding for amino acids and the number
of times each base appear in them.
The Pythagorean Tetrad also determines the number 61 because the number 4 is
symbolized by a square, the Type B version of which has 61 yods (Fig. 29).
The 64 mRNA codons comprise (3^{3}=27) codons made
up of A, C & G bases and (64–27=37) codons containing the U base.
Similarly, the 64 tRNA anticodons comprise 37 anticodons containing the A
base. Hence, there are 74 such codons and their anticodon counterparts. 74 is the
73rd integer after 1, where 73 is the 21st
prime number. The Godname EHYEH with number value 21 prescribes the codons
containing the U base and their anticodons. Bases appear in them (3×74=222) times. The pair of
separate Type A hexagons of the inner Tree of Life, which we saw earlier correspond to the
eight diagonal hexagrams of the I Ching table, contains 74 yods (Fig. 30). As they comprise
50 vertices, edges & triangles, the Godname ELOHIM with number value
50 assigned to Binah in the Tree of Life
28
prescribes the geometrical representation of the 37 codons that contain
uracil and their anticodons. As 37 is the 36th integer after 1, ELOHA, the
Godname of Geburah next below Binah on the Pillar of Judgement with number value
36 prescribes the 37 codons that contain the U base (there are
36 such codons other than the start codon). Of these, three are the stop
codons UAA, UAG, UGA. So there are 34 mRNA codons containing the U base that code only for
amino acids. They are symbolized by the 34 yods in a pair of joined, Type A triangles
(Fig. 31).
A type B triangle has 46 yods (Fig. 32a). 46 is the diploid number for the human cell, namely, its number
of chromosomes. The endpoints of the root edge denote the X and Y sex chromosomes. This
fundamental information is defined by the Tetrad because two joined, Type A squares have 46
yods (Fig. 32b).
The diploid number of the human cell is embodied in the last polygon itself
as the number of hexagonal yods lying on the boundaries of the two joined, Type A dodecagons
(Fig. 33). The two hexagonal yods on the root edge denote the two sex
chromosomes. A hexagonal yod on the edge of one dodecagon and its counterpart on the other
dodecagon symbolize a pair of autosomes.
The Godname ELOHIM prescribes all the codons in human RNA in the
following way: the parameter 384 of holistic systems is displayed not only by the complete pair
of seven separate Type polygons but also by the two sets of the first six enfolded Type A
polygons as their 384 yods other than the endpoints of the root edge (Fig. 34). The 192 yods associated with each set symbolize the 192 instances
of the bases in the 64 codons in mRNA and in the 64 codons
of tRNA. The 12 polygons symbolizing the 384 instances of all bases constitute a holistic
system in themselves
because they are prescribed by the ten Godnames.^{17} For example, they have 50 corners, where
50 is the number value of ELOHIM, each set having 26
corners, where 26 is the number value of YAHWEH. Noting that a corner of
the triangle is the centre of the hexagon and that a corner of the pentagon is the centre of
the decagon, the numbers of yods in each
29
enfolded polygon making up the 192 yods associated with each set are:
(the hexagonal yods on the root edge are regarded here as belonging to the
pentagons, the corners they share with decagons being counted as belonging to the latter).
Associated with each set of the first four enfolded polygons are 90 yods. 36
yods belong to the first two polygons and 54 yods belong to the second pair. This
36:54 division is characteristic of sacred geometries, as well as being
arithmetically expressed in the Lambda Tetractys extrapolation of Plato’s Lambda:^{18}
The clearest illustration of this division in holistic systems is provided
by the disdyakis triacontahedron. When its 120 faces and 180 internal triangles formed by its
edges are Type A triangles, their 900 sectors comprise (120×3=360) sectors in the faces and
(180×3=540) sectors in the interior. Confirmation that the 36:54 division in
the yod populations of the first four polygons is an expression of the archetypal pattern of
numbers in the Lambda Tetractys is that the individual populations are the sums of the
integers in its diagonal rows:
The 192 yods associated with each set of the first six enfolded polygons
comprise 90 yods in the first our polygons and 102 yods in the last two polygons. The former
has nine corners outside the root edge.
The three stop codons & anticodons
There are (27×3=81) instances of A, C & G bases in the 27 codons that contain only them.
There are (37×3=111) instances of the remaining 37 codons that contain at least one U base.
These codons comprise the three stop codons UAA, UAG & UGA, which have four instances of A,
three instances of U and two instances of G. There are (34×3=102) instances of bases in codons
that include the U base and which code solely for amino acids. Hence, the 192 instances of
bases consist of 81 instances of A, C & G bases in the 27 codons containing only them, nine
instances of bases in the three stop codons and 102 instances of bases in the 34 other codons.
Compare this 81:9:102 division with the pattern of yods associated with each set of the six
polygons shown in Fig. 34. The 90 yods associated with a set symbolize the 90 instances of
either A, C & G bases in the 27 codons or the bases in the stop codons. The 102 yods of
the octagon and decagon symbolize the 102 instances of bases in the 34 codons containing a U
base. That this is not just a highly improbable coincidence is indicated by the fact that
the nine corners consist of the four corners of the hexagon, the two corners of the pentagon
and the three corners of the triangle and square — precisely, the numbers of instances of,
respectively, A, G & U in
30
the three mRNA stop codons (Fig. 35). The same conclusions apply to the 90 yods in the first four
polygons enfolded on the other side of the shared root edge. The nine corners that belong
solely to this set symbolize the nine instances of the A, C & U bases in the three tRNA
stop anticodons AUU, AUC & ACU. The mirror symmetry of the two sets is the geometrical
representation of base complementary.
The mRNA start codon AUG, the tRNA start anticodon UAC, the three mRNA stop
codons UAG, UGA & UAA and their anticodons AUC, ACU & AUU have nine A, nine U, three G
and three C bases. The 60 other codons and the 60 other anticodons have 360
(=36×10) bases (180 per set). The Godname ELOHA with number value
36 prescribes how many instances of the bases occur in mRNA codons and tRNA
anticodons other than the start and stop ones. They are made up of 87 A,
87 U, 93 G & 93 C bases. 87 is the number value of
Levanah, the Mundane Chakra of Yesod, which is the penultimate Sephirah.
We saw in Section 5 that the disdyakis triacontahedron has 180 edges and 120
faces. With each edge regarded as a doubleheaded arrow, each half of the Catalan solid has 60
faces with 180 arrows. Compare this with the 60 nonstop/start codons and their 180 bases. The
60 faces in one half of the solid correspond to the 60 mRNA nonstop/start codons and the 60
faces in its inverted half correspond to their 60 tRNA anticodons (Fig. 36). A face and its mirror image are analogous to an mRNA codon and its
anticodon in tRNA. Their counterparts in the inner Tree of Life


are the 120 yods on the sides of the seven enfolded polygons (Fig. 37). The 60 red yods in the triangle, square, pentagon & dodecagon
symbolize the 60 mRNA codons and the 60 blue yods in the hexagon, octagon & decagon
denote their 60 anticodons in tRNA. The 180 arrows in one half of the disdyakis
triacontahedron correspond to the 180 bases in the 60 mRNA codons and the 180 arrows in the
other half correspond to the 180 bases in the 60 tRNA anticodons. The surface of the
disdyakis triacontahedron is a geometrical representation of all the instances of the bases
that occur in mRNA & tRNA that do not initiate or stop translation. The 24 bases making
up the four mRNA start/stop codons and their anticodons in tRNA are represented by the 24
arrows on the internal edges of the sectors of the central, 12sided polygon, nine
arrows and their reverses symbolizing the pair of complementary bases A & U and three
arrows and their reverses symbolizing the pair of complementary bases C & G:
(×9): AU
(×3):
CG
Twelve inwardpointing arrows denote either the nine A and three C bases or
the nine U and three G bases. The twelve outward pointing arrows denote, respectively, either
the nine U and three G bases
31
or the nine A and three C bases. The bidirectionality of an arrow expresses
the complementary nature of bases whereby they bind in pairs, so that a given doubleheaded
arrow denotes a base and its complement. This correlation between the geometry of the disdyakis
triacontahedron and the distinction between codons and anticodons that start or stop
translation is highly evidential because, although there should be equal numbers (n) of A &
U and equal numbers (12–n) of C & G in them, there is no a priori reason why n
should be a multiple of 3 so as to match the three types of internal edges ending on A, B &
C vertices. The geometry of the polyhedron is consistent with scientific facts about the base
composition of the start and stop codons.
The two Type A dodecagons of the inner Tree of Life encode the 60 codons and
60 anticodons because each has 60 hexagonal yods (Fig. 38a). The 180 yods surrounding the centre of each Type B dodecagon
encode the 180 instances of their bases (Fig. 38b). They can be grouped into 60 triplets of red, green & blue
yods. The yod composition of the pair of dodecagons is consistent with the codon structure
of mRNA and tRNA.
9. The 7tree encodes the 384
instances of the bases
The Tree of Life is the template for representing holistic systems. The
socalled ‘ntree’ is defined as the lowest n Trees of Life of any set of overlapping Trees of
Life. When the (12n+7) triangles in the ntree are converted into tetractyses, they
have^{19}
Y(n) = 50n + 30
yods. The 7tree has Y(7) = 380 yods. The 7tree is but part of the 91
overlapping Trees of Life of the Cosmic Tree of Life (CTOL). Figure 39 shows that four yods belonging to the eighth Tree of Life lie
outside the 7tree (two on each side of the central Pillar of Equilibrium). There are
therefore 384 yods up
32
to the apex of the 7tree. In previous articles, the 7tree has been
interpreted as mapping the 26dimensional spacetime predicted by string
theory. It maps the purely physical level of the spiritual cosmos. We find therefore that the
number 384 characterizes the Tree of Life representation of the physical universe. The
number of yods below Chesed of the nth Tree of Life is^{20}
Y'(n) = 50n – 8.
Hence Y'(4) = 192. The 384 yods as far as the top of the 7tree divides into
192 yods below Chesed of the fourth Tree of Life (denoted in Fig. 39) by red yods) and 192 blue yods down to this point. They are the
counterpart of the 192 yods in each half of the Sri Yantra. The former symbolize the 192
instances of the bases in the mRNA codons and the latter symbolize the 192 instances of the
bases in the tRNA anticodons. The two halves of the 7tree, the inner Tree of Life and the
Sri Yantra represent the mRNA and the tRNA molecules. It is the molecular
manifestation of the fundamental yin and yang polarities in nature, as
manifested in the electrostatic attraction between the positively charged proton of the
hydrogen atom in the nitrogen bases and the electrons in the nitrogen and oxygen atoms of their
complementary bases (see Fig. 23).
The 192 red yods belong to 46 triangles with 23 vertices &
62 edges, i.e., 131 geometrical elements.
131 is the number value of Samael, the Archangel of Geburah.
62 is the number value of Tzadkiel, the Archangel of Chesed, which is
the Sephirah preceding Geburah. 131 is the 65th odd integer
after 1. Hence, ADONAI, the Godname of Malkuth with number value 65,
prescribes the geometry whose 192 yods symbolize the 192 instances of the bases of the
64 tRNA anticodons. The 384 red & blue yods belong to 93 triangles with 47
vertices & 123 edges, i.e., 263
geometrical elements. 47 is the 15th prime number. Hence,
YAH, Godname of Chokmah with number value 15, prescribes the 7tree embodying
as yods the 384 instances of the bases in the 64 codons of mRNA and the
64 anticodons of tRNA. YAH prescribes the 7tree also because it has
15 SLs on each side pillar. There are 216 edges &
triangles, where 216 is the number value of Geburah. Each set of 192 yods
belong to sections with 85
33
vertices & edges, where
85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}
is a parameter of holistic systems because it is the number of yods in the
next higher differentiation of the tetractys (1storder tetractys), namely, the 2ndorder
tetractys (Fig. 40). The Tetrad (4) expresses the properties of holistic systems. Its
symbol — the square — has 25 yods when it is Type A (see Fig. 32). It is the polygonal counterpart of the 25 SLs up to Chesed of the
fourth Tree of Life that divides the 384 yods up to the top of the seventh Tree of Life into
two sets of 192 yods.
The Tree of Life basis of the pattern of 64 codons and
64 anticodons can be demonstrated as follows: if each Sephirah is replaced by
a Tree of Life, the resulting ten Trees of Life (Fig. 41) have 64 ‘Sephirothic levels’ (SLs). This is how
the number 64 parameterizes the next higher order differentiation of the
Tree of Life. They comprise Kether, Chokmah & Binah of the tenth Tree of Life and 61
SLs, starting with Chesed of that Tree of Life. They correspond, respectively, to the three
stop codons and the 61 other codons. The matching demonstrates the Tree of Life basis of the
DNA/RNA molecule. Below this Supernal Triad are 20 SLs on each side pillar. It is an
indication that there are 20 amino acids coded by at least 20 codons.
Now suppose that each of the 64 SLs is replaced by a Tree
of Life. There are then 64 overlapping Trees of Life and 385 SLs up to Chesed
of the 64th one, where
385 =

1^{2}
2^{2} 3^{2}
4^{2} 5^{2} 6^{2}
7^{2} 8^{2} 9^{2} 10^{2}
.

This shows beautifully how the Decad determines the 384 SLs needed to
emanate the 64 Trees of Life, starting from the first Sephirah of Construction
of the highest one, before reaching the nadir at Malkuth of the first one. They are arranged in
32 pairs of triads in the uppermost 32 Trees of Life and in 32 pairs of triads in the lowest 32
Trees of Life. The former set corresponds to the 32 mRNA codons (a,b,c) (a, b, c = A, U, C
& G) and their 32 complements (a*,b*,c*), where a* is complementary to a, b* is
complementary to b and c* is complementary to c. The latter set
corresponds to the 32 tRNA anticodons (a*,b*,c*) and their complements
(a,b,c). In other words, the two triads of Sephiroth of Construction within each Tree of Life
in the uppermost 32 Trees of Life denote a codon and its complement in mRNA, whilst in the
lowest 32 Trees of Life they denote a codon and its complement in tRNA. These two sets of
64 triads of SLs are the counterpart of the 64 trigrams in
each half of the I Ching table, which represent mRNA and tRNA. The 64 triads
of SLs in each set of 32 Trees of Life have their counterpart also in the 192 edges &
triangles belonging to S and the 192 edges & triangles belonging to S', discussed in
Section 1. The triangle and pentagon have 64 edges & triangles, whilst the
square and dodecagon have twice as many. The counterpart of the former in the
64 triads of SLs in the upper 32 Trees of Life are the 64 SLs
on the central pillar; the counterpart of the latter are the 64 pairs of SLs
on the side pillars. For S', the octagon, which has the same number of sectors as the triangle
& pentagon, possesses 64 edges & triangles, whilst the hexagon &
decagon, which has the same number of sectors as the square & dodecagon, has twice as many.
The counterpart of the former in the 64 triads in the lower set of 32 Trees of
Life are the 64 SLs on the central pillar; the counterpart of the latter are
the 64 pairs of SLs on the side pillars. The fact that the 192 edges &
triangles are divided into three groups reflects the fact that the Sephiroth are arranged on
three pillars. The analogy between the 64 codons and the geometry of the inner
Tree of Life implies that the triple nature of codons is a direct reflection of the three
pillars of the outer Tree of Life. This is not a metaphysical speculation. Instead, it is
a mathematical deduction from this analogy.
34
It is appropriate that Nogah, the Mundane Chakra of Netzach, which
is the Sephirah that expresses the richness and variety of biological species, should have the
gematria number value 64 (see Table 1). It is the number of corners (red yods) of the two sets of seven
enfolded polygons that do not coincide with Sephiroth, for Figure 42 shows that the former
have 70 corners, of which three corners (white yods) of the hexagon in one set coincide with
Chokmah, Chesed & Netzach and of which three corners (also white yods) of the hexagon in
the other set coincide with Binah, Geburah & Hod. The endpoints of their shared root
edge do not coincide with Tiphareth and Daath because the latter do not lie in the plane of
the two side pillars of the 3dimensional Tree of Life. The 64 corners (32
in each set of polygons) that the inner Tree of Life does not share with its outer form
represent new degrees of freedom characterizing any system that embodies this universal
paradigm. Similarly, the outer form of the Tree of Life has 70 yods, when its 16 triangles
turn into tetractyses, so that it, too, has 64 black yods unshared with its
inner form. We may regard the 64 unshared corners of the latter as the
representation of the 64 types of codons of mRNA that can potentially bind
to their corresponding anticodons in tRNA symbolized by the 64 yods in the
outer Tree of Life. The conjoined geometries of the inner and outer forms of the Tree of
Life embody the very fundamental numbers involved in the replication of the DNA molecule.
The 20 corners outside the shared root edge of the pair of dodecagons (the last polygon)
symbolize the minimum number of codons that encode for amino acids. The inner
TOL predicts how many amino acids exist. Their counterpart anticodons in the outer Tree
of Life are the 20 yods making up the tetrahedron, which is the last stage in the
sequence:
point → line → triangle → tetrahedron
that collectively constitutes its trunk. This was shown in Article 33 to
represent the 80 bones of the axial skeleton.^{21} We saw earlier in Fig. 33 that the 46 hexagonal yods lying on
the boundary of the two dodecagons symbolize the 46 chromosomes in the human cell.
Three of the 64 mRNA codons — the stop codons — are special
in that they do not encode for amino acids. If the 64 corners of the polygons
unshared with the Tree of Life really do symbolize codons, then there must be three of them
that differ from the rest in some way. One corner of the pentagon coincides with the centre of
the decagon and the lower endpoint of the root edge is the only one of the 64
unshared corners that coincides with the projection of the locations of Sephiroth onto the
plane of the 14 polygons. Hence, there are three corners that are different from the
rest. Admittedly, they have not been independently proven to symbolize these codons. But the
criticism that interpreting them this way is ad hoc misses the point. What is being
pointed out here is simply that (as one would expect) the sacred geometry of the inner TOL
naturally exhibits a distinction between three of the corners and the other 61 that mirrors the
difference between the stop codons and the codons that encode only for amino acids. Whether
these three corners do symbolize the stop codons still has to proved, although,
intuitively speaking, it is appropriate that the lower endpoint of the root edge, which marks
the completion of the generation of the 14 polygons, should denote a stop codon. It is
also intuitively appropriate that the start codon should be identified with the upper endpoint
of the root edge, which is the starting point for the generation of the inner TOL. The
counterparts in the outer Tree of Life of the three corners denoting the stop codons in mRNA
are the yods at Tiphareth, Yesod & Malkuth, which symbolize the three stop anticodons in
tRNA. Appropriately, the yod at Kether denotes the start anticodon in tRNA. The counterpart of
the three stop codons in the ten overlapping Trees of Life with 64 SLs shown
in Figure 38 are Kether, Chokmah & Binah of the tenth Tree of Life, the
remaining 61 SLs corresponding to the 61 codons. The fact that there are 20 SLs on each side
pillar reflects the fact that there must be a minimum of 20 codons that encode for the 20
amino acids. The start codon AUG would correspond to the first of these 61 SLs, namely,
Chesed of the tenth Tree of Life.
It is remarkable that there are 128 vertices, edges & faces of the first
four Platonic solids, which were believed by the ancient Greeks to be the shapes of the
particles of the Elements Earth, Water, Air & Fire:

vertices

edges

faces

Total

tetrahedron 
2+2

3+3

2+2

7+7 = 14

octahedron 
3+3

6+6

4+4

13+13 = 26

cube 
4+4

6+6

3+3

13+13 = 26

icosahedron 
6+6

15+15

10+10

31+31 = 62

Total =

15+15

30+30

19+19

64+64 = 128

The 64 mRNA codons correspond to the 64
geometrical elements in half of the four polyhedra and their counterpart 64
anticodons in tRNA correspond to the 64 elements in their other halves. It is
evidence that the pattern of codons in human DNA is not merely a product of random,
evolutionary change over millions of years. Instead, it is a manifestation of sacred geometries
that embody the divine archetypes. Each half
35
Table 7. DNAtype patterns of equivalence in representations of holistic
systems.
(7+7) separate Type A
polygons

7 Type B
polygons

I Ching
table

Sri Yantra

7 musical
scales

disdyakis
triacontahedron

mRNA & tRNA

384 vertices, edges & triangles surround centres of two sets of 7
polygons

384 edges & triangles

384 lines & broken lines

384 yods in 43 triangles

384 intervals between notes

384 doubleheaded arrows along edges

384 instances of 4 bases

192 vertices, edges & triangles surround centres of 7 polygons in each
set

192 edges & triangles in triangle, square, pentagon & dodecagon or
in hexagon, octagon & decagon

192 lines & broken lines in each diagonal half

192 yods in each half

192 rising or falling intervals

192 doubleheaded arrows on edges in each half

192 instances of 4 bases in mRNA or tRNA

128 triplets of geometrical elements surround centres of 14 polygons

128 triplets of edges & triangles

128 trigrams

128 triplets of yods

128 triplets of rising/falling intervals

128 triads of arrows

128 codons/anticodons

64 triplets of geometrical elements surround centres of 7
polygons in one set

64 triplets of edges & triangles in triangle, square,
pentagon & dodecagon (S)

64 trigrams in upper, diagonal half

64 triplets of yods in upper half

64 triplets of rising intervals

64 triads of arrows in one half

64 codons

64 triplets of geometrical elements surround centres of 7
polygons in other set

64 triplets of edges & triangles hexagon, octagon
& decagon (S')

64 trigrams in lower, diagonal half

64 triplets of yods in lower half

64 triplets of falling intervals

64 triads of arrows in other half

64 anticodons

64×3 geometrical elements in each set of 7 polygons

64 triplets of edges & triangles in either S or S'

64×3 lines & broken lines in each half

64 triplets of yods in each half

64×3 rising/falling intervals

64×3 arrows in each half

64×3 bases

A boundary and an internal geometrical element

Geometrical element and its diametric opposite

Line (yang) and broken line (yin)

Corresponding yods in each half

Rising & falling intervals

An arrow and its reverse

A base & its complement

36
comprises (15+19=34) vertices & faces and 30 edges. We
pointed out earlier that the 64 mRNA codons comprise 27 codons made up of A, C
& G bases and 37 codons containing the U base, 34 of which are nonstop codons. This 27:34
division of codons that encode only for amino acids manifests in the first four Platonic solids
as the 27 edges of the octahedron, cube & icosahedron and as the 34 vertices & faces of
all four solids. It means that the six edges of the tetrahedron must correspond to the three
stop codons of mRNA and the three stop anticodons of tRNA (they cannot correspond to either the
six vertices of the octahedron or to the six faces of the cube because this would not generate
the numbers 27 & 34). The 27 remaining edges in each half correspond either to the 27
codons or to their 27 anticodons made up of, respectively, A, C & G or U, G & C bases.
The 34 vertices & faces in each half correspond to the 34 nonstop codons in mRNA that
contain at least one U base and their anticodon counterparts.
The tetrahedron, octahedron, cube & icosahedron embody the number 128
because — as the first four regular polyhedra associated with the Elements of Earth,
Water, Air & Fire — they constitute a holistic system that is prescribed by the
Godnames.^{22} For example, the Godname YAH (YH) with number value
15 prescribes the 15 vertices in each half of the solids, the
value 5 of H being the number of vertices in each half of the tetrahedron and octahedron and
the value 10 of Y being the number of vertices in each half of the cube and icosahedron. The
number value 49 of the Godname EL ChAI is the number of edges & faces in
half of these solids. The number value 31 of EL is the number of geometrical
elements in the icosahedron and the number value 21 of EHYEH is the number of
its vertices & edges. The number value 26 of YAHWEH is the number of
elements in the octahedron, as well as the cube. When their faces are divided into their
sectors, the four solids have 120 triangles with 248 corners &
sides.^{23} This shows how they embody the dimension
248 of the superstring gauge symmetry group E_{8}. It is the number
value of Raziel, the Archangel of Chokmah (Table 1).
The number 128 is the sum of the number values of the first four
Godnames (see Table 1):
128 = 21 + 26 + 50 +
31.
As many previous articles have demonstrated, the Pythagorean Tetrad
determines and expresses all parameters of holistic systems, such as 64
(=4^{3}) and 128. As Fig. 29 shows, its symbol — the square — possesses 61 yods, when
its sectors are Type A triangles. They symbolize the 61 codons encoding only for the amino
acids. Surrounding its centre are 20 yods on its edges & diagonals. This expresses the fact
that there has to be at least 20 codons with this function. A Type A square also embodies this
information as its 20 hexagonal yods. The fourth Godname, EL (“God”), which has the
number value 31, prescribes the 61 nonstop codons because 61 is the
31st odd integer, whilst 128 (the sum of the first four Godname
numbers) is the 127th integer after 1, where 127 is the 31st prime number.
Such arithmetic connections cannot arise by coincidence. Instead, they indicate that numbers
characterizing the human DNA molecule are prescribed by Godnames because man ultimately has a
divine origin that is indicated by the way in which this articles has proved the architecture
of his DNA molecule conforms to sacred geometries found in Judaism, Taoism and Hindu
Tantra.
Table 7 lists equivalent features of the (7+7) separate Type A polygons, the
seven separate Type B polygons, the I Ching table, the Sri Yantra, the seven musical scales,
the disdyakis triacontahedron and the RNA codons & anticodons.
References &
notes
^{1} All numbers from this table mentioned in the text will be written in
boldface.
^{2} Phillips, Stephen M. Article 4: “Godnames prescribe inner Tree of
Life,” (WEB, PDF), pp. 1–2.
^{3} Ibid.
^{4} Phillips, Stephen M. Article 3: “The sacred geometry of the Platonic
solids,” (WEB, PDF), pp. 9, 20.
^{5} Ibid., p. 10.
^{6} Phillips, Stephen M. Article 40: “The unification of all sacred
geometries and its implication for particle physics,” (WEB, PDF), pp. 2326.
^{7} Phillips, Stephen M. Article 20: “Algebraic, arithmetic & geometric
interpretations of the I Ching table;” Article 21: “Isomorphism between the I Ching table,
the 3×3×3 array of cubes and the Klein Configuration,” (WEB, PDF).
^{8} Phillips, Stephen M. Article 26: “How the seven musical scales relate
to the disdyakis
37
triacontahedron,” (WEB, PDF), p. 13.
^{9} Phillips, Stephen M. Article 39: “The correspondence between the inner
Tree of Life, the Sri Yantra & the I Ching diagram and their realization in the seven
musical scales,” (WEB, PDF), p. 4.
^{10} Phillips, Stephen M. Article 35: “The Tree of Life nature of the Sri
Yantra and some of its scientific meanings,” (WEB, PDF).
^{11} Phillips, Stephen M. Article 22: “The disdyakis triacontahedron as
the 3dimensional counterpart of the inner Tree of Life,” (WEB, PDF).
^{12} Phillips, Stephen M. Article 23: “The Polyhedral Tree of Life,”
(WEB, PDF).
^{13} Phillips, Stephen M. 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).
^{14} A triangular prism is two similarly orientated, parallel, identical
triangles with a straight edge joining their corresponding vertices. A square antiprism is
two similar, parallel squares orientated at an angle of 45° with two edges of the antiprism
joining each corner of one square to the two nearest corners of the other square.
^{15} Phillips, Stephen M. Article 43: “The Tree of Life nature of the
{3,7} tessellation of the 168 automorphisms of the Klein quartic on the
3torus,” (WEB, PDF).
^{16} There are other amino acids, such as selenocysteine. However, it is
not coded for directly in the genetic code. Instead, it is encoded in a special way by a UGA
codon, which is normally a stop codon.
^{17} Ref. 2, pp. 4–5.
^{18} Phillips, Stephen M. Article 11: “Plato’s Lambda — its meaning,
generalisation and connection to the Tree of Life,” (WEB, PDF).
^{19} Proof: the ntree has (12n+7) triangles with (6n+5) corners and
(16n+9) edges. Two hexagonal yods lie along each edge and one hexagonal yod is at the centre
of each triangle when it becomes a triangle. The number of yods in the ntree ≡ Y(n) = 6n +
5 + 2(16n+9) + 12n + 7 = 50n + 30.
^{20} Proof: there are 12 yods below Chesed of the nth Tree of Life that
are outside the (n–1)tree. The number of yods in the latter is Y(n–1) =
50(n–1) + 30 = 50n – 20. Therefore, the number of yods
below Chesed of the nth Tree of Life ≡ Y'(n) = 50n – 20 + 12 =
50n – 8.
^{21} Phillips, Stephen M. Article 33: “The human axial skeleton is the
trunk of the Tree of Life,” (WEB, PDF).
^{22} Ref. 4, pp. 9–10.
^{23} Ref. 4, p. 10.
Information embedded in DNA proves the
existence of God
38