Stephen M. Phillips
Flat 4, Oakwood House, 117-119 West Hill Road. Bournemouth. Dorset BH2 5PH.
well as having an outer form, the Tree of Life has an inner form hitherto unknown to
students of Kabbalah as far as the author is aware. It consists (Fig. 1) of two identical sets of seven regular polygons: triangle, square,
pentagon, hexagon, octagon, decagon and dodecagon, all
polygons in each set being enfolded in one another and joined at a common,
so-called ‘root edge.’ In Articles 4–7 it was shown that dynamic and structural parameters
of the superstring are encoded in several sections of this inner form of the Tree of Life.
This is because the geometrical properties of these polygons and their yod populations
generated by conversion of their sectors into tetractyses are determined by the number
values of the Godnames assigned to the ten Sephiroth of the Tree of Life. This means that
they constitute different but equivalent Tree of Life patterns embodying information about
the microcosmic manifestation of this universal blueprint in the space-time continuum,
namely, the superstring. This article will examine another section of the polygonal form of
the Tree of Life, namely, its first four regular polygons. It will show how their
properties, too, are prescribed by the Godnames. The way in which this section encodes
certain superstring parameters will then be compared with how other sections discussed in
previous articles embody them. This set has particular importance because the Tetrad
Principle formulated in Article 1 states that numbers of universal (and therefore
scientific) significance are the property of either the fourth member or the first four
members of a class of mathematical objects (1). This is illustrated by the
first four Platonic solids, which embody the numbers 240 and
248 (group parameters of the symmetry group
E8 predicted by superstring theory to describe the unified force between
superstrings), the structural parameter 168 and the number 137 that
determines the fine-structure constant.
2. Properties of the first four
The first four regular polygons enfolded in the Tree of Life are
the triangle, square, pentagon and hexagon (Fig. 2). The number of corners and yods in the first four polygons when their
sectors are tetractyses are
set out below:
Number of corners =
Number of yods =
The geometrical properties and yod populations of the first four polygons
and the first (4+4) polygons are analysed by considering them firstly as separate and then
as enfolded. Numbers appearing in boldface in the text indicate various number
values of the Sephiroth, their Godnames, Archangels, Orders of Angels and Mundane Chakras.
These are tabulated below.
Table 1. Number values of the Sephiroth.
(Cited numbers are in coloured cells)
4 separate polygons
- Number of corners of polygons = 3 + 4 + 5 + 6 = 18.
- Number of sides of polygons = 18.
- Number of corners & sides of polygons = 18 + 18 = 36.
- Number of sectors = 18.
- Number of corners of sectors = 18 + 4 = 22.
- Number of sides of sectors = 18 + 18 = 36.
- Number of corners & sides of sectors = 22 + 36 = 58.
- Number of corners, sides & sectors = 58 + 18 = 76.
- Number of yods = 19 + 25 + 31 + 37 = 112.
- Number of hexagonal yods = 15 + 20 + 25 + 30 = 90.
- Number of yods on boundaries of polygons = 18 + 2×18 = 54; (54–18=36) are
- Number of yods on boundaries of tetractyses = 18 + 2×18 + 2×18 + 4 = 94;
(94–22=72) are hexagonal.
(4+4) separate polygons
- Number of corners of polygons = 2×18 = 36.
- Number of sides of polygons = 2×18 = 36.
- Number of corners & sides of polygons = 2×36 =
- Number of sectors = 2×18 = 36.
- Number of corners of sectors = 2×22 = 44.
- Number of sides of sectors = 2×36 = 72.
- Number of corners & sides of sectors = 2×58 = 116.
- Number of corners, sides & sectors = 2×76 = 152.
- Number of yods = 2×112 = 224. Number of yods other than centres of polygons
= 224 – 4 – 4 = 216.
- Number of hexagonal yods = 2×90 = 180.
- Number of yods on boundaries of polygons = 2×54 = 108; (2×36=72) are
- Number of yods on boundaries of tetractyses = 2×94 = 188; (2×72=144)
4 enfolded polygons
- Number of corners of polygons = 3 + (4–2=2) + (5–2=3) + (6–2=4) = 12; (12–2=10) are
outside root edge (“external”).
- Number of sides of polygons = 3 + (4–1=3) + (5–1=4) + (6–1=5) = 15;
(15–1=14) are external.
- Number of corners & sides of polygons = 12 + 15 = 27; (27–3=24) are
- Number of sectors = 18 – 1 = 17 (the triangle fills one sector of the hexagon).
- Number of corners of sectors = (3+1=4) + (2+1=3) + (3+1=4) + 4 = 15;
(15–2=13) are external.
- Number of sides of sectors = (3+3=6) + (3+4=7) + (4+5=9) + (5+4=9) = 31;
(31–1=30) are external.
- Number of corners & sides of sectors = 15 + 31 = 46;
(46–3=43) are external.
- Number of sectors & their sides = 17 + 31 = 48.
- Number of corners, sides & sectors = 46 + 17 = 63; (63–3=60) are external.
- Number of yods = 19 + (25–4=21) + (31–4=27) + (37–4–6=27) = 94; (94–4=90)
are external. Of the 90 yods outside the root edge, (90–3=87) are not Sephirothic
points of the outer Tree of Life.
- Number of hexagonal yods = 94 – 15 = 79; (79–2=77) are external.
- Number of yods on boundaries of polygons = 12 + 2×15 = 42; (42–4=38) are
- Number of yods on sides of tetractyses = 15 + 2×31 = 77;
(77–4=73) are external.
(4+4) enfolded polygons
- Number of corners of polygons = 2×10 + 2 = 22; (22–2=20) are
- Number of sides of polygons = 2×14 + 1 = 29; (29–1=28) are
- Number of corners & sides of polygons = 22 + 29 = 51;
(51–3=48) are external.
- Number of sectors = 2×17 = 34.
- Number of corners of sectors = 2×13 + 2 = 28; (28–2=26) are
- Number of sides of sectors = 2×30 + 1 = 61; (61–1=60) are
- Number of corners & sides of sectors = 28 + 61 = 89; (89–3=86) are
- Number of sectors & their sides = 34 + 61 = 95.
- Number of corners, sides & sectors = 89 + 34 = 123; (123–3=120) are
- Number of yods = 2×90 + 4 = 184; (184–4=180) are external.
- Number of hexagonal yods = 2×77 + 2 = 156; (156–2=154) are
- Number of yods on boundaries of polygons = 2×38 + 4 = 80;
(80–4=76) are external.
- Number of yods on sides of tetractyses = 2×73 + 4 = 150;
(150–4=146) are external. Number of yods on edges of tetractyses other than corners of
polygons outside root edge = 150 – 20 = 130.
3. Shared yods & geometrical
In this section we shall analyse the properties that both the four polygons
and the (4+4) enfolded polygons share with the Tree of Life or with the 1-tree.* We shall illustrate how the number 4, the Pythagorean Tetrad, expresses these shared properties, as it does all the
properties of the outer and inner forms of the Tree of Life. Fig. 3 shows that three external corners, four external sides and one
triangular sector belonging to each set of the four enfolded polygons are shared with the
Tree of Life (or, rather, its projection onto the plane of the polygons). On each side of
the root edge are seven shared external corners & sides (7 = 4th odd integer) and eight
shared external corners, sides & sectors (8 = 4th even integer). 16 (= 42)
geometrical elements outside the root edge of both sets of polygons are shared with the Tree
of Life. 14 of them are external corners & sides. The root edge shares its lower end
point with the Sephirah Tiphareth — or, rather, its projection. The root edge itself is not
shared because it is only part of the projection of the Path connecting Kether
and Tiphareth. The number of corners & sides of both sets of polygons which are shared =
14 + 1 = 15. The Godname YAH with number value 15 therefore prescribes
how many corners & sides the (4+4) enfolded polygons share with the Tree of Life.
Fig. 3 also shows that three external corners, five external
* The n-tree is the lowest n trees of the Cosmic Tree
of Life (see Article 5 for definition of the latter).
sides and one sector belonging to each set of polygons are shared with the
1-tree, whilst their root edge shares two corners and one side. The total number of geometrical
elements shared with the 1-tree = 3 + 2×(3+5+1) = 21. The Godname EHYEH (AHIH) with
number value 21 prescribes how many geometrical elements the (4+4) enfolded
polygons are shared with the 1-tree. Indeed, its letter values denote different groups of
A = 1:
H = 5:
H = 5:
5 external edges on one side of root edge;
8 corners + 2 triangles;
5 external edges on other side of root edge
When the (4+4) separate polygons become enfolded, the 224 yods in the former
become the 184 yods in the latter, i.e.,
yods disappear. Fig. 4 shows the 70 yods in the 16 triangles of the Tree of Life when
they are turned into tetractyses. 20 yods in a set of four polygons are shared with the Tree
of Life (36 yods in both sets), of which 15 yods lie on Paths.
Thirteen red yods outside the root edge in each set lie on Paths, making a total of
26 such yods for both sets, where 26, the number value of YAHWEH, is the
number of combinations of (1+2+3+4=10) objects arranged in a tetractys:
D E F
G H I J
number of combinations = 2n – 1
21 – 1 = 1
22 – 1 = 3
23 – 1 = 7
24 – 1 =
In the case of the 1-tree, 15 yods outside the root edge in each
set of polygons are shared with Paths of the 1-tree (34 yods in total). The number of yods
outside the root edge not lying on Paths of the Tree of Life = 180 – 26 = 154.
Including the two black yods on the root edge which do not lie on Paths (see Fig. 4), there are 156 such yods, where 156 is the 155th integer after
1. The number of yods unshared with Paths of the 1-tree = 184 – 34 = 150 = 15×10. Of
the 79 hexagonal yods of the four enfolded polygons, 14 are shared with Paths of the 1-tree,
leaving 65 unshared, hexagonal yods, where 65 is the number value of
ADONAI, Godname of Malkuth.
The 16 triangles of the Tree of Life have 10 corners and 22 sides, that is,
48 geometrical elements. Of these, seven corners, eight sides and two triangles (17
elements) are shared with the (4+4) enfolded polygons, leaving 31 unshared,
geometrical elements, where 31 is the number value of EL, Godname of
The number of sides of the four enfolded polygons = 15. The number of
such sides of the 4n polygons enfolded in the n-tree = 15n. The four enfolded polygons
have 27 corners & sides. The topmost corner of the hexagon is shared with the lowest corner
of its counterpart enfolded in the next higher tree. The number of corners & sides of the
4n polygons enfolded in the n-tree = 26n + 1. As each root edge comprises two endpoints
(corners) and one side, the number of corners & sides of the other set of 4n polygons
outside their root edges = 26n + 1 – 3n = 23n + 1. The number of corners
& sides of the 8n polygons enfolded in the n-tree = 26n + 1 + 23n + 1 = 49n +
2. Each set of four polygons has therefore 15 sides and 26 corners
& sides that are intrinsic to it, whilst every set of (4+4) enfolded polygons has
49 intrinsic corners & sides. The 17 tetractyses in each set of four polygons
have 31 sides (30 external) and 15 corners with (2×31 +
15) = 77 yods lining their sides, i.e., 73 yods outside their root edge,
where 73 is the number value of Chokmah. The 34 tetractyses of the (4+4) enfolded
polygons have (2×30 + 1 = 61) sides with (2×73 + 4 = 150 =15×10) yods on
80 yods are on the boundaries of the polygons and
76 are outside their root edge. Of the 61 sides, 11 are shared with the 1-tree,
leaving 50 sides that are unshared, where 50 is the number value of
ELOHIM, Godname of Binah. The 1-tree also shares eight corners of its 19 triangles with both
sets of polygons (see Fig. 3), whose 34 tetractyses have
89 corners & sides. There are therefore (89–11–8=70) unshared corners & sides, of
which 20 are corners and 50 are sides. The 35 unshared corners and sides in each
set of polygons comprise 10 corners and 25 sides.
4. How Godnames prescribe the first four
Set out below are ways in which properties of the four polygons
and the (4+4) polygons are prescribed by the number values of the Godnames assigned to the
ten Sephiroth of the Tree of Life:
||21 geometrical elements in the (4+4)
polygons are shared with the 1-tree.
4 enfolded polygons have 15 sides and
15 corners of their 17 tetractyses. The number of yods on the
boundaries of the 34 tetractyses of the (4+4) polygons = 150 = 15×10.
This is also the number of yods in the (4+4) enfolded polygons that do not
lie on Paths of the 1-tree. The (4+4) polygons share 15 corners
& sides with the Tree of Life.
(4+4) enfolded polygons have 26 corners outside
their root edge. Every 4 enfolded polygons have 26 corners &
sides. Outside their root edge, the (4+4) enfolded polygons share with the
Tree of Life 26 yods on its Paths.
Number of corners and sides of (4+4) enfolded polygons = 51 =
50th integer after 1. The 34 sectors of the (4+4) enfolded polygons
have 50 sides unshared with the 1-tree;
Number of sides of 17 tetractyses in 4 enfolded polygons =
31. The Tree of Life has 31 geometrical elements unshared
with (4+4) enfolded polygons;
4 separate polygons have 36 corners & sides and
36 sides of their 18 tetractyses. The polygons have
36 hexagonal yods on their boundaries. The (4+4) separate
polygons have 36 corners and 36 sides;
4 separate polygons have 76 geometrical elements.
(4+4) enfolded polygons have 76 yods outside their root edge on
Tetractyses of (4+4) enfolded polygons have 130 yods on their
boundaries other than corners outside their root edge, where 130 =
129th integer after 1;
(4+4) enfolded polygons have 154 hexagonal yods outside their
root edge, where 154 = 153rd integer after 1;
||Every (4+4) enfolded polygons have 49
intrinsic corners & sides;
65 hexagonal yods in the 4 enfolded
polygons are unshared with Paths of the 1-tree. (4+4) enfolded polygons have
156 yods unshared with Paths of the Tree of Life, where 156 =
155th integer after 1.
The natural way in which the Godname numbers appear in the
above analysis of the geometrical properties of the polygons and their yod populations refutes
the argument that their presence lacks real significance because it was contrived by various
selections of these properties.
5. Connections between the 1-tree and the
first four polygons
Having established that
the ten Godnames prescribe the first four of the seven polygons and therefore define it as a
‘Tree of Life pattern,’ we will now explore their correspondence to the 1-tree.
The four enfolded polygons have 12 corners, of which ten are outside the
root edge, one of them — the uppermost corner of the hexagon — being shared with the lowest
corner of the hexagon enfolded in the next higher tree. Each set of 4n polygons enfolded in
the n-tree has (9n+1) corners outside their n root edges. (10n+1) corners are associated
with each set of 4n polygons. ADONAI, the Godname of Malkuth, prescribes the 10-tree because
its number value 65 is the number of Sephirothic emanations in the 10-tree (what
in previous articles were called ‘Sephirothic levels,’ or SLs). Enfolded on either side of
its central pillar are 40 polygons of the first four types associated with which are (10×10
+ 1 = 101) corners, 91 of them being outside their root edges. 101 is the
26th prime number and the number value of Michael, the Archangel of Tiphareth.
The 25-tree is prescribed by ADONAI MELEKH, the full Godname of Malkuth, because its number
value 155 is the number of SLs in the 25-tree. 100 polygons of the first four
types with (10×25 + 1 = 251) associated corners are enfolded on either side of the central
pillar of the 25-tree. The two words ADONAI and MELEKH prescribe its division into the
10-tree and the 15 trees above it:
251 = 101 + 150,
the number 101 denoting the 10 endpoints of root edges
associated with the 40 polygons and their 91 external corners. Eleven of the latter are the
highest and lowest corners of the 10 joined hexagons,
Figure 5. Correspondence between the 251 yods in the
1-tree and the 251 corners of the first four
types of polygons enfolded in the 25-tree prescribed by ADONAI MELEKH, the
Godname of Malkuth.
Figure 6. Correspondence between the 251 yods in the
1-tree and the 251 corners of
the first six types of polygons enfolded in the 10-tree that is prescribed
10 of them belonging exclusively to the hexagons enfolded in
the 10-tree and one being also the lowest corner of the hexagon enfolded in the 11th tree.
The number 101 therefore has
the geometrical differentiation:
101 = 10 + 91 = 10 + 11 + 80,
80 = 10×(1+2+3+2)
is the number value of Yesod and ‘1’ denotes the corner of
the triangle in each tree outside their root edges, ‘2’ denotes the two corners of the
square, ‘3’ denotes the three external corners of the pentagon and ‘2’ denotes the two
external corners of the hexagon that are unshared with adjoining hexagons.
80 = 10×(1+2) + 10×(3+2) = 10×3 + 10×5 = 30 +
where 30 is the number of external corners of the first two
polygons enfolded in the 10-tree and 50 is the number of corners of the last two polygons
enfolded in the 10-tree. Fig. 5 displays the types of
corners of the 100 polygons of the first four types enfolded on one side of the 25-tree
prescribed by ADONAI MELEKH. It also shows that, when its 19 triangles are turned into
tetractyses, the 1-tree contains 80 yods (30 yods belong to the Lower Face formed by
Tiphareth, Netzach, Hod, Yesod and Malkuth, leaving 50 yods in the Upper Face) and that, when the three
sectors of each triangle are turned into tetractyses, the 1-tree contains 251 yods — the
same as the number of corners of the 100 polygons. The reason why the two remarkable
251 yods in 1-tree 251 corners of first 4 polygons enfolded in 25-tree
(30+50= 80) yods in 1-tree (30+50=80) outer corners
of first 4 polygons enfolded in 10-tree
exist is that, being the lowest of the 91 overlapping trees
making up the Cosmic Tree of Life and therefore its most ‘Malkuth’ level, the 1-tree
embodies the same parameters as any section of CTOL that bears a formal correspondence to
Malkuth — in this case, the 25-tree, whose 25 trees are the counterpart of the 25 tree
levels of the 7-tree mapping the physical plane, the lowest of the seven planes that
corresponds to Malkuth. In Articles 2 and 5, these tree levels were interpreted as the 25
spatial dimensions that quantum mechanics predicts for spinless strings. This is why the
Godname ADONAI MELEKH assigned to Malkuth (physical universe) refers to the 25-tree and why
the Godname ADONAI prescribes its ten lowest trees, corresponding to which are the 10 tree
levels signifying the ten spatial dimensions of 11-d supergravity
space-time. Analogous structures defined by the set of Godname
numbers — whether of the outer or the inner form of the Tree of Life — must embody
the same numbers and display
the same pattern of
differentiation of whatever these numbers signify because they are equivalent, holistic
objects that embody the divine paradigm. This is why the
first six polygons enfolded on either side of the 10-tree have 251 corners (Fig. 6), for the (6+6) polygons also constitute a Tree of Life pattern (see Article 4
(2) for their prescription by the ten Godnames).
have 30 corners, the squares &
pentagons have 50 corners and the hexagons,
octagons & decagons have 171 corners. This is the same 30:50:171 pattern as displayed by
the yods in the 1-tree (see Fig.
6. Encodings of 10-whorl structure of
What is the meaning of the ubiquitous, geometrical encoding of the
number 251 and its division into the numbers 80 and 171? These numbers have a
remarkable interpretation in terms of the ten-fold structure of the basic unit of matter
described (3) by the Theosophists Annie Besant
and C.W. Leadbeater over a
century ago with the aid of a yogic siddhi (psychic ability)
with the Sanskrit name of ‘anima.’ Moreover, they support the author’s theory of
superstrings (4) derived from higher-dimensional,
extended objects called ‘D-branes,’ as will be explained
shortly. En passim, it should be
pointed out that the theory has not been tailored in order to procure this agreement. It
was conceived by the author for purely scientific reasons long before he discovered that
these numbers characterise the outer and inner forms of the Tree of Life.
Magnified with what the author has called ‘micro-psi’
(5), the basic constituent of atoms,
which Besant and Leadbeater called the ‘ultimate physical atom’ (UPA), were seen to consist
of ten closed curves, or ‘whorls’ (Fig. 7). These spiral in 2½
revolutions in parallel tracks and separate at the bottom of the particle into sets of seven
and three curves, which then twist 2½ times in opposite directions about the axis of spin of
the UPA before returning to its top. Besant and Leadbeater noticed two types of UPAs: a
‘positive’ variety in which the whorls spiral downwards clockwise as observed from its top
and a ‘negative’ type in which they wind around their axis in an anticlockwise sense. Each
is the mirror image of the other. Three, so-called ‘major’ whorls appear thicker than the
remaining seven, so-called ‘minor’ whorls. The reason for this is as follows: each circular
turn in a stringy whorl is a circular helix made up of seven smaller turns spaced the same
distance apart. Each of these is another helix with seven turns, and so on. There are seven
orders of helices. Every 25 helical turns of a given order in a major whorl comprise 176
turns of the next higher order, whereas in a minor whorl they consist of 175 such turns.
This augmentation of one extra turn in every 25 of the next lower order extends throughout
the seven orders of helices in a major whorl, making it consist of more higher-order helices
and appear thicker than a minor whorl.
Each of the ten whorls was found to be essentially a circular
helix with 1680 turns or coils (Fig. 8). Leadbeater said
(6) that he checked his count of
these coils by studying 135 different UPAs, which were found to have the same number of
turns in their whorls whatever the elements in which they were found.
Statistical analysis of the UPA populations determined by
Besant and Leadbeater for all 111 of what they assumed were chemical atoms, as well as
detailed correlation of their constituent particles with predictions based upon facts about
nuclei and their quark composition, established (7) that the UPA is a constituent of
the up and down quarks making up protons and neutrons in atomic nuclei. The string-like
nature of the whorls is self-evident. In fact, were it not for the fact that the UPA
comprises ten stringy whorls,
not one whorl, its identification with what physicists call the ‘superstring’ would be just
as obvious. Superstring theory predicts that space-time has ten dimensions, so that a
microscopic, 6-dimensional space exists beyond ordinary, large-scale space. One of the
models for this space that string theorists have considered is the so-called ‘6-d torus.’
The 2-torus, or doughnut, is the surface generated when the centre of a circle moves around
another circle (1-torus) in a plane at right angles to it. The 6-torus is its 6-dimensional
version. The six higher orders of helical spirillae in each whorl represent the winding of a
closed string around successively smaller, mutually perpendicular circles, each a
1-dimensional space. In other words, Leadbeater’s description of the higher-order structure
of the UPA is consistent with this type of space. However, he described the UPA not as one
closed string but as ten closed curves. If superstrings were fundamental, he
would have observed only one closed whorl. This indicates that the current picture of
superstrings as simple loops winding around some compact, 6-dimensional space is just that —
a simplistic version of the truth. Instead, they must be derived from more general, extended
objects called ‘D-branes.’
Some string theorists have suggested that 1-dimensional
strings may result from the wrapping of D-branes around a curled-up dimension. But this
cannot be one of the six curled-up dimensions predicted by superstring theory because each
string-like whorl winds itself around all six of these circular dimensions, not five of
them. Hence space-time must have more than ten dimensions. There are five types of
superstrings, and one of them has been shown (8) to result from the wrapping of a
2-dimensional sheet (2-brane) around one of the ten spatial dimensions predicted by
supergravity theories. But this still creates only one string, whereas Leadbeater’s investigations imply
that superstrings actually consist of ten separate, closed curves. The only possibility is
for space to have more than ten dimensions, the wrapping of a D-brane around the extra
dimensions being responsible for these curves. The only candidate available is
the 26-dimensional space-time
predicted for spinless strings by quantum mechanics but rejected by physicists for many
years until the so-called ‘heterotic superstring model’ was proposed. In Article 2
(9), it was proposed that a 11-brane
(a 11-dimensional object) existing in 26-dimensional space-time wraps itself around ten of
the 15 higher, curled-up
dimensions beyond supergravity space-time, the topology of this 10-dimensional space
creating ten non-intersecting curves whose separation is an illusion because they are
simply the projection into superstring space-time of a single, higher-dimensional,
extended object. Imagine a 2-dimensional being living on a sheet. As he is unaware of the
third dimension of space, he would perceive a cylinder with thick walls that penetrated
the sheet at right angles to it as two concentric circles that would move together but
keep separate. He would have no way of knowing that they were part of one object.
Instead, he would believe that they were different objects. In the same way, the ten
whorls of the UPA exist as separate objects only in the 11-dimensional space of
supergravity space-time; they are
really part of one object that extends
into 15 higher
The author’s theory has the following consequence: just as
the position of a point in large-scale space is defined by three numbers — its spatial
co-ordinates — so a point in 25-dimensional space is located by 25 numbers. Any point on a
curve in 10-dimensional space is located by ten co-ordinates. But if the curve has been
created by a D-brane wrapping itself around the curled-up dimensions of a higher space, then
there are 15 hidden co-ordinate
variables defined for that point. Ten different curves will have (10×10=100) spatial
co-ordinate variables in supergravity space-time and (10×15=150) higher
co-ordinate variables.Including the time co-ordinate, which is common to all ten curves,
100 + 150 + 1 = 251
co-ordinate variables defining the ten curves, of
which 101 variables define them
in 11-dimensional, supergravity space-time and 150 variables remain hidden because they
refer to the space beyond this space-time.
This explains why there are 251 yods in the 1-tree with its
triangles turned into three tetractyses and why the 60 polygons enfolded in ten overlapping
Trees of Life have 251 corners, as described earlier. Each yod
or corner symbolises one of the numbers or co-ordinate variables needed to define ten
separate points in 26-dimensional space-time — the geometrical origin of the superstring as the Malkuth
manifestation of the Tree of Life blueprint. The reason why the
1-tree with single tetractyses
contains 80 yods is that each
of the ten curves that comprise the superstring has eight transverse spatial co-ordinates,
so that the superstring itself has (10×8=80) such variables or geometrical degrees of
freedom. A single Tree of Life has 70 yods. Its ten Sephirothic yods denote the ten
longitudinal coordinates of points on ten closed curves and its 60 other yods denote their
60 coordinates in the 6-dimensional, compactified space in which superstrings exist as
We saw earlier that the 251 corners of the first four
polygons enfolded in the 25-tree split up into the ten corners of the root edges in the
10-tree, the 11 uppermost and lowermost corners of the hexagons enfolded in the
10-tree, 80 external corners
and 150 corners of the 60 polygons enfolded in the 15 trees above the 10-tree. The 11 hexagonal corners
symbolise the time co-ordinate and the ten longitudinal co-ordinate variables of the ten
curves comprising the superstring. The ten corners of the root edges denote their coordinate
variables defined with respect to the tenth dimension of supergravity space-time and the 150
corners signify the 10×15 = 150 co-ordinate variables ‘hidden’ so to speak in
the ten curves because they refer to the space whose 15 dimensions beyond supergravity space-time correspond
to the 15 trees in the 25-tree
above the 10-tree. The ten independent corners in each set of four polygons symbolise the
ten curves, whist similar corners denote different co-ordinates of the same curve.
The 101 corners of the polygons
enfolded in the 10-tree denote the (10×10 + 1 = 101) space-time co-ordinate variables of the
Figure 9. The seven enfolded polygons have 251 yods outside the
root edge that are either not Sephirothic points or centres (O).
ten curves of the superstring in 11-dimensional space-time and the 150
corners of the polygons enfolded in the 15 trees of the 25-tree beyond the
10-tree symbolise the (10×15=150) co-ordinate variables of the ten curves defined
with respect to the 15-dimensional space beyond supergravity space-time.
The number 251 is encoded in the seven enfolded polygons as follows: this
set of polygons contains 260 yods outside their root edge (10). Of these, three are located
at the positions of Chokmah, Chesed and Netzach in the Tree of Life and six are centres of
the polygons ((the yod coinciding with Chesed is the centre of the hexagon). There are
therefore (260–3–6=251) yods in the seven enfolded polygons outside their root edge that are
not Sephirothic points or centres (Fig. 9).
7. Encoding of 168 as the structural
parameter of the superstring
We found in Section 2 that the first four
enfolded polygons have 90 yods outside their root edge. Of these, three are located at
Sephiroth and three are centres of these polygons. The number of their yods outside the root
edge which are not Sephirothic points or centres of these polygons = 90 – 3 – 3 = 84,
84 = 12 + 32 + 52 +
i.e., the sum of the squares of the first four odd integers,
showing how the Pythagorean Tetrad determines this number. The two sets of four polygons
therefore have (84+84=168) such yods. This is the number value of Cholem
Yesodeth, the Mundane Chakra of Malkuth. (90–3=87) yods outside the root edge
of the first four enfolded polygons do not coincide with Sephiroth. This is the number
value of Levanah, the Mundane Chakra of Yesod, which is the Sephirah next above
Malkuth in the Tree of Life. That this particular Sephirah is involved is highly
significant and yet more evidence of how information about the subatomic world is encoded
in the Tree of Life and its equivalent sections. This is because Malkuth signifies the
outer, physical form of whatever is designed according to the blueprint of the
Tree of Life. It is therefore appropriate that 168 is the kernel of the number
1680 — the number of coils in each helical whorl of the UPA described by Besant and
Leadbeater with a form of remote-viewing and proved (11) by the author to be the
superstring constituent of up and down quarks — for this number quantifies the
form of the superstring — the basic unit of physical matter. Ten
overlapping Trees of Life have 80 polygons of the first four types containing
(10×168=1680) yods that are not Sephirothic points or centres. This demonstrates that
the number 1680 is truly a parameter of the Tree of Life, for it quantifies a property of a
section of the inner form of ten Trees of Life, each a representation of a
Sephirah. In fact, as the UPA/superstring is the microphysical manifestation of the Tree of
Life blueprint, each whorl is the corresponding manifestation of a Sephirah, the three major
whorls corresponding to the Supernal Triad of Kether, Chokmah and Binah and the seven minor
whorls corresponding to the seven Sephiroth of Construction. As the microscopic
manifestation of a Sephirah, a whorl is a Tree of Life in itself, so that it is represented
by ten Trees of Life. As a circularly polarised standing wave, its 1680 oscillations are the
manifestation of the 1680 yods in the first (4+4) polygons enfolded in ten overlapping Trees
of Life other than its centres or corners that coincide with SLs of the ten trees.
The significance of the excluded yods is that they belong to the
21 yods prescribed by the Godname EHYEH that are either Sephiroth, Daath or
centres of the two sets of five independent polygons whose centres do not coincide with any
of their corners (Fig. 10). In fact, the letter values of EHYEH denote the various classes of
such yods. There are seven of these yods per set of four enfolded polygons (five outside the
root edge), so that the set of 40 polygons enfolded on each side of the central pillar of
the ten trees have 71 yods that are either Sephirothic points or centres (the significance
of this number will be revealed shortly). Seventy of these yods are intrinsic to the ten
trees because the topmost corner of the hexagon enfolded in the tenth tree coincides with
the lowest corner of the hexagon enfolded in the eleventh tree. 50 of these yods
are outside the ten root edges and 20 are their endpoints. ELOHIM, Godname of Binah with
number value 50, prescribes the yods that are either Sephirothic points or centres of
polygons (see Fig. 4). The counterpart of this
50:20 division in a single Tree of Life is the 20 yods in the tetrahedron with
corners at Netzach, Hod, Yesod & Malkuth and the 50 yods outside it. The
division, which (as later articles will
Figure 11. 168 yods in the first (4+4) enfolded polygons
and 168 yods
in the last (3+3) polygons are not centres or Sephirothic points (O).
demonstrate), is characteristic of holistic systems, appears in the formula
for the number Y(n) of yods in n overlapping Trees of Life:
Y(n) = 50n + 20.
We saw earlier that there are 251 yods outside the root edge of
the seven enfolded polygons that are not Sephirothic points or centres. 84 yods outside the
root edge of the first four enfolded polygons are not centres of
their polygons but a corner of the pentagon is the centre of
the decagon, so that there are 83 yods outside the root edge that not Sephirothic points or
centres of any of the seven polygons. There are therefore (251–83=168) yods in the
last three enfolded polygons outside their root edge that are not Sephirothic points or
centres, whilst (including the root edge) there are (84+84=168) yods in the first
(4+4) polygons that are not such. Now consider the root edges in overlapping trees. The yod
at their lower ends is at the position of Tiphareth of that tree and the yod at their upper
ends coincides with Daath, i.e., Yesod of the next higher tree. One of the two remaining
yods of the root edge may be considered to be associated with one set of polygons and the
other may be considered to be associated with the other set. This means that there are
168 yods in the first (4+4) enfolded polygons that are not Sephirothic points or
centres of any polygon. There are therefore (168+168+168=504) yods in
the (7+7) enfolded polygons that are not Sephirothic points or centres (Fig. 11). The (70+70) polygons enfolded in ten overlapping trees have
(10×504=5040) such yods. This number has the property:
5040 = 712 – 1 = 3 + 5 + 7 + …. + 141.
In other words, 5040 is the sum of the first 70 odd integers,
starting with 3. This number, which we shall shortly show is a structural parameter of
superstrings, is prescribed by ELOHA, the Godname of Geburah with number value
36 because 71 is the 36th odd integer. As a Tree of Life contains 70 yods
16 triangles are turned into tetractyses (see Fig. 4), we discover the amazing
property that the number of yods other than Sephirothic points or centres in the polygons
enfolded in ten trees is the sum of the odd integers that can be assigned to the yods in a
single Tree of Life (Fig. 12). Its Lower Face (shown
shaded) has 30 yods, the rest of the Tree of Life having 40 yods. The sum of the 40 odd
integers 3, 5, … 81 outside the Lower Face is 412 –1 = 1680, so that the sum
of the 30 integers composing the Lower Face is 5040 – 1680 = 3360 = 2×1680. Numerically,
therefore, the encoding of the number 5040 in the Tree of Life causes it to split into the
numbers 1680 and 3360. Compare this with the fact that the 5040 yods in the polygons
enfolded in 10 trees which are not Sephirothic points or centres comprise the 1680 such yods
in the first (4+4) polygons enfolded in each tree and the 3360 yods of the last (3+3)
polygons (see above). The Lower Face of the Tree of Life creates the same split
(3360+1680) as that created by the division of the seven polygons into, respectively, the
last three ones and the first four ones. Notice that this has not been concocted, for the
integers are assigned in Fig. 12 sequentially from left to right, running down the page. Also,
notice that the sum of the integers at the position of the ten Sephiroth is
49 59 83
107 97 125 141
= 700 =
70 70 70
70 70 70 70,
i.e., the sum of the Pythagorean Decad assigned to each of the 70 yods
in the Tree of Life! This exemplifies the beautiful, mathematical design of the Tree of
What the replication of the pattern of encoding of the number 5040 is
telling us (quite apart from the importance of the number itself) is that the numbers 1680,
3360 and 5040 must have significance vis-à-vis the superstring as the microphysical
actualisation of the Tree of Life blueprint. In fact, 1680 is the number of helical turns of
each whorl component of the superstring, being the number of oscillations of the circularly
polarised waves running around each closed curve. 3360 is the number of such turns per
revolution of all ten whorls (each whorl makes five revolutions, comprising 336 turns per
revolution), whilst 5040 (=3×1680) is the number of turns in the three major whorls of the
UPA. The 1680 yods in the 80 polygons of the first four types enfolded in the
ten trees symbolise the 1680 turns of the first major whorl, which corresponds to Kether in
the Tree of Life. The 1680 yods in the 30 polygons of the last three types that are enfolded
in the ten trees signify the 1680 turns of the second major whorl, which corresponds to
Chokmah. The 1680 yods that are their mirror images in the 30 polygons of the last three
types enfolded on the
Figure 13. The number (3360) of 1st-order spirillae in each revolution of
the 10 whorls of the UPA/superstring is the number of yods in the seven
enfolded polygons with 2nd-order tetractyses as their sectors. They
comprise 1680 hexagonal (black) yods of tetractyses denoting Sephiroth of
Construction inside the 40 sectors of all polygons except the hexagon, as
well as 1680 yods that either line its edges or belong to the hexagon or
tetractyses at corners of each 2nd-order tetractys.
other side of the central pillar in the ten trees correspond to the 1680
turns of the third major whorl, which corresponds to Binah. The two mirror-image halves of
the inner Tree of Life are the manifestation of the opposite polarities of Chokmah and
Binah, which have the Kabbalistic titles of Abba and Aima, the Cosmic
Father and Cosmic Mother, representing the male and female principles (what are called
‘yang’ & ‘yin’ in Taoism).
The structural parameter 5040 is embodied in the inner form of a single
Tree of Life. When the sectors of its
seven enfolded polygons are transformed into 2nd-order tetractyses, each with
the set of polygons contain 3360 yods (Fig. 13) (12). 1680 yods either lie on edges
of sectors or belong to 1st-order tetractyses at the corners of each 2nd-order tetractys.
There are 840 yods inside the sectors of the triangle, octagon & decagon and 840 yods
inside the sectors of the square, pentagon & dodecagon, each of these sets of polygons
having 21 sectors. Transformed by the 2nd-order tetractys, the seven enfolded
polygons exhibit the same 1680:1680 division of yods as found in the last (3+3) polygons
enfolded in ten overlapping Trees of Life. The same pattern occurs in the first (6+6)
polygons enfolded in ten trees (Fig. 14). Each set of six
enfolded polygons has 195 yods, of which 26 are corners, leaving 169 yods.
Associated with each set are 168 yods, so that there are (1680+1680=3360) yods
in the 120 polygons of the first six types enfolded in ten trees other than their 482
corners. Each set of 60 polygons has 251 corners that symbolise the 251 space-time
coordinates of ten independent points in 26-dimensional spacetime. The Tree
of Life parameter 251, which was found in Section 5 to be embodied in the 1-tree as its 251
yods and in Section 6 to be the number of yods outside the root edge of the seven enfolded
polygons other than Sephirothic points or centres, reappears again in a new Tree of Life
pattern as the number of corners of the 60 polygons enfolded in 10 overlapping Trees of Life
that contain 1680 yods.
The Tree of
Life has an inner form defined by its geometry and prescribed by the number values of the
ten Kabbalistic Godnames. As demonstrated in earlier articles, various sections of this
inner structure are also prescribed by the Godname numbers and encode the same set of
parameters quantifying their geometrical properties and yod populations. This article has
analysed one such section — the first four regular polygons enfolded in the outer Tree of
Life — and has proved that it encodes a number embodied in both the outer and inner forms of
the Tree of Life as the number of degrees of freedom or co-ordinate variables characterising
ten curves in the 26-dimensional space-time predicted by quantum mechanics for
spinless strings. This agrees with the century-old, paranormal description of the basic
constituent of matter by the Theosophists Annie Besant and C.W. Leadbeater and with its
interpretation by the author as the superstring constituent of up and down quarks.
Independent confirmation of this came from the appearance of the paranormally obtained
number 1680 as a natural property of both sets of the first four polygons and as
a similar property of the last three polygons. Such simultaneity cannot plausibly be due to
coincidence because it is obvious that the chance of the same number happening to appear in
two different sets of polygons making up the seven polygons is extremely small — even more
so when choice of both combinations is restricted by the number values of the ten Godnames.
What this and previous articles have presented is evidence of an enormous ‘conspiracy’ that
theologians might prefer to call ‘divine design’ whereby number and geometry join together
in the mathematical design of the cosmic blueprint called the Tree of Life and its
microscopic manifestation in space-time as the superstring. However, as we have outlined
here and as Article 5 discussed in more detail, the superstring is only the end of the
story, not its beginning……
- Phillips, Stephen M. Article 1: “The Pythagorean nature of superstring and bosonic string
theories,” (WEB, PDF).
- Phillips, Stephen M. Article 4: “Godnames prescribe the inner Tree of
Life,” (WEB, PDF).
- Occult Chemistry, Annie Besant and C.W. Leadbeater (1st ed.,
1908; 2nd ed., 1919; 3rd ed., 1951; reprinted 3rd ed., 1994).
- For a non-technical overview, See Article 2 at the author’s website.
- Extra-sensory Perception of Quarks, Stephen M. Phillips
(Theosophical Publishing House, Wheaton, Ill., U.S.A., 1980).
- Occult Chemistry, 3rd ed., p. 23.
- Ref. 5 and ESP of Quarks and Superstrings, Stephen
M. Phillips (New Age International, New Delhi, India, 1999).
- For example, see: “Superstrings in D = 10 from supermembranes in D =
11,” M.J. Duff, P.S. Howe, T. Inami and K.S. Stelle, Phys. Lett. B, vol. 191
(June 1987), 70–74, and: “Heterotic and type I string dynamics from eleven dimensions,”
P. Horava and E. Witten, Nucl. Phys. B 460 (1996), 506–524.
- Phillips, Stephen M. Article 2: “The physical plane
and its relation to the UPA/superstring and spacetime,” (WEB, PDF).
- Ref. 2, p. 2.
- Ref. 7.
2nd-order tetractys has 85 yods, of which 13
yods line each of its sides. When each of the n triangular sectors of an
n-sided, regular polygon is turned into a 2nd-order tetractys, there are
) independent yods per sector of the polygon. Its
yod population = 72
n + 1, where “1” denotes the yod at the
centre of the polygon. The polygonal form of the inner Tree of Life consists
of a triangle, square, pentagon, hexagon, octagon, decagon and dodecagon. They
are enfolded in one another and share the same base, or what the author has
called the “root edge,” as they should be thought of as growing out of this
fundamental line joining Daath and Tiphareth in the Tree of Life. When the
seven separate polygons are superposed on one another in their enfolded state,
corresponding members of the set of 13 yods forming what becomes their shared
side coincide and therefore must not be counted separately in a calculation of
their yod population. Below are listed the yod populations of each polygon and
(except for the triangle) their numbers of yods outside the root edge:
Number of yods = 72n + 1
Number of yods outside root edge
289 – 13 = 276
361 – 13 = 348
433 – 13 = 420
577 – 13 = 564
721 – 13 = 708
865 – 13 = 852
Total = 3385
Inspection of Fig. 2 reveals that the
tip of the triangle opposite the root edge is also the centre of the hexagon (the
triangle is simply a triangular sector of the hexagon).
Similarly, the tip of the pentagon is the centre of the decagon. With
2nd-order tetractyses as their sectors, the centre of the triangle where corners of its
three 2nd-order tetractyses meet is also the central yod of the tetractys at the centre
of the 2nd-order tetractys constituting a sector of the hexagon (see diagram opposite).
The 11 yods between corners on each of the two sides of the triangle outside its shared
base coincide with yods on the sides of this sector of the hexagon. There are (1 + 1 + 1
+ 2×11 = 25) yods in the total population calculated above that coincide with yods
belonging to other polygons (these are the only yods occupying the same positions). In
determining the yod population when the separate polygons are superposed, these yods
must be subtracted in order to avoid double-counting. Therefore, the yod population of
the seven enfolded polygons constructed from 2nd-order tetractyses = 3385 – 25 =