ARTICLE 15
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH.
England.
Website: http://www.smphillips.mysite.com
Abstract
In their search for Mtheory, some physicists have
recently linked the Lie algebra of E_{8}, the rank8, exceptional gauge
symmetry group governing superstring interactions, to the 8dimensional,
division algebra of octonions. This algebra can be represented by the Fano
plane, which is the simplest projection plane. Its group of automorphisms is
SL(3,2), which is isomorphic to PSL(2,7), the group of 168 automorphisms of the
Klein quartic equation. This number appears in the description of the basic
units of matter given 108 years ago by Annie Besant and C.W. Leadbeater, who
claimed to use a yogic siddhi to observe highly magnified images of atoms and
subatomic particles. 168 is also the number value of Cholem Yesodeth, the name
assigned in Kabbalah to the most physical, cosmic aspect of the Tree of Life,
which it regards as the universal blueprint governing how God manifests in all
things. This article will demonstrate remarkable connections and
correspondences between beautiful mathematical ideas, a scientific theory of
matter, paranormal descriptions of the latter, the Jewish mystical tradition
and the Pythagorean basis of music. As well as providing convincing support for
Besant’s and Leadbeater’s claim, these links give clues to the holistic nature
of the ideas that underlie Mtheory.

1
1. Introduction 121
years ago, the two Theosophists Annie Besant and Charles W. Leadbeater claimed to
undertake a series of investigations into the nature of atoms, using one of the
siddhis, or paranormal abilities, that can, according to yogic tradition, be gained
intentionally or unintentionally through meditation. Called ‘anima’ in Sanskrit and
discussed over 2000 years ago in the Yoga Sutras of Patanjali, this supposed
ability to experience highly magnified images of microscopic objects has been given
by the author the modern name of ‘micropsi.’ In parapsychological terms it is a type
of ‘remote viewing,’ although a more accurate term is ‘clear cognition,’ as the
objects being scanned need be only a few inches from the micropsi observer, who does
not need to have his eyes open because he does not ‘see’ with them. Besant &
Leadbeater published their observations of atoms in 1908 in their book Occult
Chemistry.^{1} All the material that had
accumulated over 38 years of intermittent study
finally appeared in 1951 in its third edition.
According to Besant & Leadbeater, atoms are ultimately made up of
indivisible units that they called ‘ultimate physical atoms,’ or UPAs. They had two
forms (Fig. 1). Each consisted of ten closed, nonintersecting curves that
spiralled 2½ times around the axis about which the UPA spun, maintaining parallel paths,
and then twisted 2½ times in tighter spirals upwards through the core of the UPA, each
curve completing its circuit at the top. In the ‘positive’ variety, the curves, or
‘whorls,’ spiralled clockwise when viewed from the top; the ‘negative’ type spiralled
anticlockwise. For the purpose of the present discussion, the most important details
that Leadbeater gave
2
about the UPA were:

each whorl in either type was essentially a closed, helical coil ( Fig. 2) with 1680 turns. Leadbeater meticulously checked this
number by examining 135 different UPAs. It was the same whatever the element
whose atoms he thought he was observing;

each whorl made five revolutions about the axis of the spinning UPA, twisting 2½
times in an outer spiralling movement and 2½ times in an inner circuit. This means
that a whorl coils (1680/5 = 336) times each time it completes one revolution, that
is, 168 times in every halfrevolution through 180°.
By analysing selfconsistently the many thousands of details recorded in
Occult Chemistry for 111 purported atoms, the author proved^{2} in a modelindependent way that the UPA is an as yet
undiscovered constituent of the up and down quarks making up the protons and neutrons in
atomic nuclei. He also pointed out features of the UPA that are consistent with their
interpretation as closed superstrings. This article will focus on those features listed
above in order to establish rigorous mathematical contact with grouptheoretical aspects of
the unified superstring force and their connection to octonions, the Fano plane and the
Klein Quartic, an equation wellknown to mathematicians. Through the pivotal role of the
number 168, it will establish mathematically the ten possible links between the following
five subjects:
Their validity does not depend on the invoking of dubious, metaphysical
ideas or speculations based upon some untested model or theory other than superstring theory
itself. Reference to the Jewish mystical doctrine of Kabbalah will make use only of
mathematical aspects of the Tree of Life diagram at the heart of its teachings.
This article represents work in progress and does not offer any final ‘theory of everything.’
Rather, it provides a few paving stones for the path that will lead to it.
2. Octonions
In 1878 and 1880, Frobenius and Peirce^{3} proved that the only associative^{4} real division
3
algebras are real numbers, complex numbers, and quaternions.
Adams^{5} proved that ndimensional vectors form an algebra in which
division (except by 0) is always possible only for n = 1, 2, 4, and 8. Bott and
Milnor^{6} proved that the only finitedimensional real division algebras
occur for dimensions n = 1, 2, 4, and 8. Each gives rise to an algebra with particularly
useful physical applications (which, however, is not itself necessarily nonassociative), and
these four cases correspond to real numbers, complex numbers, quaternions, and Cayley
numbers, respectively. The Cayley algebra is the only nonassociative algebra.
Hurwitz^{7} proved in 1898 that the algebras of real numbers, complex
numbers, quaternions, and Cayley numbers are the only ones where multiplication by unit
"vectors" is distancepreserving. These most general numbers are also called “octonions.”
The mathematical fact that ndimensional, division algebras are allowed only for n =
2^{0} = 1, 2^{1} = 2, 2^{2} = 4 and
2^{3} = 8 gives meaning to these powers of 2 on one slope of the Platonic
Lambda (Fig. 3). It
is a powerful example of the ‘Tetrad Principle’ formulated by the
author^{8} wherein the fourth member of a class of
mathematical object (in this case, even numbers) has fundamental significance to physics (in
this case, the relevance of octonions to superstring theory). In the musical context of
Plato’s cosmological treatise, Timaeus, the numbers of his Lambda generate the
musical proportions of the Pythagorean musical scale, successive octaves of which have
pitches 2^{0}, 2^{1}, 2^{2}, 2^{3}, etc. This demonstrates
the archetypal role played by these powers of 2, for they define not only successive musical
octaves but also the dimensions of the four possible division algebras. It intimates a
connection between the Pythagorean basis of music and octonions and therefore with
superstring theory, as was discussed in Article 13. This will be explored in Section 7
An octonion has the form:
N = a_{0}+ a_{1}e_{1} +
a_{2}e_{2} + a_{3}e_{3} +
a_{4}e_{4} + a_{5}e_{5} +
a_{6}e_{6} + a_{7}e_{7},
where the a_{i} (i = 0, 1, 2, 3, 4, 5, 6, 7) are real numbers,
the unit octonions e_{i} (i = 17) are
4
imaginary numbers:
e_{i}^{2} =
1
(i = 17)
and are anticommutative:
e_{i} e_{j} = e_{j}e_{i}
(i ≠ j)
Shown below is the multiplication table for the 8tuple of
unit octonions: (1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5},
e_{6}, e_{7}):
It comprises (8×8=64=4^{3}) entries, of which
eight are diagonal and real (8 = fourth even integer) and (64–8=56) are offdiagonal and
imaginary (28 on one side of the diagonal and 28 on the other side with opposite signs due to
their anticommutativity). 28 is the seventh triangular number, where 7 is the fourth odd
integer and the fourth prime number. There are 36 entries with a positive sign (one real and 35
imaginary) and 28 entries with a negative sign (7 real and 21 imaginary). 36 is the sum of the
first four even integers and the first four odd integers:
36 = (2+4+6+8) + (1+3+5+7).
This illustrates how the Pythagorean Tetrad (4) defines
the properties of the octonion multiplication table. The 35 positive imaginary entries consist
of five copies of each of the seven imaginary unit octonions and the 21 negative entries
comprise three copies of these, that is, the multiplication table contains (ignoring their
signs) eight copies of each imaginary octonion. Multiplication yields seven new copies, that
is, (7×7=49) imaginary octonions made up of 28 positive in four sets and 21 negative in three
sets. Multiplication also generates seven, new, identical, negative numbers 1, so the 56 new
entries comprise equal numbers with positive and negative signs (four positive sets
of
5
seven and (3+1) negative sets).
As the realisation of the most general type of numbers
showing neither associativity nor commutativity, the eight unit octonions are the
mathematical counterpart in the Tree of Life of Daath and the seven Sephiroth of
Construction:
The differentiation between the nonSephirah Daath and the Sephiroth of
Construction corresponds to the distinction between the real unit octonion 1 and the seven
imaginary octonions. The 1:3:4 pattern of new numbers created by their multiplication in
pairs (see
previous paragraph) corresponds, respectively, to Daath, the triangular
array of Chesed, Geburah and Tiphareth and the quartet of
6
Netzach, Hod, Yesod and Malkuth situated at the corners of the
tetrahedron at the base of the Tree of Life (Fig. 4).
The multiplication table for the seven, imaginary, unit octonions
(e_{1}e_{7}) (Table 2) has (7×7=49) entries, of which 42 are imaginary (21 in three
positive copies and 21 in three negative copies of the set) and seven are real (1). As
(e_{i} )×(e_{j} ) = (e_{i} )×(e_{j} ),
the multiplication table for the negative, imaginary unit octonions is the same as Table 2. This 7:42 pattern conforms to the tetractys, the Pythagorean
archetypal pattern of wholeness, for the following reason: the set
(e_{1}e_{7}) can be assigned to what the author calls the seven ‘hexagonal
yods’* of the tetractys (Fig. 5), the sets (e_{1}, e_{2}, e_{3} ) and
(e_{4}, e_{5}, e_{6})
_______________________________
* Socalled because they are located in the tetractys at the corners and
centre of a hexagon.
7
being located at the corners of the two intersecting equilateral triangles
forming a Star of David, whilst e_{7} is at the centre of the tetractys because
it corresponds to Malkuth, which the rules of correspondence between the Tree of Life and
the tetractys require to be the central yod of the tetractys. The tetractys of the next
higher order (Fig. 6) contains yods, of which 15 (the fourth triangular number after
85 = 4^{0} + 4^{1} + 4^{2} +
4^{3}
1 and the sum of the first four powers of 2 in the Lambda) are corners of
tetractyses and (85–15=70) are hexagonal yods. The seven hexagonal yods of the tetractys now
become (7×7=49) such yods belonging to their corresponding seven tetractyses. The 49 entries
of the multiplication table for the seven imaginary unit octonions can be assigned to them
in a way consistent with their 7:42 pattern. The set (1,1,1,1,1,1,1) corresponds to
the central set of seven hexagonal yods, the three sets
(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e
_{7}) correspond to one triangular array of seven yods and the three sets
(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e
_{7}) correspond to the other triangular array of seven yods. The original set of
seven octonions, represented by the Pythagorean symbol of wholeness, yields on
multiplication another pattern that is really just a more differentiated form of the latter:
unity remains unity as new levels of complexity emerge.
3. Fano projective
plane
In mathematics, a projective plane consists of a set of "lines" and a set of
"points" with the following properties:
Given any two distinct points, there is exactly one line incident with
both of them.
Given any two distinct lines, there is exactly one point incident with both of them.
There are four points such that no line is incident with more than two of them.
The last condition simply excludes some degenerate cases.
A projective plane is an abstract mathematical concept, so the "lines" need
not be anything resembling ordinary lines, nor need the "points" resemble ordinary
points.
It can be shown that a projective plane has the same number of lines as it
has points. This number can be infinite (as for the real projective plane) or finite. A
finite
8
projective plane has (n^{2}+n+1) points, where n is an integer
called the order of the projective plane. A projective plane of order n has
(n+1) points on every line, and (n+1) lines passing through every point. For all known
finite projective planes, the order is a prime power. The existence of finite projective
planes of other orders is an open question. The smallest possible projective plane has only
seven
points and seven lines. It is often called the “Fano plane,” and is shown in
Figure 7. The seven points are shown as small blobs, and the seven lines are
shown as six line segments and a circle. However, we could equally consider the blobs to be
the "lines" and the line segments and circle to be the "points" — this is an example of the
duality of projective planes: if the lines and points are interchanged,
the result is still a projective plane. The Fano plane with seven points
therefore has order 2 because 2^{2} + 2 + 1 = 7.
The Fano plane is known to generate the multiplication table for the seven
imaginary
9
unit octonions if they are assigned to its points in the way shown in
Figure 8. The multiplication of two units is the third on the same straight
or curved line provided that their multiplicative ordering follows the arrow on the line.
Their product is minus the third if multiplication is opposite to the sense of the arrow.
For example, e_{5}e_{2} = e_{3} and
e_{1}e_{2} = e_{4}. But e_{6}e_{4} =
e_{4} and e_{1}e_{4} = e_{2}.
Having pointed out earlier that a fundamental correspondence exists between
the seven imaginary octonions and the seven Sephiroth of Construction of the Tree of Life,
it should come as no surprise that the geometry of the Fano plane representing the algebra
of octonions is implicit in the construction of the Tree of Life from identical circles
overlapping centre to circumference (Fig. 9). They are inscribed by the same circle. It is a simple exercise in
trigonometry to prove that the straight line passing through the two points of intersection
of the black and red overlapping circles shown in Figure 9 meets the larger circle at a point such that the middle point
divides the straight line in the Golden Ratio 1:φ = 1: 1.618… . The path joining Chesed to
Geburah is the base of a Golden Rectangle whose sides are collinear with the Pillars of
Mercy and Judgement and extend slightly above the path joining Chokmah to Binah. The
dashed lines in Figure 9 are the upper side of the Golden Rectangle and the sides of the
squares adjoining its vertical side. It is of great significance to the relevance of
octonions to elementary particle physics that the Golden Ratio, which appears in the growth
patterns of many living things, should be so naturally present in the geometry of the
representation of these numbers. It is evidence that, as the fourth and last class of
division algebras, octonions are nature’s numbers.
Each line in the Fano plane joins three octonions that form a 3tuple
(e_{i} , e_{i}+_{1}, e_{i}+_{3}), where the
index is defined modulo 7. Each of these seven 3tuples (Fig. 10) is a quaternionic triple, obeying the same algebra as the three
quaternions I, j & k:
{i, j} = {j, k} = {k, i} = 0
and ij = k, jk = I and ki = j, where i^{2} =
j^{2} = k^{2} = 1. A 3tuple has seven combinations:
A. 
e_{i}, e_{i+1},
e_{i+3} 

B. 
e_{i}e_{i+1},
e_{i}+1e_{i+3}, e_{i+3}e_{i} 
(6 permutations) 
C. 
e_{i}e_{i+1}e_{i+3} 
(6 permutations) 
10
Associated with each 3tuple are (3+6+6=15) elements that are made up of the
three original octonions (A), one copy of them and their negatives (B) arising from
anticommutativity of pairs, three 1’s and three 1’s (C), that is, 12 new elements. The
seven 3tuples have (7×7=49) combinations of their 21 elements, among which are three of
each octonion from all the 3tuples, that is, two new copies of the seven octonions, so that
there are really (49 – 2×7 = 35) distinct combinations. Seven of these are the octonions
themselves and 28 are new combinations (three pairs and one triple for each
3tuple) generating (7×12=84) new elements made up of 42 imaginary numbers and 42 real ones,
or alternatively, 42 positive elements (21 real, 21 imaginary) and 42 negative elements (21
real, 21 imaginary). They are made up of 16 distinct elements: (±1, ±e_{1},
±e_{2}, ±e_{3}, ±e_{4}, ±e_{5}, ±e_{6},
±e_{7}).
As
e_{i}_{+}_{3}e_{i}_{+}_{1} =
e_{i} =
(e_{i}_{+}_{3})(e_{i}_{+}_{1}),
(e_{i}_{+}_{3}, e_{i}_{+}_{1},
e_{i} ) is a 3tuple. This means that there are 35 distinct combinations:
A. 
(e_{1}), (e_{2}),(e_{3}), …
(e_{7}) 

B. 
(e_{1})(e_{2}),
(e_{2})(e_{4}), etc 
(7×6 = 42 permutations) 
C. 
(e_{1})(e_{2})(e_{4}),
(e_{2})(e_{4})(e_{1}), etc 
(7×6 = 42 permutations) 
with (42+42=84) permutations of two or three octonions, and 35 distinct
combinations:
A. 
(e_{1}), (e_{2}), …
(e_{7}) 

B. 
(e_{4})(e_{2}),
(e_{2})(e_{1}), etc 
(7×6 = 42 permutations) 
C. 
(e_{4})(e_{2})(e_{1}),
(e_{2})(e_{1})(e_{4}), etc 
(7×6 = 42 permutations) 
with (42+42=84) permutations of two or three negative octonions. There are
(35+35=70)
11
combinations of one, two and three octonions from the two sets of seven
3tuples with (84+84=168) possible orderings of the pairs and triplets in them. As
1/e_{i} = e_{i} , each 3 tuple in one set is the inversion of
its counterpart in the other. Both set of results may be summarised by saying that there are
84 permutations of pairs and triples of octonions obeying
e_{i} e_{j} = e_{k} and 84 such permutations for the
inverse octonions obeying (1/e_{i} )(1/e_{j} ) = 1/e_{k},
that is, e_{i} e_{j} = e_{k}, where the indices i, j
& k are in cyclic order. The seven 3tuples and their inverses display (84+84=168)
permutations of their members. The octonion algebra can be represented by the Fano plane
because SL(3,2) the simple group of automorphisms (incidencepreserving bijections) of the
latter is of order 168.
Figure 11 shows the isomorphism between the outer and inner forms of
the Tree of Life. Constructed from 16 triangles, the outer form contains 70 yods when these
triangles are turned into tetractyses. 35 () yods belong to the sequence of the first four simplexes: point,
line, triangle and tetrahedron that constitute the ‘trunk’ of the Tree of Life, leaving 35
() yods belonging to the 11 other triangles. The inner form of the Tree of
Life is two identical sets of seven regular polygons: triangle, square, pentagon, hexagon,
octagon, decagon and dodecagon. They are enfolded in one another and share the same ‘root
edge.’ The 14 polygons possess 70 corners, 35 of them being associated with one set of
seven polygons and the remaining 35 corners being associated with the mirror image set of
seven polygons on the other side of the root edge. We see that both the outer and inner
forms of the Tree of Life have 70 degrees of freedom represented by yods or corners of
polygons and that these are divided up into two sets of 35. This 35:35 division
corresponds to the octonions in the seven 3tuples having 35 combinations and their
inverses also having 35 combinations. Once more, the analogy
is evidence that the mathematics of octonions is an intrinsic
aspect of the cosmic blueprint of the Tree of Life.
12
The same 35:35 division is displayed by the first six polygons, each set of
which has 35 triangular sectors (Fig. 12). When the latter are converted into tetractyses, one finds that there are 168 yods on
their boundaries outside their root edge, 84 creating the edges of 35 tetractyses on one
side and 35 forming the edges on the other side. So the 12 polygons with (35+35) sectors
have (84+84) yods on their edges outside their shared edge. The tetractyses in one set of
polygons symbolise distinct combinations of octonions in the seven 3tuples, their 84
boundary yods signifying the possible permutations of two and three octonions, whilst the
tetractyses in the mirror image set of regular polygons represent combinations of the
seven inverse (negative) octonions, all their 84 boundary yods denoting corresponding
permutations of two and three such octonions.
Each 3tuple (e_{i}, e_{i}_{+}_{1},
e_{i}_{+}_{3}) forms three pairs with six permutations and one
triplet with six permutations, i.e., a total of 12 permutations. Similarly for each inverse
3tuple (e_{i}_{+}_{3}, e_{i}_{+}_{1},
e_{i}). A 3tuple and its inverse provide (12+12=24) permutations. All seven
3tuples and their inverses create (7×24=168) permutations of two and three octonions. That
this number includes permutations of the inverses of imaginary unit octonions
reflects the fact that every automorphism of the Fano plane must have its own inverse
because PSL(3,2) is a group. As will be discussed later, the factorisation 7×24 is of
fundamental significance to the structure of superstrings.
According to Kabbalah, everything is the coming into being of the perfection
of Adam Kadmon, or ‘Heavenly Man,’ as symbolised by the ten ‘divine qualities,’ or
‘Sephiroth,’
13
of the Tree of Life. There are four great stages in the physical realisation
of this universal blueprint. They are the ‘worlds’ of Atziluth (archetypal level embodied in
the divine names), Beriah (archangelic level), Yetzirah (angelic hierarchies) and Assiyah
(physical universe). Every Sephirah exists at each level as the Godname, Archangel, Angelic
Order and Mundane Chakra, which is its cosmic, physical manifestation. The Mundane Chakra of
Malkuth, the outer, physical form of Adam Kadmon, is Cholem Yesodeth (“breaker
of the foundations”), traditionally referred to as the four elements of earth, water, air
and fire. In the ancient practice of gematria, numbers were assigned to letters of the
alphabet in order to identify words or phrases in religious texts that have hidden
connections because they have the same number values. Table 3 lists the 27 letters of the Hebrew alphabet and their associated
numbers. The
sum of the values of the letters of Cholem Yesodeth is 168
(Fig. 13). Had this been any of the 39 other number values of the Godnames,
Archangelic Names, Angelic Names and Mundane Chakras of the ten Sephiroth, its appearance
might plausibly be attributed to chance. But the Mundane Chakra of Malkuth signifies the
most physical, cosmic aspect of the Tree of Life. So a number that characterises the Fano
plane representation of the
14
algebra of octonions and which may be related to the group theory of
superstrings is found to be the gematria number value of the Kabbalistic term for the most
appropriate level of the most appropriate Sephirah!
Confirmation that this is not coincidental but indicative of the
transcendental truths contained in Kabbalah is the fact that the sum of the letter values of
Malkuth is 496 (Fig. 14). This is the very number discovered in 1984 by physicists Michael
Green and John Schwarz to be the number of nonabelian gauge fields transmitting the unified
force between 10dimensional superstrings if it is free of quantum anomalies^{9}. Again, it is not merely that this number appears among the
number values of the Kabbalistic words for the manifestations of the ten Sephiroth in each
of the four Kabbalistic worlds. Given that there are 40 such numbers, this, arguably, could
be coincidental on purely statistical grounds. What makes the presence of the number 496 so
highly significant is that this dynamical parameter of
superstrings refers in the context of physics to the most appropriate Sephirah, namely,
Malkuth — the physical universe of subatomic particles and their forces. Had it been the
number value of any other Sephirah or of its aspect in one of the four worlds, it would
have been difficult to argue the case for its appearance being noncoincidental. The
joint association of 496 and 168, which we pointed out in Section 1 is the
structural parameter of superstrings, with the most apt Sephirah is amazing evidence for
the mystical doctrine of Kabbalah possessing scientific truths.
As each 3tuple obeys the multiplication rule:
e_{i}e_{i}_{+}_{1} =
e_{i}_{+}_{3},
the seven imaginary, unit octonions, when assigned to the corners of a
heptagon, form
15
seven triangles whose corners are the members of each 3tuple (Fig. 15). Successive clockwise rotations of a triangle by 2π/7 turn it into
triangles representing successive
16
3tuples. The fourth (green) triangle is the turning point in the 7fold
cycle of rotation because the last octonion e_{7} is at its corner and further
rotation generates triangles that include the octonions that started the cycle.
Divided into tetractyses, a heptagon contains 42 yods surrounding its centre
(Fig. 16). In other words, starting from unity symbolised by the central yod,
42 ‘bits of information’ and degrees of freedom express the 7fold differentiation of unity.
Each tetractys represents a 3tuple and the six yods per tetractys symbolise the six
permutations of products of imaginary unit octonions within each 3tuple. With each sector
divided into three tetractyses, a heptagon contains 106 yods,* that is, 15 per sector surround its centre. We saw earlier that
multiplication of the octonions in each 3tuple generates 12 permutations of pairs and
triplets, so that a 3tuple can be associated with (12+3=15)
___________________________________________________
* An nsided regular polygon with its n sectors divided into three
tetractyses has 15n + 1 yods.
17
elements. The seven 3tuples therefore possess (7×15=105) elements. Amongst
these are three copies of each octonion. Assigning the real octonion 1 to the central yod,
Figure 17 demonstrates that an isomorphism exists between the yods
within each sector of the heptagon and the octonions within each 3tuple and their
permutations. The octonion indices 1, 2, 3, 4, 5, 6, 7 follow the sequence of colours of the
rainbow, starting with red. To every one of the 91 hexagonal yods in the heptagon there
corresponds an octonion, a pair of octonions or a triplet of octonions. The 49 coloured yods
denote singles or pairs and the 42 white yods denote triplets.
Work by the author reported in previous articles proved that the inner form
of the Tree of Life encodes the selfreplication of the latter to span 91 trees. They
comprise the 49 trees that map the 49 subplanes of the cosmic physical plane and the 42
trees that map the 42 subplanes of the six cosmic superphysical planes. Figure 16 is their tetractys representation, coloured triangles
signifying subplanes. A remarkable parallelism exists between this 49:42 pattern and the
49:42 pattern of octonions and their pairs and triplets:
It exists because the seven imaginary unit octonions are the algebraic
counterpart of the seven Sephiroth of Construction, which define the seven planes of
consciousness and their sevenfold division into subplanes. Thought by physicists
untilrecently to have no relevance to subatomic particle physics, these constructs
of the largest possible division algebra are the mathematical
realisation of the sevenfold objective nature of God. Their grouping
into seven 3tuples represented by the Fano plane is another manifestation of the seven
Divine qualities. We have shown how 168 — the kernel of the number 1680 — the number of
circularly polarised oscillations that Leadbeater counted in a whorl of what the author
interpreted as a superstring constituent of quarks — is the order of the symmetry group
SL(3,2) of the Fano plane representing the imaginary octonions. What, if at all, is the
octonion basis of 1680 itself? Imagine the number 1, the real octonion, at a point
directly above the centre of a heptagon to whose
18
corners the seven imaginary octonions are assigned. It is shown in Figure 19 connected by dotted lines to these corners. The number of
combinations of four octonions from the set of eight is [^{8}_{4}] = 70.
They comprise [^{7}_{3}] = 35 combinations of three imaginary octonions and
the real octonion 1 at the corners of a tetrahedron, as well as [^{7}_{4}] =
35 combinations of four imaginary octonions at the corners of a quadrilateral in the plane
of the heptagon. Each combination of four octonions has (4!=24) permutations. The 70
combinations of four octonions have (70×24=1680) permutations. In terms of graph theory,
there are 1680 ordered 3simplexes made up of (35×24=840) ordered simplexes in the plane of
the heptagon and 840 nonplanar, ordered 3simplexes. In fact, there are 70 3simplexes
in a
7simplex made up of eight points and therefore 1680 ordered 3simplexes.
Notice the appearance once again of the number 70 and its split into two 35’s, one providing
840 permutations of the set (1, e_{i}, e_{j}, e_{k}) (i ≠ j ≠ k) and
840 permutations of the set (e_{i}, e_{j}, e_{k}, e_{l}) (i
≠j ≠k≠l), just as we found earlier that the seven 3tuples have 35 distinct combinations
with 84 different permutations and the seven 3tuples made up of the negative octonions have
35 combinations with 84 permutations. But now the inclusion of the real octonion 1 and the
consideration of permutations of any three or four octonions instead of just the seven
3tuples have replaced the factor of 7 by 7×10. In other words, there are five times as many
combinations of three octonions as 3tuples and five times as many combinations of four
octonions. Compare these results with the following details about the UPA:
 each whorl has 1680 turns;
19
 each whorl makes five revolutions, 2½ times in an outer spiral and 2½
times in a tighter, inner spiral;
 a whorl makes 1680/5 = 336 turns in one revolution, that is, 168 turns
in half a revolution;
 a whorl makes 840 turns in its outer 2½ revolutions and 840 turns in its
inner 2½ revolutions.
Each helical turn is therefore the spacetime manifestation of a permutation
of four octonions. Permutations including the real octonion result in either the outer or
the inner revolutions. Which one can be determined by deeper analysis that will be presented
elsewhere. The analysis presented here is obviously not a complete explanation of the
spatial structure of the UPA. It serves, however, to demonstrate how the octonions have the
potential to provide a natural explanation for the micropsi description of the UPA, as,
indeed, one would expect if they are nature’s numbers. It may be argued that, because
this number might have explanation in terms of not octonions but another set
of eight mathematical objects. But this misses the point, which is that the Fano plane
representing the algebra of octonions has 168 automorphisms, which makes it implausible in
the extreme that it is merely coincidental that Leadbeater counted 1680 turns in each whorl
of the UPA, given the established connection between the Lie algebra of
E_{8} and octonions and the evidence that the UPA is a superstring. Notice also
that the 168element group PSL(2,7), which is isomorphic to SL(3,2), is the central quotient
group of SL(2,7), the 336element group of 2×2 matrices with determinant 1 whose entries are
elements of the finite cyclic group Z_{7} of order 7. This will be discussed in
the next section. Compare this with the fact that each revolution of a whorl has 336 turns.
So it is not one but two numbers implicit in Leadbeater’s description whose
connection to octonions would have to be coincidental. This is even more improbable.
Finally, the fact that 168 is also, aptly, the number value of the most physical
manifestation of a Sephirah — the Mundane Chakra of Malkuth — would have to be a matter of
chance as well! The reader must decide whether the highly improbable possibility of
coincidence is more acceptable an explanation than what may be just as hard to believe,
namely that over a century ago Leadbeater described with micropsi features of subatomic
particles that conform qualitatively not only to the general picture of superstrings but
also quantitatively to mathematical concepts that are beginning to be
20
seen by some physicists to underlie Mtheory.
4. Encoding of 168 in Tree of
Life
We found in Section 3 that the Fano plane representation of the octonions
implied the factorisation:
168 = 7×24,
where 7 is the number of 3tuples and 24 is the number of permutations of
pairs and triplets within each 3tuple and their negatives. This may be expressed more
precisely as:
168 = 7×12 + 7*×12*,
where ‘7’ is the number of 3tuples (e_{i},
e_{i}_{+}_{1}, e_{i}_{+}_{3}), 7* is the
number of 3tuples (e_{i}, e_{i}_{+}_{1},
e_{i}_{+}_{3}) and 12 and 12* are their respective numbers of
permutations. Therefore,
336 = 2×168 = 14×12 + 14*×12*,
where 14* = 2×7*. This factorisation of 336 is encoded in the inner form of
the Tree of
21
Life in the following way: the dodecagon is the seventh and last of the
regular polygons making up this inner form (see Fig. 11). Transformation of the 12 sectors of each dodecagon into three
tetractyses generates 181 yods, that is, 169 yods other than their corners. There are
therefore 168 yods associated with each dodecagon sharing one side, 14 per sector* (Fig. 21). This is the
factorisation given above. Hidden within the pair of dodecagons sharing one
side are 336 new degrees of freedom revealed by their construction from tetractyses —
the template that builds sacred geometry.
That this is not due to chance is proved by the construction of all seven
enfolded polygons from the higherorder tetractys shown in Figure 6. This is what the author has called the “2ndorder tetractys,” the
1storder one being the wellknown tetractys and the 0thorder tetractys being the
mathematical point. When each of their 48 sectors are turned into this type of tetractys,
one finds that they contain 3360 yods, that is, the number of yods in 336 tetractyses
(Fig. 22). The very order of SL(3,2) or PSL(2,7) is embodied in the sacred
geometry of the inner form of the Tree of Life! Moreover, as the UPA is made up of 1680
turns in each of its ten helical whorls, which revolve five times around its axis, the
number of turns in each revolution of all ten whorls is 10×1680/5 = 3360. What this is
telling us is that one revolution of the UPA constitutes a whole in
_________________________________________
* Of the 15n + 1 yods in an nsided polygon with its sectors divided
into three tetractyses, n are corners, leaving 14n + 1 yods, 14 per sector surrounding its
centre.
22
some sense that is repeated five times but that this whole is itself a
doubling of a basic pattern characterised by the number 168, itself quantifying another
whole. In terms of group theory, this is simply the fact that SL(2,7) is the double cover of
PSL(2,7), which is isomorphic to SL(2,3), the group of automorphisms of the Fano plane.
We found earlier that there are 1680 permutations of the 70 sets of four
octonions that can be selected from the eight octonions, each set having 24 permutations.
The factorisation:
1680 = 70×24
is geometrically realised in the pair of dodecagons. The 2ndorder tetractys
contains 85 yods, where
85 = 4^{0} + 4^{1} + 4^{2} +
4^{3}.
Surrounding its centre are
84 = 1^{2} + 3^{2} + 5^{2} +
7^{2}
yods. We saw earlier that this is the number of permutations of octonions
belonging to
23
the seven 3tuples. As 13 yods lie along each edge of a 2ndorder tetractys,
there are (71n+1) yods in an nsided polygon surrounding its sectors when they are turned
into 2ndorder tetractyses, that is, (71n+1–13=71n–12) such yods outside one edge. A
dodecagon (n=12) has 840 such yods and two dodecagons sharing one side have (2×840=1680)
yods outside it surrounding the centres of their 24 sectors (Fig. 23). Once again the number 1680 is found to be associated with the
number 24, 70 being the average number of yods per sector surrounding its centre (average
only because counting yods outside the shared edge means that the two sectors sharing it do
not have the same number of yods as the 22 other ones). Another association of the number
168 with the number 24 will be discussed in the next section.
5. The Klein
Quartic
The mathematician Felix Christian Klein (1849–1925) is wellknown to
mathematicians for his work in group theory, function theory, nonEuclidean geometry and on
the connections between group theory and geometry. 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
has the 336fold symmetry of the group SL(2,7). This curve is known to
mathematicians as the “Klein Quartic.” It is a specific occurrence of a quartic curve, whose
general form for two variables x & y is
Ax^{4} + By^{4} + Cx^{3}y +
Dx^{2}y^{2} + Exy^{3} + Fx^{3} +
Gy^{3} + Hx^{2}y + Ixy^{2} + Jx^{2} +
Ky^{2} + Lxy + Mx + Ny + O = 0,
where the letters are real numbers. The “Klein surface” is the Riemann
surface of the Klein quartic (Fig. 24). Klein showed that it is mapped onto itself (hence “automorphisms”)
by 168 analytic transformations. Schwarz^{10} proved at the end of the nineteenth century that the
automorphism group of a compact Riemann surface of
24
genus g≥ 2 is finite. Hurwitz^{11} showed that this surface has at most 84(g1) automorphism
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. Accola and Maclachlan^{12} found a lower bound for the number μ(g) of automorphisms
for a surface of genus g:
8g + 8 ≤ μ(g) ≤ 84(g1)
More recently, Belolipetsky and Jones^{13} showed that, for every g≥2, there is a compact arithmetic
Riemann surface of genus g with at least (4g1) automorphisms, the least value of g
attaining the lower bound being g = 24.
Macbeath showed that there is an infinite number of Hurwitz curves.^{14} No curves other than the Klein curve with g = 3 and one
with genus 7 have equations known to mathematicians.^{15} The Riemann surface of the Klein Quartic. It can also be
represented by the Klein Configuration (Fig. 25). It has 168 coloured, hyperbolic triangles and 168 grey triangles.
Each of the 14 slices shown numbered has 12 coloured triangles. They can be grouped in
groups of seven, each group forming one of 24 heptagons:
25
 three heptagons are coloured cyan — the central heptagon
(1) and two
(2) composed
of the 14 outermost, cyan triangles;
 seven red heptagons (7) ;
 seven yellow heptagons (sets of four (4) and three
(3)
triangles in each sector);
 seven green heptagons (sets of four (4) and three (3) triangles in each
sector);

This composition can be represented as:
168 coloured triangles = 7×24,
where
24 = (1+2) + (4+3) +
(4+3) + (4+3).

The numbers within each bracket denote the triangles in the two slices
of each sector of the Klein configuration. Each sector is made up of two
different slices, each with 12 triangles, i.e., the 168 triangles can be
sorted into 84 from one type of slice and 84 from the other type. This 84:84 division is
the same as that shown in Figure 12 to be embodied in the first six polygons of the inner form of the Tree of Life.
It is mirrors the 84 permutations in the seven 3tuples, 12 per 3tuple, and the 84
permutations of the negative octonions, 12 per 3tuple. In other words, the triangles
correspond to permutations of octonions, the two slices of each of the seven sectors
corresponding to a 3tuple and the 3tuple of its negative octonions. An exact parallel
exists between the permutation properties of the octonions and the automorphisms of the
Riemann surface of the Klein Quartic. This is, of course, simply because the former
belong to the group SL(2,3) and the latter belong to PSL(2,7), which is isomorphic to
SL(2,3).
Because the Klein Quartic is a Riemann surface of genus 3, it can be
realised as a regular map in an orientated 2manifold of genus 3, i.e., the 3torus
(Fig. 26), either with 24 heptagons, three meeting at each of its 56
vertices, or with 56 triangles, seven meeting at each of its 24 vertices. According to
the Heawood Conjecture, the maximum number of colours sufficient to colour a map on a
surface of genus g is
where denotes the floor function.* Ringels and Young proved in 1968^{16} that this
______________________________
* The floor function is the largest integer less
than or equal to x.
26
number is also sufficient except for the sphere (and plane) and the Klein
Bottle. Nine colours are needed to colour maps on the 3torus.
6. Connection between Klein
Configuration and E_{8}
Let us now examine the 14 slices of the Klein Configuration in more detail
and compare the composition of coloured triangles with the root structure of the Lie algebra
of the superstring gauge symmetry group E_{8}. The 168 automorphisms of the Klein
Quartic represented by the 168 coloured hyperbolic triangles of the Klein configuration
are:
168 = 7×24 = 7×[(1+2) + (4+3) + (4+3) + (4+3)]
= 7(4+4+4) + 7(3+3+2) + 7(3+1)

= (28+28+28) + (56+28) = 84+84.
Seven slices of one type have 84 triangles made up of 28 red, 28 yellow
and 28 green triangles and the seven slices of the other type have 84 triangles made up
of 21 yellow, 21 green, 21 red and 21 cyan triangles. The roots of the
E_{8} algebra can be described in terms of eight orthonormal unit vectors
{u_{i}}. Eight zero roots correspond to points at the centre of the root diagram
and 240 nonzero roots all have length √2. They are given by
The 240 nonzero roots of E_{8} comprise 168 made up of four
sets of 28 and one set of 56, one set of 70 and two single ones. There are as many sets of
28 and 56 in the 168
27
nonzero roots as there are groups of 28 and 56 coloured triangles in the
Klein configuration. This is not a coincidence, for the 168 automorphisms of the Klein
Quartic are in onetoone correspondence with the group of 168 automorphisms of the Fano
plane that represents the 8dimensional octonion algebra, which in turn is connected to the
8dimensional Lie algebra of E_{8}. It, too, may not be coincidental that
E_{8} has 168 more nonzero roots than E_{6} (240 compared with
72) because E_{6} is an exceptional subgroup of E_{8} that has
been shown to accommodate the Standard Model of particles and their interactions other than
gravity.
7. The octonions & Pythagorean
octaves
Compare the eight notes of the Pythagorean musical scale:
C

D

E

F

G

A

B

C'

1

9/8

(9/8)^{2}

4/3

3/2

27/16

243/128

2

with the eight unit octonions:
1

e_{1}

e_{2}

e_{3}

e_{4}

e_{5}

e_{6}

e_{7}

Although it is tempting to make the correspondence between the tonic with
tone ratio 1 and the real octonion 1, this is incorrect because the first seven notes
correspond to the seven Sephiroth of Construction,^{17} the octave C' being a musical repetition of the
Table 4. Table of tone ratios of the notes in the seven church modes.
Lydian 
1

9/8

81/64

729/512

3/2

27/16

243/128

Phrygian 
1

256/243

32/27

4/3

3/2

128/81

16//9

Dorian 
1

9/8

32/27

4/3

3/2

27/16

16/9

Hypolydian 
1

9/8

81/64

4/3

3/2

27/16

243/128

Hypophrygian 
1

256/243

32/27

4/3

1024/729

128/81

16/9

Hypodorian

1

9/8

32/27

4/3

3/2

128/81

16/9

Mixolydian 
1

9/8

81/64

4/3

3/2

27/16

16/9

sequence that they underlie, whilst, as was pointed out earlier, the real
octonion corresponds to Daath in the Tree of Life and the seven imaginary octonions
correspond to the seven Sephiroth of Construction. Hence, note C corresponds to
e_{1}, D corresponds to e_{2}, etc. Each type of church mode^{18} has five tone intervals and two leimmas. They are
generated^{19} by starting with successive intervals of the Pythagorean scale
until one completes the cycle of seven intervals by returning to the first sequence:
28
1. 
T 
T 
L 
T 
T 
T 
L 






2. 

T 
L 
T 
T 
T 
L 
T 





3. 


L 
T 
T 
T 
L 
T 
T 




4. 



T 
T 
T 
L 
T 
T 
L 



5. 




T 
T 
L 
T 
T 
L 
T 


6. 





T 
L 
T 
T 
L 
T 
T 

7. 






L 
T 
T 
L 
T 
T 
T, 
where the Pythagorean whole tone ratio T = 9/8 and the Pythagorean leimma L
= 256/243. Table 4 shows the tone ratios of the (7×7=49) notes of the seven modes.
Similarly, successive rotation through 2π/7 of the triangles shown in Figure 15 moves through the seven 3tuples, creating (7×7=49)
elements:
Just as each mode has eight notes separated by seven intervals, i.e., it
comprises 15 notes and intervals, so each 3tuple is associated with three octonions, six
permutations of pairs of them and six permutations of all three, i.e., 15 elements. The
3tuples can be regarded as the octonion counterpart of the seven Greek musical modes, the
seven combinations of octonions within each 3tuple being
analogous to the seven notes of its corresponding mode. Figure 27 compares the seven notes of the Pythagorean musical scale and
the seven imaginary octonions.
We saw earlier that the Klein Configuration factorises the 168 automorphisms
of the
29
Klein Quartic as 7×24. We also saw that the 8tuple (1, e_{1},
e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7})
has 70 combinations of four octonions with (70×24=1680) permutations. The number 24 appears
in the Pythagorean musical scale as the tenth overtone (Table 5).
The Pythagorean significance of the number 10 is that it is the fourth
triangular number — the tetractys.
This overtone is also the 33rd tone, where
30
33 = 1! + 2! + 3! + 4! = 33,
that is, it is the sum of the permutations of 10 objects arranged in the
pattern of a tetractys. In other words, starting from the number 1, the sequence of
successive integers: 1, 12, 123, 1234 has 33 permutations such that the 33rd one 1234 for which the integers are monotonically ascending has the value
24 when the integers are multiplied by one another. That this is not just a coincidence
but an indication of deep connections between the Pythagorean basis of both music and
superstrings is the fact that the ten overtones and 22 fractional tone ratios up to the
perfect fifth of the fifth octave shown in Table 5 correspond precisely to the 10 Sephiroth and 22
Paths of the Tree of Life (Fig. 28), whilst we saw in Figure 11 that the Tree of Life has 16 triangles which, when turned
into tetractyses, contain 70 yods. In other words, the geometry of the Tree of Life embodies
both the factors 70 and 24 of the number 1680 — the number of circularly polarised
oscillations made by each whorl of the UPA. What this remarkable correlation between
Pythagorean tone ratios and the structure of the Tree of Life is indicating^{20} is that, as the tenth overtone and the 32nd note, the tone
ratio 24 must have physical significance because it corresponds to Malkuth,
which is the tenth and last Sephirah, as well as the 32nd Path in the Kabbalistic sense. The
number 24 certainly does have such significance in the context of the
octonions underlying the superstring gauge symmetry group E_{8} and in the
context of the Klein Configuration. A spinless string in the 26dimensional spacetime
predicted by quantum mechanics for such strings can oscillate in 24 orthogonal directions
that are transverse to its length at any point. These 24 vibrational degrees of freedom in
the bosonic string field are known to play a role in heterotic superstring theory. Their
full significance for Mtheory awaits elucidation. The musical counterpart of the number 24
as the tenth overtone would indicate that the 10fold Tree of Life paradigm becomes fully
realised in spacetime with the generation, relative to the vacuum ground state of a string
(the first overtone of one full oscillation) of 24 independent waves. Each oscillation
creates a nonabelian gauge charges of E_{8} in a way that generalises to the
nonabelian context Kaluza’s identification of oscillation along a circular, fifth dimension
with the electric charge of a particle. The 240 such charges, which are spread as separate
oscillations along the ten stringlike projections of a certain brane^{21} into 4dimensional spacetime, 24 to a string (namely, the
ten whorls of the UPA), would then correspond to the 240 nonzero roots of
E_{8}.
31
Conclusion
A continuous chain of mathematical connections has been
established between the group E_{8} describing superstring interactions and the
number 168 implicit in the paranormal, 108years old description of the building blocks of
matter by Annie Besant and C.W. Leadbeater. Remarkable confirmation that this number is a
structural parameter of the superstring is provided by the Kabbalistic system of knowledge,
whilst the fact that the number value of the very Sephirah signifying the physical universe
is the very dimension of a gauge group describing anomalyfree interactions between
superstrings removes any doubt concerning whether the Kabbalistic appearance of 168 could be
a coincidence.
E_{8}: rank8 exceptional group describes superstring forces
Octonions: eight unit octonions form a division algebra isomorphic to E_{8}
SL(3,2): group of automorphisms of Fano plane representing
octonions is of order 168
PSL(2,7): group of 168 automorphisms of Klein Quartic is isomorphic to SL(3,2)
One halfrevolution of the whorl of UPA has 168 turns
The form of the UPA reflects the holistic nature of the Mtheory currently
being sought by many theoretical physicists throughout the world. As the author has
proved^{22} that the UPA is not a quark but its subquark constituent, their
search is being hampered by their placing false constraints on Mtheory, namely, to find a
theory that not only unifies the five superstring theories and supergravity but also
predicts the existence of three generations of quarks with interactions that conform to the
gauge symmetry of U(1)×SU(2)×SU(3). But neither quarks nor this gauge symmetry are
fundamental , and the correct Mtheory will make this revolutionary prediction. This article
has discussed some of its ingredients, particularly, octonions, the Klein Quartic and
PSL(2,7).
References
^{1} Besant, Annie, and Leadbeater, Charles W.
Occult Chemistry, Theosophical
32
Publishing House, Adyar, Chennai, India, 1951.
^{2} Phillips, Stephen M. Extrasensory Perception of Quarks, Theosophical
Publishing House, Wheaton, Ill. USA, 1980; Anima: Remote Viewing of Subatomic
Particles, Theosophical Publishing House, Adyar, Chennai, India, 1996; ESP of
Quarks and Superstrings, New Age International, New Delhi, India, 1999.
^{3} Mishchenko, A. and Solovyov, Y. “Quaternions.” Quantum 11, 47, and 18,
2000.
^{4} An associative algebra is one for which a(bc) = (ab)c. If this is
untrue, it is nonassociative.
^{5} Adams, J. F. “On the NonExistence of Elements of Hopf Invariant One.” Ann. of
Math. 72, 20104, 1960.
^{6} Bott, R. and Milnor, J. “On the Parallelizability of the Spheres.” Bull. Amer.
Math. Soc. 64, 8789, 1958.
^{7} Hurwitz, A. “Ueber die Composition der quadratischen Formen von beliebig
vielen Variabeln.” Nachr. Königl.
^{8} Phillips, Stephen M. Article 1: “The Pythagorean Nature of Superstring
and Bosonic String Theories,” (WEB, PDF).
^{9} Green, M.B. & Schwarz, J.H. “Anomaly cancellations in supersymmetric d = 10
gauge theory and superstring theory.” Physics Letters, B149, 117.
^{10} Schwarz, H.A. “Ueber diejenigen algebraischen Gleichungen zwischen
zwei veränderlichen Grössen, welche eine Schaar rationaler, eindeutig umkehrbarer
Transformationen in sich selbst zulassen.” J. reine angew. Math. 87, 139–145; Ges. Math.
Abh. II, pp. 285–291 (1890, reprinted Chelsea, 1972).
^{11} Hurwitz, A. “Über algebraische Gebilde mit eindeutigen
Transformationen in sich,” Math. Ann. 41 (1893), 403442.
^{12} Accola, R.D. “On the number of automorphisms of a closed Riemann
surface.” Trans. Amer. Soc. 131 (1968), 398408; Maclachlan, C. “A
bound for the number of automorphisms of a compact Riemann surface.” J. London Math. Soc.
44 (1968), 265272.
^{13} Belolipetsky, Mikail and Jones, Gareth. “A bound for the number of
automorphisms of an arithmetic Riemann surface.” http://arxiv.org/PS_cache/math/pdf/0306/0306105.pdf.
^{14} Murray Macbeath, A. “On a theorem of Hurwitz.” Proc. Glasgow Math.
Assoc. 5 (1961), 9096.
33
^{15} Murray Macbeath, A. “Hurwitz groups and surfaces.” The Eightfold
Way. MSRI Publications, vol. 35, 1998, 11.
^{16} Ringel, G. and Youngs, J.W.T. “Solution of the Heawood MapColoring Problem.”
Proc. Nat. Acad. Sci. USA 60, 438445,
1968.
^{17} Phillips, Stephen M. Article 14: “Why the Ancient Greek Musical Modes are
Sacred,” (WEB, PDF).
^{18} Ibid.
^{19} Ibid.
^{20} See also Article 13: “The Music of Superstrings,” (WEB, PDF).
^{21} This is discussed in detail in the author’s book The Mathematical Connection between Religion and Science
(Antony Rowe Publishing, 2009) and in Article 2 (WEB, PDF).
^{22} Reference 2.
34
