ARTICLE 18
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2
5PH. England.
Website: http://www.smphillips.mysite.com
Abstract
The ancient Chinese divinatory system called ‘I Ching’
uses a table of 64 hexagrams and their commentaries. This pattern expresses
all possible permutations of lines (yang) and broken lines (yin), grouped in
pairs of triplets, or trigrams. The eight possible trigrams are shown to
symbolize the binary number representations of the integers 0–7. A hexagram
represents the sum of pairs of these integers. The 21 hexagrams on each side
of the diagonal of the table denote 21 integers that add up to 168. Previous
articles discussed this number as the structural parameter of superstrings.
There are also 168 yin/yang lines outside the diagonal in each half of the
table. Remarkable correspondence exists between this pattern and the
encoding of the number 168 in the inner form of the Tree of Life. Starting
with the three lines of the Heaven trigram, which symbolizes the Bode number
3 of Venus, the doubling of planetary Bode numbers according to the
TitiusBode law has as its I Ching counterpart the progressive doubling of
yin/yang lines and rows of hexagrams. For example, the I Ching table has 384
such lines. 384 is the Bode number for the planet Pluto predicted by the
author’s logarithmic spiral theory of planetary formation. The table is
isomorphic to the Klein Configuration representing the 168 automorphisms of
the Klein Quartic equation described by PSL(2,7). This group is isomorphic
to SL(2,3), the group of automorphisms of the Fano plane, which represents
the algebra of octonions. Yin and yang lines symbolize the 168 nonzero
roots of the superstring symmetry group E_{8} that are not also
nonzero roots of E_{6}, one of its exceptional subgroups. The I
Ching table therefore encodes planetary distances, the vibrational form of
superstrings and their unified force.

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1. Tai Chi
In Chinese Taoism, founded by the great spiritual teacher Lao Tzu,
reality or nature — both earthly and cosmic — is called the ‘Tao.’ Originally meaning
“the Way,” Tao signifies the noumenal order underlying all phenomena. It is not a static
state but a dynamic process — a ceaseless cycle of ebb and flow like the tides, of
expansion and contraction like the heart, of outgoing and return like the systolic and
diastolic phases of the circulation of blood in the body, of building up and breaking
down like the anabolic and catabolic stages of metabolism, where complex molecules are
built up or broken down with the release of energy, and of perpetual appearance and
disappearance, such as the creation and annihilation of virtual subatomic particles in
the vacuum.
All manifestations of Tao are generated by the dynamic interplay of two
polar opposite forces or tendencies — yang and yin. Yang is the male, fecundating,
creative power, the generative principle associated with Heaven, the ultimate source and
cause of all things. Yin is the dark, receptive, female and maternal element represented
by the Earth. It is the formative, reproductive principle that transmutes the amorphous,
undifferentiated energy of creation into a multiplicity of forms. In the ancient
geocentric view of the universe, Heaven was the realm of movement and change, whilst
earth below was resting. Therefore, yang came to symbolize movement and yin rest. The
Taoist symbol of these opposite (but not opposing) tendencies is the Tai Chi Tu,
called the “Diagram of the Supreme Ultimate” (Fig. 1). The bright half represents yang and is regarded as the outgoing,
positive aspect of a cycle. The dark half denotes yin and is the returning negative
phase of the cycle. The two dots in the diagram express the idea that yin is latent in
yang and vice versa; each time each force reaches its extreme, it contains in itself
already the seed of its opposite. Yang turns into yin and yin into yang in a perpetual
cycle of metamorphosis. For example, the energy of a mechanical system consists of its
kinetic energy and its potential energy. If the system is one that is isolated from its
environment, the law of conservation of energy requires only that the sum of these two
types of energies remains constant over time as each varies. Consider the simple
harmonic motion of a swinging pendulum. As it rises, it slows because of the opposing
force of gravity acting on the bob and it loses kinetic energy (its yang quality). At
the same time, it gains potential energy (its yin quality) as it works against gravity
until it comes momentarily to rest at the highest point of its swing when its kinetic
energy is zero and its potential energy is a maximum. Then it swings back, increasing
its kinetic energy but losing potential energy until it reaches the lowest point again,
when it regains all its initial kinetic energy and loses all its potential energy.
During its upward and downward motion, potential energy — its yin quality — changes into
kinetic energy — its yang quality and vice versa. Neither kind of energy is constant.
Only their sum is, and there is continual interchange between them.
Subatomic particles, atoms and even molecules behave as either particles
or waves. Their observation forces them to act as either one or the other. This
dualistic nature of matter was epitomised by the great Danish physicist Niels Bohr in
his Principle of Complementarity. The particle picture and the wave picture are ideas of
classical physics. Bohr pointed out that each is only partially correct in the real,
quantum world and that which aspect is observed depends upon how the observation is
carried
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out. It is not that only one of the two opposite possibilities always
exists for a given subatomic object. Instead, waves transform into particles and
particles change into waves depending on the mode of observation. Before being observed,
the object is neither wave nor particle (the term ‘wavicle’ has been coined to express this
unobservable, noumenal aspect of quantum objects). Called into phenomenal being, so to
speak, by being observed, the object is forced by the mode of observation to behave
either as a wave or as a particle — it cannot act as both at any given instance. The
great mystery still for many physicists is how this choice actually comes about,
although Bohr would have dismissed the issue as meaningless for the positivistic
reason that it is not an observable transformation.
The complementarity of opposite modes of existence is the manifestation
in the microscopic world of the interchangeable duality of yang and yin. When Bohr
visited China in 1937, he was deeply struck by the ancient Chinese notion of the polar
opposites of yang and yin. Knighted ten years later, he chose the symbol of tai chi tu
as the motif for his coatofarms (Fig. 2). Its inscription Contraria sunt complementa (Opposites
are complementary) expresses his principle of complementarity between the particle and
wave aspects of matter. This became an essential component of the ‘Copenhagen school’ of
interpretation of quantum mechanics founded by Bohr. This was the orthodox understanding
for many years. However, it has come under attack in recent years.^{1}
The Chinese calendar divides the year into 24 equal periods marked by
the Sun’s position against the tropical zodiac. Each of the twelve Chinese months
comprises two periods marked by the Sun moving
through 30°, so that the length of each period is not a
fortnight. The ancient Chinese recorded the beginnings of the four seasons by noting the
positions of the shadow cast by a vertical pole in the ground, the shortest shadow
occurring on the summer solstice and the longest marking the winter solstice.^{2} They drew six concentric circles around the pole, divided
them into 24 segments and marked on the circles the increase in length of shadow,
measuring from the outermost circle to
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the centre from summer solstice to winter solstice and from the centre
of the circle to the boundary from winter solstice to summer solstice (Fig. 3). Subtracting from all measured lengths the minimum length of the
shadow at summer solstice, they found that the increase in length of the shadow followed
a curve like that dividing the Tai Chi in half. As summer was yang and winter was yin,
the origin of their symbolic representation is believed to be this ancient observation
of the periodic change of the length of shadows cast by the sun throughout the year.
2. The I Ching
Pronounced “Ee Jing”, I Ching (易經), the Book of Changes, is about 5,000
years old and, as such, is one of the oldest available sources of spiritual wisdom. The scholar Kung MganKwoh
in the second century B.C.E. described the legend that the first Chinese Emperor, Fu
Hsi, was responsible for the original creation of I Ching, although there are
different stories concerning where he found his inspiration. The most poetic and
inspiring concerns a dragonlike creature that climbed from the water near him as he
was meditating one day by the River Ho (the Yellow River). He noticed the lines on the
creature’s scales, and decided these were to help him in his quest for knowledge. He
set about drawing diagrams representing the patterns on the scales of this creature,
believing that the diagrams would be sufficient to encompass all wisdom. This article
confirms his belief by showing that it encodes the threedimensional form of
superstrings and the distances of the planets from the Sun.
I Ching symbols are composed of lines representing yin (female) or yang
(male) properties. The original I Ching, as produced by Emperor Fu Hsi, consisted of
eight symbols called ‘Kwa’ or ‘trigrams,’ each made up of three lines. A line could be
either ‘yin,’ the female power, or gentleness: – –, or ‘yang,’ the male power, or
strength: ^{__}. The eight combinations (2×2×2=8) of yin and
yang in the three lines of a trigram are:

Chien
Heaven

Chên
Thunder

K’an
Water

Kên
Mountain

Tui
Lake

Li
Flame

Sun
Wind

K’un
Earth

In 1143 B.C.E., King Wen, whilst under sentence of death, placed the
eight trigrams in pairs to produce the 64 ‘hexagrams’ (sixlined symbols) with which we
are now familiar. His son, the Duke of Chou, added a commentary on each line in each
hexagram and on the symbolism (known as the ‘Hsiang Chuan,’ or the ‘Image’) of a
hexagram). This produced (64×6=384) commentaries that still form an essential part of I
Ching.
Centuries later, Confucius added more commentary, known as the ‘Ten
Wings.’ The commentary states that “Change has an absolute limit: This produces two
modes: The two modes produce four forms, the four forms produce eight trigrams; the
eight trigrams determine fortune and misfortune.”
Over the centuries, I Ching continued to be recognised, and even given
religious significance, before it came to France in the early 19th century. In the early
20th
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century, a German translation of I Ching was completed and published by
Richard Wilhelm, who had lived in China for many years. This translation still forms the
basis of many of the published I Ching texts.
Creation of
hexagram
I Ching is used as a method of divination in the following way: keeping
firmly in his mind the question whose answer he seeks (one is advised to phrase the
question unambiguously and to write it down beforehand on a piece of paper), a person
tosses three coins.* The results — head or tail — are noted and the
procedure repeated to generate six sets of three outcomes. Each result is interpreted
according to the rules listed below:
1. 
three tails 

changing yang line called “moving yang” 
2. 
three heads 

changing yin line called “moving yin” 
3. 
two heads + tail 

stable yang line ^{___} 
4. 
two tails + head 

stable yin line – – 
Only changing lines are reversed, i.e., the moving yang becomes the
stable yang line and the moving yin becomes the stable yin line. The hexagram is built
up line by line by successive throws of three coins, starting from the bottom line and
working upwards towards the top line. If any toss yields three tails or three heads, a
separate hexagram is built up from the changing lines. In both cases, the six lines
are then divided into two sets of three — the upper and lower trigrams — and the
commentary listed in the I Ching for this so called ‘resolved’ hexagram is consulted,
as well as that for the socalled ‘moving’ hexagram containing one or more changing
lines, should these appear.^{3} There are 2^{6} = 64 possible hexagrams (Fig. 5). As each line may be stable or moving, there are 64^{2} =
2^{12} = 4^{6} = 4096 pairings of a resolved and moving hexagram, that
is, possible outcomes when the hexagram is generated.
It is not a purpose of this article to judge the validity (or lack
thereof) of the I Ching as a system of divination. Rather, it will show how real
information about the natural world is embodied in its pattern of lines, just as
the Chinese have always claimed.
3. The form of the subquark
superstring
In his Yoga Sutras, the first systematic exposition of the
philosophy and practice of
__________________________
*The original divinations, upon which the Book of Changes was written,
were performed exclusively with yarrow stalks. Some scholars believe using coins
produces inaccurate results because it changes the probabilities and statistics for
individual types of lines. This begs the question whether the type of random process
generating hexagrams influences the accuracy of I Ching as a divinatory tool. Many
methods are used by I Ching practitioners (e.g., see: http://way.to/iching), showing the lack of agreement
between them on this issue.
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yoga, Patanjali discussed the various paranormal powers that the yogi can
acquire. In Aphorism 3.26 of the Sutras, it is stated that the yogi can develop
“knowledge of the small, the hidden or the distant by directing the light of superphysical
faculty”:
The Sanskrit name for this siddhi is ‘anima,’ meaning ‘mindfulness.’ The
author has given it the modern name of ‘micropsi.’ The ability to experience mental
images of objects too small to be seen by the naked eye that are the same as what would be seen in a microscope was claimed by Annie Besant
and Charles W. Leadbeater.^{4} These two early leaders of the Theosophical Society published in
1908 their observations of the atoms of many elements using the yogic micropsi ability
that they claimed to have been trained to develop. According to Besant &
Leadbeater, chemical atoms were composed of sometimes thousands of basic particles,
which they called ‘ultimate physical atoms,’ or UPAs. They noticed two types of UPAs: a
positive variety from which the force binding them together flows outwards and a
negative type into which this force is directed (Fig. 6). Each type is composed of ten closed curves that spiral side by
side 2½ times down towards the pointed nadir of the object. They then separate
into strands of three and seven, each twisting upwards 2½ times in a narrower spiral
around the central axis of the UPA towards its top. These curves (called ‘whorls’) never
touch and spiral clockwise as seen from the top in the positive UPA and anticlockwise in
the negative UPA. The UPA spins about its axis, pulsates and precesses like a spinning
top.
The author showed^{5} that the UPA is a subquark — an as yet undetected constituent
of the up and down quarks making up the protons and neutrons in atomic nuclei.
Furthermore, their stringlike appearance and their winding in a series of progressively
smaller circles are consistent with their being superstrings, which are currently
believed by many physicists to be the fundamental units of matter. As mirror images of
each other, the positive UPA is a subquark carrying positive magnetic charge and the
negative variety carries negative magnetic charge. Each whorl is a closed helix
with 1680 turns (Fig. 7). Leadbeater checked this number carefully 135 times. As a whorl
makes five complete revolutions whilst winding around the outer and inner halves of the
UPA, there are on average 1680/5 = 336 turns in each revolution. A halfrevolution
comprises 168 turns and a quarterrevolution consists of 84 turns. These numbers are
structural parameters of the subquark superstring.
The positive and negative UPAs (subquarks carrying positive and negative
magnetic charges) are the fundamental manifestation in the subatomic world of the
opposite but
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complementary yangyin polarities symbolized by the Tai Chi. The (+) UPA
is a righthanded form whose whorls spiral clockwise. It manifests the yang property of
matter (Fig. 8) because it is the source of outwardly directed, hypercolour SU(3)
gauge fields that bind the UPAs in groups of three as quarks. The (–) UPA has a
lefthanded form whose whorls spiral anticlockwise. It manifests the yin quality because
it is the sink into which these fields converge.
A plane wave represents a field whose oscillations take place in some
fixed direction (Fig. 9). Two plane waves whose oscillations take place in perpendicular
planes with the same amplitude and whose phases differ by 90° add up to form a
circularly polarised wave whose amplitude is constant and rotates at a constant rate in
a plane perpendicular to the direction of both waves (Fig. 10). The circular turns of the whorl shown in Figure 7 are the circularly polarised vibrations of a standing wave. It
makes 1680 oscillations, 336 per revolution. 168 oscillations take place in a
halfrevolution of each whorl of the UPA and 84 occur in a quarterrevolution. In the
(–) UPA the amplitude of the circularly polarised wave rotates in the opposite sense.
4. Trigrams symbolize binary
numbers
In the binary number system, Arabic numerals are represented by their
binary counterparts. A binary number can be represented by any sequence of bits (binary
digits), which in turn may be represented by any mechanism capable of being in two
mutually exclusive states. Binary numbers are commonly written using the symbols
0 and 1. The binary forms of the integers 0, 1, 2, … 7
are:
7
Any positive integer N is a linear combination of successive integer
powers of 2:
where c_{n} = 0 or 1. The binary digits making up the binary
form of an integer are the coefficients c_{n} with the rule that the digit at
the extreme right of the binary number is c_{0}, the one to the left next to it
is c_{1}, etc. The binary digit at each position has two possible values:
0 or 1. Compare this property with the trigrams as
combinations of yang or yin lines. Making the identification:
yang line


=

0

yin line


=

1

(for clarity, binary numbers are written henceforth in red) and defining
the convention that the line at the base of a trigram corresponds to the binary digit to
the far right of its binary number representation, the binary numbers symbolized by the
eight trigrams and their equivalent
Arabic numerals are shown in Fig. 11. Notice that the same set of numerals would have resulted by
identifying a yang line with 0 and a yin line with
1. As the numbers to be discussed later turn out to
correspond to equal numbers of yin and yang lines, as the balance of yang and yin in
Nature and Bohr’s Complementarity Principle would lead one to expect, the conclusions
that will be drawn do not depend upon which identification scheme is correct. However,
Figure 1 indicates that the traditional ordering of the trigrams starts
with the Heaven trigram comprising three yang lines and ends with the earth trigram
consisting of three yin lines. As the spiritual realms are the origin of the physical
world, it is, intuitively speaking, more correct to assign the first number 0 to the
Heaven trigram and the last number 7 to the Earth trigram rather than the reverse. This
means that the binary number 0 must correspond to the yang
line and the binary number 1 must correspond to the yin
line, as shown above.
Associated with each pair of trigrams in a hexagram is a corresponding
pair of the integers that they represent. Figure 12 shows their sums. They range from 0 (the sum of the two Heaven
trigrams in the top lefthand corner) to 14 (the sum of the two Earth trigrams in the
bottom righthand corner). The integers along the diagonal of the 8×8 square array of
integers are even because each is a sum of identical integers, the diagonal hexagrams
being composed of like trigrams. Integers in the same column in successive rows increase
by 1. The sum of the 64 integers is 448 = 2^{6} + 2^{7} + 2^{8},
which has the binary representation 111000000. The sum of the eight diagonal integers =
56 and the sum of the two sets of 28 integers that are off the diagonal = 448 – 56 =
392, i.e., each set adds up to 196. This is also the sum of the 28 integers on the
boundary of the array. The sum of the (6428=36) integers inside the boundary = 448 –
196 = 252. This is the same as the sum of the 36 integers forming one diagonal half of
the I Ching
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array. The sum of the six internal, diagonal integers = 42, so that the
sum of the (36–6=30) internal, offdiagonal integers = 252 – 42 = 210. The superstring
and astronomical significance of some of these numbers will be discussed later.
5. Encoding of superstring
structural parameters 168 & 336
The seven integers 1–7 can be paired in 21 different ways. The sum of
these combinations is 168. The pairs of integers correspond to the 21 pairs of trigrams
on either side of the diagonal of the I Ching table that are enclosed in the black or
white triangles shown in Fig. 13. So 21 hexagrams on each side of the diagonal encode the number
168, which, according to Section 4, is the number of circularly polarised oscillations
made by each whorl as it twists through 180°. This is summarised in Table 1. The outer
and inner halves of the UPA are the counterpart of the yang/yin duality, just as the
righthanded, positively charged UPA and the lefthanded, negatively charged UPA are
another manifestation of the yang and yin aspects of matter. The sum of the 22
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Table 1
The number values of the 64 hexagrams. The sum of the 21
integers on either side of the diagonal in a black or white triangle is 168.
This corresponds to the 168 circularly polarised oscillations made during half a
revolution of an outer or an inner segment of each whorl of the UPA. The sum of
the 22 integers outside the triangles is 84, which is the number of such
oscillations made during a quarterrevolution of a whorl.
10
integers outside both triangles is 84, which is the number of
oscillations made by a whorl during one quarter of a revolution around the spin axis of
the UPA. We find that the numbers of oscillations made in one revolution (336), in one
halfrevolution (168) and in onequarter revolution (84) of each whorl of the UPA are
encoded in the I Ching table of hexagrams symbolising binary number representations of
the integers 0–7. This confirms the archetypal nature of the venerable Chinese diagram.
The numbers 84 and 336 are good examples of the author’s Tetrad
Principle^{6} because
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2},
that is, 84 is the sum of the squares of the first four odd
integers, whilst
336 = 4×84 = 2^{2} ×(1^{2} + 3^{2} +
5^{2} + 7^{2} )
= 2^{2} + 6^{2} + 10^{2} + 14^{2},
i.e., 336 is the sum of the squares of four even integers spaced 4 units
apart, starting with 2^{2} = 4. Notice also that the number of yang and yin
lines in the eight trigrams = 24 = 1×2×3×4 and that the number of hexagrams in the I
Ching table = 64 = 4^{3}.
The number of yang and yin lines in the I Ching matrix = 64×6 = 384. The
number of lines of each type = 192. The number of lines in the eight diagonal hexagrams
= 6×8 = 48. The number of lines in the 56 offdiagonal hexagrams = 384 – 48 = 336.
There are 168 lines in the 28 hexagrams on either side of the diagonal.
They comprise 84 yang lines and 84 yin lines. As each whorl makes 84 circularly
polarised vibrations during onequarter of its revolution, each resulting from two plane
waves oscillating 90° out of phase, the latter represent the yang and yin phases of a
cycle that is half of a larger cycle of 168 oscillations made as the whorl winds through
half a revolution (Fig. 13).
The number of lines and broken lines making up the 28 hexagrams on the
boundary of
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the I Ching matrix is 6×28 = 168. The square is the ancient symbol of
the four elements of Earth, Water, Air and Fire. The twelve odd integers 3, 5, … 25
arranged on the sides of a square, four per side, add up to 168 (Fig. 14). This parallelism between the properties of
numbers in the context of the geometrical symbol of the number 4
demonstrates its profound role in determining numbers of universal significance — in
this case the superstring structural parameter 168.
The number of circularly polarised waves in a whorl as it makes 2½ outer
revolutions and 2½ inner revolutions is 1680. The number of waves
in each half is 840 (Fig. 15). As a remarkable arithmetic counterpart to the 28 hexagrams on
the boundary of the I Ching matrix with 168 lines and broken lines, the sum of the 28
odd integers that can be arranged along the boundary of a square, eight per side, is 840
(Fig. 16).
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6. The I Ching table as the
Chinese Tree of Life
At the heart of the Jewish mystical tradition called ‘Kabbalah’ is the
Tree of Life, or Otz Chiim. It is a representation of Adam Kadmon, or Heavenly Man, the
divine prototype or archetypal blueprint governing the nature of God. His ten divine
qualities, or Sephiroth, are symbolized by spheres or circles and connected by 22 Paths.
They are arranged in space on the three vertical pillars of the
tree
(Fig. 17). The pillars do not lie in the same plane because the tree is a
threedimensional network of lines, some of which cross over one another, as indicated
in Fig. 17 by the broken lines. The Sephiroth Chokmah, Chesed and Netzach
lie on the righthand Pillar of Mercy, Binah, Geburah and Hod are on the lefthand
Pillar of Severity and Kether, Tiphareth, Yesod and Malkuth lie on the central Pillar of
Equilibrium. The socalled ‘invisible Sephirah’ Daath (the circle with a broken line
shown in Fig. 17) appears to lie on the central pillar halfway between Kether and
Tiphareth. Actually, Daath hovers in front of it because it is not connected by Paths
and only does so when it becomes Yesod of the next higher overlapping tree and this
Sephirah lies in front of the plane containing the side pillars, whereas the path
joining Kether and Tiphareth projects behind this plane. Each Sephirah is negative, or
13
receptive, to the preceding one and positive to, or having an influence
on, the following one. Metaphysically and psychologically speaking, the Sephiroth on the Pillar of Mercy embody dynamic, active,
creative aspects of the divine life, those on the Pillar of Severity manifest
introspective, passive and nourishing qualities, whilst the Sephiroth on the middle
pillar represent the harmony or balance of these complementary qualities. The kinetic
energy of a dynamical system subject to conservative forces for which the Law of
Conservation of Energy applies is not necessarily constant but changes into potential
energy. Kinetic energy — the yang aspect of the system, as discussed in Section 1 — is
an attribute of the Pillar of Mercy and potential energy — its yin aspect — is a
quality of the Pillar of Severity. One of the titles of Chokmah, which heads the
Pillar of Mercy, is ‘Abba’ (the Supernal Father) and Binah heading the Pillar of
Severity has the title “Aima” (the Supernal Mother). These titles demonstrate that the
male and female aspects of God manifest equally in the side pillars of the Tree of
Life. The yang or active, outgoing aspect is expressed in the Pillar of Mercy and the
yin or reflective, ingoing aspect is realised in the Pillar of Severity. The Tao is
the balance of these complementary opposites. It corresponds to the central Pillar of
Equilibrium.
Figure 18 depicts how the Tree of Life is formed from the centres and
points of intersection of four identical circles. The circle of the Tai Chi symbol has a
diameter that is four times their radii. The black and white circles in the yang and yin
halves coincide with Daath and Yesod of the Tree of Life.
Earlier work^{7} by the author has shown that the geometry of the Tree of Life
— its outer form — implies a geometrical structure within it that has been unrecognised
hitherto by Kabbalists. This inner form of the Tree of Life consists of two similar sets
of seven regular polygons. They are the triangle, the square, the pentagon, the hexagon,
the octagon, the decagon and the dodecagon. The 14 polygons share one side. Figure 19 shows both forms of the Tree of Life superimposed on each other. As the outer form is
threedimensional, what is depicted is its projection onto the plane containing the
two sets of polygons. The leftright symmetry of the outer form of the Tree of Life
generates a similar symmetry in its inner form: one set of polygons is the mirror
image of the other set, with their shared edge extending from Daath to Tiphareth
(actually, their projection onto the polygonal plane). This is how Nature’s yin/yang
polarity manifests in the inner form of the Tree of Life.
The symbol of the tetractys at the heart of the number philosophy
of Pythagoras:
14
15
embodies sacred geometry like the Tree of Life with numbers
characterizing how this blueprint becomes actualized in the universe. This is achieved
by converting the triangular sectors of each polygon into tetractyses. This turns them
into various numbers — the population of dots making up each one. Information about
actual manifestations of this cosmic blueprint is encoded numerically in the geometry of
both its outer and inner forms. The same information has analogous encodings in subsets
of the seven regular polygons. The most obvious example of this is the two sets of the
first six polygons (Fig. 20). The twelve polygons have 50 corners. 48 of them are outside
their shared edge, 24 per set. The number of dots in the (6+6) polygons is 384, that is,
there are (384–48=336) dots other than their external corners. This means that 48 dots
(24 on each side of their shared side) are needed to shape through their corners the two
sets of polygons made up of 336 dots, each set containing 168 dots. Compare these
properties of the inner Tree of Life with the fact that the I Ching table comprises 48
lines and broken lines in its diagonal hexagrams (24 yang lines and 24 yin lines) and 28
hexagrams on either side of the diagonal with 168 yang lines and 168 yin lines. The
number (192) of dots in one set of polygons is equal to the number of yang lines in the
I Ching table and the number of dots in the similar set on the other side of the shared
edge is equal to the number of yin lines, whilst the number of corners on each side is
the same as the number of yin or yang lines in the diagonal hexagrams. This remarkable
equivalence exists because the I Ching table is, as Chinese tradition states, a
representation of the Tao — reality — and so it encodes the superstring structural
parameters 168 and 336 in a fashion analogous to how the inner Tree of Life does this.
The Kabbalistic names of the Sephiroth, their Godnames, Archangels,
Orders of Angels and Mundane Chakras can be converted into numbers through the ancient
practice of gematria, whereby the number value of a word is the sum of its letter
values. This system of 50 numbers represents a complete prescription of the properties
of the outer and inner forms of the Tree of Life as the universal blueprint. This
prescription applies
16
also to certain subsets of the seven polygons. For example, Yahweh, the
Godname of Binah with number value 26, prescribes the set of the first six polygons,
which have 26 corners, and Elohim, the Godname of Binah with number value 50, prescribes
both sets of polygons, which have 50 corners. They have 48 corners outside their shared
edge and 48 is the number value of Kokab, the Mundane Chakra of Hod. 168, the
number of dots associated with each set of six polygons other than their corners, is the
number value of Cholem Yesodoth, the Hebrew name of the Mundane Chakra of
Malkuth. That it is not just coincidence that the Sephirah signifying physical reality
should aptly have a Mundane Chaka whose number value characterizes the structure of a
superstring can be confirmed by looking at the populations of dots in the six polygons
(Fig. 21).
The number value 78 of the word ‘Cholem’ is the number of dots in the
triangle, pentagon and octagon, and the number value 90 of the word ‘Yesodoth’ is the
number of dots in the square, hexagon and decagon. It is highly improbable that the
number values of both words should coincide with dot populations of whole
groups of polygons. This view is strengthened by the fact that, of the 168 yin/yang
lines on the boundary of the I Ching square, 90 lines (coloured blue in Fig. 21) are on two of its adjacent sides and 78 lines
17
(coloured red) make up the remainder of the hexagrams on the boundary.
This remarkable correlation between the properties of geometrical figures belonging to
the mystical traditions of two different cultures indicates their transcendental origin.
The number 168 is embodied in not only the first six polygons but also
the seventh polygon — the dodecagon. Dividing its 12 sectors into three triangles and
then converting them into tetractyses generates 169 dots other than the 12 corners of
the dodecagon. Associated with each dodecagon are 168 dots (Fig. 22). The next type of transformation of the seventh polygon yields
just as many new dots as the first kind does for the first six polygons.
The two sets of the first six polygons have 168 dots on their boundaries
outside the shared edge (Fig. 23), that is, 168 dots are needed to delineate the shapes of the 12
polygons. This demonstrates in a clear way the shapedefining character of this number,
which is realised in the subquark superstring as the number of turns in the closed helix
of a whorl as it revolves through 180°. The I Ching counterpart of this property is
shown in Fig. 14 as the 168 yin and yang lines of the 28 hexagrams on the
boundary of the I Ching pattern.
When the outer form of the Tree of Life is projected onto the plane
containing its inner form so that the central edge shared by the polygons coincides with
the line joining Daath to Tiphareth, one finds that Chokmah and Binah coincide with the
uppermost corners of the two hexagons,
Chesed and Geburah coincide with the exterior corners of the two
triangles and Netzach and Hod are located at the lowest corners of the two hexagons
(Fig. 24). Six of the 70 corners of the two sets of seven polygons
therefore coincide with positions of Sephiroth on the side pillars. This leaves 64
corners, of which two are the endpoints of the shared edge and 62 are truly new
geometrical degrees of freedom. The upper endpoint coincides with the nonSephirah
Daath, meaning ‘knowledge.’ Just as the seven Sephiroth of Construction emerge from the
Abyss of Daath separating the objective aspect of God from the subjective Godhead, so
each hexagram emerges with each corner of the polygons, starting from the pair of Heaven
trigrams and finishing the combinatorial cycle with the pair of Earth trigrams.
The
18
first and last hexagrams play the role of the endpoints of the root
edge. The two sets of 31 corners correspond to the two ways of stacking a pair of
trigrams. The number value of the Godname El (meaning ‘God’), which is assigned to
Chesed, the first Sephirah of Construction, is 31. This is the number of hexagrams other
than the first and last ones in each half of the I Ching table. The number 64 is the
number value of Nogah (‘glittering splendour’), the Mundane Chakra of Netzach,
and 62 is the number value of Tzadkiel (‘Benevolence of God’), the Archangel assigned to
Chesed.
There are 168 yin/yang lines in the 28 hexagrams on either side of the
diagonal set of eight hexagrams comprising 48 such lines. The number of lines in the
(28+8=36) hexagrams forming a complete half of the I Ching table = 168 + 48 = 216. The
number 36 (=6^{2}) is the number value of the Godname Eloha assigned to Geburah,
the sixth Sephirah, whose number value is 216 = 6^{3} . In conformity
to the restrictive nature of this Sephirah, its Godname delineates the set of 36
distinct hexagrams, the 28 others being just those on the other side of the
diagonal with their trigrams stacked in reverse order.
Given that the eight trigrams can be regarded as symbolic binary
representations of the integers 0–7 and that the Kabbalistic system of Godnames, etc
define aspects of the I Ching table, it should not be very surprising that Yahweh
(יהוה) — the most wellknown Godname — has a simple
representation in terms of binary numbers. Its number value is 26, which is the sum
of the powers of 2 in the following tetractys array:
26 =

2^{0}

=

1

=

1

2^{1 } 2^{0}

1 1

3

2^{2} ^{ }2^{1 } 2^{0}

1 1 1

7

2^{1} ^{ }2^{2}
^{ }2^{1 } 2^{0}

1 1 1 1

15

Each row comprises successive powers of 2, starting with 2^{0} =
1. Remarkably, the number value 26 of the Godname of Chokmah, expressing the creative
aspect of God, is the sum of the four odd integers 1 = 1, 3
= 11, 7 = 111 & 15 =
1111. These integers are known in mathematics as the first
four Mersenne* numbers, the nth Mersenne number being defined as M_{n} =
2^{n} – 1. Notice also that the fourth Mersenne number M_{4} = 15 is the
sum of the first four powers of 2 representing the lowest row of the tetractys and
expressed by the binary number 1111. This is the number
value of Yah (יה), the older version of the Divine Name Yahweh. It is
the number of hexagrams forming two adjacent sides of the I Ching square. Yah
therefore prescribes the eightfold size of the square array of hexagrams. It also
defines the number of its hexagrams because
64 = 8^{2} = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15.
The number of hexagrams inside the table in one triangular half of it is
21. This is the number value of Ehieh (AHIH), the Godname assigned to Kether, the first
Sephirah. They are made up of (21×6=126) yin/yang lines. Remarkably, 126 is the sum of
the four types of combinations of the letters A, H and I in AHIH:
A = 1, H = 5, I = 10
1.
2.
3.
4. 
A + H + I
AH + HI + AI + HH
AHI + HIH + AHH
AHIH 
=
=
=
=

16
42
47
21


TOTAL

=

126

________________________
* Named after Father Marin Mersenne, who was a
natural philosopher, theologian, mathematician, musical theorist and friend of
Descartes, Fermat and Pascal.
19
The remaining number of yin/yang lines in the I Ching table = 384 – 126
= 258. This is the 129th even integer. This shows how Yahweh Sabaoth, the Godname of
Netzach with number value 129, prescribes the table. Yah defines 15 hexagrams lying on
two adjacent sides of the square, leaving 13 hexagrams with 78 yin/yang lines on the
other two sides. The number of yin/yang lines in the I Ching square other than these =
384 – 78 = 306. This is the 153rd even integer. It shows how Elohim Sabaoth, the Godname
of Hod with number value 153, prescribes the table (see also Fig. 43).
In his Timaeus Plato described how the demiurge measured the
World Soul with the harmonic proportions of the musical scale by dividing a strip of its substance into portions measured
by the numbers forming two geometric series of four terms each: 1, 2, 4, 8 and 1, 3, 9
and 27 (Fig. 25). Article 11 discussed the way in which these numbers when seen
as part of a tetractys of ten numbers generates the tone ratios of the Pythagorean
musical scale. Their sum is 90, which is the number of yin/yang lines in the 15
hexagrams on two adjacent sides of the I Ching square prescribed by the Godname Yah. In
terms of the equivalence of the tetractys and Tree of Life as representations of the
tenfold nature of God, the dot at the centre of the tetractys corresponds to Malkuth,
the Sephirah denoting the physical world, symbolized by the planet Earth. In the Lambda
tetractys, this number is 6. This is the number of yin lines in the
pair of Earth trigrams, which is the last of the hexagrams on adjacent
sides of the I Ching square (shown in Fig. 26 in red). The sum of the remaining nine integers is 84, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2} .
This is the number of yin/yang lines in the remaining 14 hexagrams. The
sum of the integers at the corners of the hexagon in the Lambda tetractys is 48. This is
the number of yin/yang lines in the eight hexagrams at the top of the square expressing
the integers 0, 1, 2, 3, 4, 5, 6 and 7. The sum (36) of the integers at the corners of
the tetractys is the number of remaining yin/yang lines. Symbolically, the integer 6 at
the centre of the tetractys defines the number of lines and broken lines in a hexagram,
whilst the sum of the integers 1, 8 and 27 at the corners of the Lambda is the number of
hexagrams in a diagonal half of the square.
20
7. Encoding of planetary distances
in the I Ching table
It was shown in Article 17 that the assumption that the primordial
planetary nebula giving birth to the Solar System had the geometry of a logarithmic
spiral explains why the first eight planets conform to the TitiusBode Law. In
accordance with the eightfold pattern often found in nature, namely, the eight notes of
the musical scale, the octets of quarks and the eightfold periodicity of elements in
the chemical periodic table, it was hypothesized that the spiral geometry of the
planetary nebula had the fractallike quality of being rescaled for every set of
eight
Table 2. Wavelengths of the Pythagorean musical undertones.
C

D

E

F

G

A

B

Number
of subharmonics

1

8/9

64/81

3/4

2/3

16/27

128/243

0

2

16/9

128/81

3/2

4/3

32/27

256/243

1

4

32/9

256/81

3

8/3

64/27

512/243

3

8

64/9

512/81

6

16/3

128/27

1024/243

5

16

128/9

1024/81

12

32/3

256/27

2048/243

7

32

256/9

2048/81

24

64/3

512/27

4096/243

9

64

512/9

4096/81

48

128/3

1024/27

8192/243

11

128

1024/9

8192/81

96

256/3

2048/27

16384/243

13

256

2048/9

16384/81

192

512/3

4096/27

32768/243

15

512

4096/9

32768/81

384

1024/3

8192/27

65536/243

17

1024

8192/9

65536/81

768

2048/3

16384/27

131072/243

19

planets that condensed out of it. This means that, as well as being
the last member of the first octet of planets, Uranus is also the first member of the
second octet, the asymptotic centre of the rescaled spiral being centred on its orbit.
In terms of a (1/10 A.U.), the mean distance of the nth planet after Mercury in the
first octet from the first asymptotic centre is:
d_{n} = (3/2)×2^{n}a

(n = 1–8)

and the mean distance of the n'th planet after Uranus in the second
octet from the second asymptotic centre (that is, from Uranus) is:
D_{n'} = (3/2)×2^{n'}a

(n' = 1–8)

The mean distance of Uranus (n = 7) from the first asymptotic centre
is
d_{7} = (3/2)×2^{7} = 192a
and the mean distance of Pluto (n' = 2) from Uranus is
D_{2} = (3/2)×2^{2}a = 192a.
The predicted mean distance of Pluto from the Sun is 38.8 A.U., which
compares well with the measured value of 39.48 A.U.
21
Table 2 above shows that 384 is the 16th subharmonic, so that it is the
last of the sub harmonics in the 4×4 square array:
2
8
32
128

3
12
48
192

4
16
64
256

6
24
96
384

Also,
384 =

4!
4!
4!
4!

4!
4!
4!
4!

4!
4!
4!
4!

4!
4!
4!
4!

(4! =1×2×3×4)

These properties are illustrations of the author’s Tetrad Principle,
proposed in Article 1,^{8} according to which integers that have cosmic significance
because they parameterize holistic systems like the Solar System and the superstring are
always defined mathematically by the first four integers: 1, 2, 3 & 4, or by the
number 4, or tetrad, as either the fourth or sixteenth of a class of numbers or as the
sum of a 4×4 square array of numbers that mathematically relate to the tetrad, such as
4!. The 64 hexagrams are an example of the lattermost because 64 is the sum of the 16
integers 4 arranged in a 4×4 square array.
A decagon divided into tetractyses contains 61 dots, 30 lying on its
boundary and 31 in
22
its interior. This property correlates with the fact that the tone ratio
384 of the perfect fifth of the ninth octave is the thirtieth overtone and the 61st note
(Fig. 27). In other words, if the tone ratio 1 of the tonic of the first
octave is assigned to the centre of the decagon, the 30 notes up to 384 with fractional
tone ratios can be assigned to the dots inside the decagon and the 30 overtones up to
384 can be assigned to dots lying on its boundary. This illustrates the Pythagorean
title “All Perfect” assigned to the number 10 represented by the decagon, for it indeed
generates through the musical tones of the Pythagorean scale the number 384 as the
number of yin and yang lines in the I Ching table and as the mean distance of Pluto from
the first asymptotic centre of the logarithmic spiralshaped planetary nebula that
spawned the ten members of the Solar System. The Godname El (לא) of Chesed with number
value 31 prescribes the number 384 because this is the tone ratio of the 61st note,
where 61 is the 31st odd integer.
It was pointed out in Section 5 that 168 dots other than corners are
associated with the first six polygons of the inner form of the Tree of Life. The two
sets of polygons have 50 corners and 2×168 = 336 dots, a total of 386 dots. Of these,
two are the endpoints of the shared edge,
leaving 384 dots, 192 dots being associated with each set of six
polygons. The predicted mean distance of the last planet Pluto from Uranus is 192a — the
same as the mean distance of Uranus from the first asymptotic centre (Fig. 28). Therefore, Uranus is midway between Pluto and this centre. In
terms of the unit of distance a, exactly the same division:
384 = 192 + 192
exists between the mean distances between Pluto and Uranus and between
Uranus and the first asymptotic centre as that between the dots in the two sets of
polygons. This demonstrates explicitly the Tree of Life nature of the Solar System.
Table 2 indicates that 192 is the wavelength of the perfect fourth of the
eighth octave of undertones and 384 is the wavelength of the perfect fourth of the ninth
octave — the next higher octave. It shows the analogy between the musical sounds of the
scale and the picture of the Solar System made up of planets belonging to two
octets. In fact, the wavelengths of successive perfect fourths of the undertones:
3, 6, 12, 24, 48, 96, 192
(shown highlighted in purple in Table 2) are just the distances (in terms of a) of the first seven
planets after Mercury from the asymptotic centre predicted by the logarithmic spiral
theory of the TitiusBode Law.
The same 192:192 division is displayed in the I Ching diagram (Fig. 29). The 64 hexagrams have 6×64 = 384 lines and broken lines, that
is, 192 of each type. The uppermost row of eight trigrams consists of 24 yin/yang lines.
The first row of
23
24
hexagrams has 48 yin/yang lines (24 of each type),
that is, 24 more lines. According to Table 13 in Article 17, the average distance (in units
of a) of the Asteroid Belt from the first asymptotic centre is 24, whilst the mean distance
of Jupiter from this centre is 48, that is,
24 units further out. The first two rows of hexagrams have 96 lines. 96
is the distance of Saturn from the centre. The first four rows have 192 lines. 192 is
the distance of Uranus from the centre. All eight rows of hexagrams have 384 lines, and
384 is the distance of Pluto from the centre. Its mean distance from Jupiter = 384 – 48
= 336, which is the number of yin/yang lines in the offdiagonal hexagrams of the I
Ching diagram, the diagonal hexagrams having 48 lines. This famous divinatory diagram
can
25
therefore be said to represent the Solar System in that, as they double
in number, the populations of yin and yang lines in the rows of hexagrams measure the
distances of successive planets from the asymptotic centre of the logarithmic spiral
from which they were born (Fig. 30). This distance is given by the TitiusBode Law:
d_{n}= 3×2^{n1},
where n = 1 for Venus, n = 2 for Earth, etc. To understand the way in
which the I Ching table encodes planetary distances, one has, firstly, to read across
the first row of trigrams, starting with the Heaven trigram, and taking subsets that
double in size. The number 3 in the TitiusBode Law denotes its number of yang lines.
Therefore, the Heaven trigram defines the mean distance of Venus from the asymptotic
centre. Then the first two trigrams have six lines. They correspond to Earth (n=2),
whose mean distance from the centre is six units. Next, the first four trigrams have 12
lines. They define the mean distance of Mars (n=3) from the centre. All eight trigrams
have 24 lines, which define the mean distance 24 of the Asteroid Belt (n=4) from the
centre.
Having counted the first row of trigrams, one proceeds to count
downwards, now doubling the number of rows each time. The first row of hexagrams has 48
lines, which defines the mean distance of Jupiter (n=5) from the centre. The first two
rows have 96 lines, which defines the mean distance of Saturn (n=6) from the centre. The
first four rows comprise 192 lines, which defines the average distance of Uranus (n=7)
from the centre. This completes the first octet of planets. Uranus is at the asymptotic
centre of the second, rescaled octet of planets whose distances from this centre are
given by^{9}
D_{n’}= 96×2^{n'1},
where n = 1 for Neptune and n = 2 for Pluto. Instead of counting two,
four and eight trigrams again, the rescaling means that the first four rows of
hexagrams replaces the Heaven trigram and we commence counting two rows of hexagrams
with 96 lines. These define the average distance 96 of Neptune from the second
asymptotic centre, that is, its distance from the first
asymptotic centre (its true Bode number) = 196 + 92 = 288. Then we count
four rows of hexagrams with 192 lines, which define the distance 192 of Pluto from the
second centre. This means that its average distance from the first asymptotic centre
(its actual Bode number) = 192 + 192 = 384.
The four successive powers of 2 in the Platonic Lambda:
2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} =
8,
express this doubling of successive planetary distances from the
asymptotic centre Similarly for the analogous progressive doubling of lines in the I
Ching table. The
26
doubling stops at the fourth term in the geometric series,
which defines both the eight trigrams, the first eight rows of trigrams making
up
the first half of the I Ching table and all eight rows of hexagrams — an
illustration of the Tetrad Principle at work. 192 — the distance from the asymptotic
centre of the last member of the first octet of planets and the number of yang or yin
lines in the I Ching table — is the wavelength of the perfect fourth of the
eighth octave of undertones, where 8 is the fourth even integer. In musical
terms, the Heaven trigram can be regarded as the octave of the Pythagorean musical scale
and the 128 trigrams of the I Ching table span a musical octave ending in the pair of
Earth trigrams of the sixtyfourth hexagram — the lowest note of the eightfold scale.
27
128 is the tone ratio of the eighth octave and the fiftieth tone or
undertone in the Pythagorean scale, showing how Elohim, the Godname of Binah with the
number value 50, prescribes this number. The sum of the binary values of the 28
hexagrams on the boundary of the table (Table 1) is 196. This is the distance of Uranus from the Sun predicted by
the logarithmic spiral theory.^{10} The boundary of the table encodes not only the superstring
structural parameter 168 but also the scale of the first planetary octet.
It is fitting that the number value of Nogah, the Mundane
Chakra of Netzach, the seventh Sephirah, is 64, for it is the number of hexagrams
resulting from the seventh doubling up (see Fig. 30). It is also fitting that the number value of Kokab,
the Mundane Chakra of Hod (the eighth Sephirah) is 48, for this is the number of
yin/yang lines in the eight hexagrams of any row of the I Ching table. That the
pattern of 64 hexagrams unfolding in the table is Kabbalistically complete is confirmed
by the fact that 64 Sephirothic emanations are needed to create ten overlapping
Trees of Life, that is, a Tree of Life differentiated by having each Sephirah
represented by a Tree of Life (Fig. 32).
As Fig. 24 indicates, there are 64 corners of the polygons making up the
inner form of the Tree of Life that do not coincide with the positions of Sephiroth on
the side pillars, that is, 64 new, geometrical degrees of freedom intrinsic to this
inner form shape its growth out of what the author calls the ‘root edge’ shared
by the polygons.
In Article 11,^{11} the Lambda tetractys was generalised to a tetrahedron of
which it is the first face. The fourth face is a tetractys array of integers the ratios
of which define the tone ratios of the perfect fourths, perfect fifths and octaves of
the Pythagorean musical scale. The sum (350) of its twenty integers is the number of
corners of the ten sets of seven polygons enfolded in ten overlapping Trees of Life.
Fig. 33 indicates that its largest integer is 64. It further illustrates
how this number characterizes archetypal objects such as the inner Tree of Life, ten
Trees of Life and the I Ching table. In fact, the geometric series of integers 8, 16, 32
and 64 lining an edge of the fourth face denotes the number of hexagrams in,
respectively, one, two, four and eight rows of the table.
8. Octonions and the I Ching
table
We found earlier that the geometric terms 1, 2, 4 and 8 in the Platonic
Lambda define the number of trigrams in the four stages of the building of the row of
eight trigrams and the factors by which the rows of hexagrams build up into the I Ching
table. Mathematicians have known for many years that there are four — and only four —
classes of normed, real division algebras. They are the 1dimensional real numbers, the
2dimensional complex numbers, the 4dimensional quaternions and the 8dimensional
octonions, or Cayley numbers.
An octonion has the form:
28
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 = 1–7) are imaginary numbers:
and are anticommutative in their multiplication:
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}):
Table 3. The octonion multiplication table.
Unlike the I Ching table, which has 64 different pairs of trigrams
symbolising in binary form the 15 different integers 0–14 created by adding together the
values of pairs of trigrams, Table 3 has 16 different entries: 1, 1, and ±e_{i}.
As the realisation of the most general type of numbers showing neither
associativity nor commutativity, the eight unit octonions are the mathematical
counterpart to Daath and the seven Sephiroth of Construction. Their correspondence with
the eight trigrams is set out below:
We saw in Section 15 that the sum of the values of the 42 offdiagonal
hexagrams whose trigrams express the numbers 1–7 is 336, the 21 values on either side
being 168. These values are the sums of the pairs of hexagrams. If the rule is imposed
that addition should be modulo 8, the sum of the 49 integers in the new table is 168,
the number value of Cholem Yesodoth, Mundane Chakra of Malkuth. This shows how
El Chai, the Godname of Yesod with number 49, prescribes this structural parameter of
superstrings. That this is not coincidence is indicated by the fact that the number
values 90 and 78 of, respectively, Yesodoth and Cholem are the sum of
the first three rows and the sum of the last four rows of the table of new numbers
(Fig. 34). This reflects the fact that the sum of the original 49
integers, of which 28 are greater than or equal to 8, is
29
392, so that addition modulo 8 yields the number 392 – 28×8 = 168.
The Godname Ehieh assigned to Kether with number value 21 prescribes the
superstring structural parameter 168 because it is the sum of the 21 different pairs of
integers that can be selected from the set of integers 1–7.
There are seven 3tuples e_{i}, e_{i+1} and
e_{i+3}, where the index is defined modulo 7. They are represented by the Fano
projective plane (Fig. 35). It generates the multiplication table for the seven imaginary,
unit octonions when they are assigned to its points in the way shown in Fig. 35. The multiplication of two octonions is the third on the same
straight or curved line provided that their multiplicative ordering follows the
arrows
on the line. Their product is minus the third if multiplication is
opposite to the sense of the arrow. Although the Fano plane is no more than a mnemonic,
the diagram is simply connected to the generation of the Tree of Life from similar
circles that overlap centre to circumference (Fig. 36). The three lines inside the triangle are paths of the Tree of
Life. The circles generate the internal contour of the Tai Chi. When the path joining
Chesed
30
and Geburah is extended to cut the circle of the Tai Chi, a square
constructed on this extension and its upper edge extended back to intersect the vertical
line along the Pillar of Mercy, the rectangle formed by the vertical and horizontal
lines is a Golden Rectangle. The Golden Ratio is therefore present in the Fano plane
when it is centred on Tiphareth (meaning “beauty”) as the focal point of the Tree of
Life and the centre of the Tai Chi.
The Fano plane has the symmetry group SL(2,3) — the special linear group
of 3×3 invertible matrices with determinant 1 over the field (1, 1). It is isomorphic
to PSL(2,7), the projective special linear group. This is the automorphism group of the
Klein Quartic equation well known to mathematicians:
X^{3}Y + Y^{3}Z + Z^{3}X = 0
A Riemann surface with the maximum number of automorphisms is a Hurwitz
curve. The Klein curve is the Hurwitz curve with the smallest genus 3.
It has 168 automorphisms. They can be represented by the Klein
Configuration (Fig. 37). It has 168 coloured, hyperbolic triangles and 168 grey
triangles, each one representing an automorphism. Each of its seven identical sectors
comprises two slices containing 24 coloured triangles and 24 grey triangles. One slice
comprises four red triangles, four yellow triangles and four green triangles. The other
slice three yellow red triangles, three yellow triangles, three green triangles and
three blue triangles. The seven former types have 28 red, 28 yellow and 28 green
triangles, that is, 28 sets of a red, blue and green triangle. The Klein Configuration
is therefore made up of 28 triplets of a red, blue and green triangle, seven triplets of
red triangles, seven triplets of yellow triangles, seven triplets of green triangles and
seven triplets of blue triangles. In other words, it comprises 28 triplets of one kind
(one red, one yellow, one green triangle) — a total of 84 triangles — and 28 triplets of
four other kinds. The colours codify different types of automorphisms. They comprise 49
red triangles, 49 yellow triangles, 49 green triangles and 21 blue triangles. This shows
how the Godnames Ehieh with number value 21 and El Chai with number value 49 prescribe
the Klein Configuration.
What is remarkable about this grouping of the 168 coloured triangles
into triplets, 28 of one kind with 84 triangles and 84 of four other kinds with 84
triangles, is that it correlates with the fact that the I Ching table is composed of
triplets of yin/yang lines — trigrams — and that there are 56 trigrams on each
side of the diagonal. It is immaterial that not all the coloured triangles cluster in
each slice in groups of 3. Instead, what is nontrivial is that the numbers of triangles
of each of the four colours are such that they permit natural division into 28 triplets
of one kind and 28 triplets of four other types. The coloured triangles correspond to
the yin and yang lines in the 28 offdiagonal hexagrams in a half of the table and the
grey triangles correspond to the yin and yang
31
32
lines in the 28 offdiagonal hexagrams in the other half. Figure 38 displays the isomorphism between the features of the I Ching
table and the Klein Configuration.
Further evidence that the I Ching table encodes the physics of
superstrings is provided by the root structure of the superstring gauge symmetry group
E_{8}. 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. The 56 permutations of
the 8tuple (1, 1, 0, 0, 0, 0, 0, 0) comprise 28 with the coordinate 1 on the
righthand of the coordinate 1 and 28 with 1 on its lefthand. The 168 permutations
therefore comprise six sets of 28. Compare this with the 28 hexagrams on either side of
the diagonal in the I Ching table. Figure 23 indicates that they have 84 yin lines and 84 yang lines. Each
yin or yang line denotes a simple root of E_{8} and a hexagram symbolizes the
six types of 8tuples, the 28 hexagrams reflecting the fact that there are 28
permutations of each type. Here is unambiguous evidence of the connection between the
group mathematics of superstrings and the pattern of the ancient I Ching table.
The division of each of the seven slices of the Klein Configuration into
two sets of 12 coloured triangles generates the division 168 = 84 + 84. Its counterpart
in the I Ching table is the fact that there are 84 yin lines and 84 yang lines making up
the 56 trigrams on side of the diagonal. Similarly, the two sets of 12 grey triangles in
each slice have their counterpart in the 84 yin lines and 84 yang lines of the 56
trigrams on the other side of the diagonal. In terms of octonions, it was shown in
Article 15^{12} that the 12 coloured triangles in one half of a slice
correspond to permutations of pairs and triplets of octonions in a 3tuple
(e_{i}, e_{i+1}, e_{i+3}), whilst the 12 coloured triangles in
the other slice correspond to the permutations of pairs and triplets of the inverses of
octonions in a 3tuple, namely, (e_{i+3}, e_{i+1}, e_{i} )
(multiplication of inverse octonions operates in the
33
34
opposite sense of the arrows of the Fano plane in Fig. 35). One can formally identify a yang line as an octonion
e_{i} and a yin line as the inverse of an octonion, i.e., e_{i} . This
is consistent with the intuitive meaning of yang and yin as positive and negative
polarity.
Figure 39 shows the correspondence between the Tree of Life and the
Klein Configuration. Associated with the righthand set of the first six regular
polygons are 168 black dots other than their 25 corners. They correspond to the 168
black hyperbolic triangles in the Klein Configuration. Associated with the lefthand set
of similar polygons are 168 white dots other than their 25 corners. They correspond to
the 168 white triangles. The Divine Name Elohim with number value 50 prescribes the 336
dots because the two sets of polygons have 50 corners.
As was pointed out in Article 15,^{13} the two sets of 12 coloured triangles in each slice of the
Klein Configuration correspond to the 12 permutations of pairs and triplets of each
3tuple of octonions (e_{i} , e_{i+1}, e_{i+3}) and to the 12
permutations of pairs and triplets of each 3tuple of their inverses (e_{i+3},
e_{i+1}, e_{i}). The (84+84) permutations of the seven 3tuples are
symbolized by the (84+84) dots on the boundaries of the first (6+6) polygons. As each
whorl of the UPA makes 168 circularly polarised oscillations in every half revolution,
84 of them in every quarterrevolution, each dot also denotes an oscillation. The
following equivalence is established:
The inner form of the Tree of Life, the I Ching and the Klein
Configuration representing the group of automorphisms of the Klein Quartic are
isomorphic versions of the same pattern, namely, the universal blueprint that determines
the nature of all things made in the biblical ‘Image of God’ — whether a superstring or
the Solar System.
In the Pythagorean counterpart of the mystical traditions of Kabbalah
and Taoism, the representation of whole (or holistic) systems is the
tetractys. Its 10 dots symbolize the tenfold nature of Divine Unity. If
one calls it the ‘1storder tetractys,’ its next higher differentiation is the
‘2ndorder tetractys’ shown in Fig. 40. Its ten tetractyses are composed of 85 dots, where 84 is the
sum of the first four powers of 4. This fact is another example of the Tetrad
Principle^{14} proposed by the author, wherein archetypal patterns or
numbers of universal (and therefore scientific) significance are expressed by the number
4. We saw in Section 6 that, when the 70 sectors of the two sets of six polygons are
turned into tetractyses, 336 dots are generated that correspond to the 336 yin/yang
lines of the two sets of 28 hexagrams on either side of the diagonal of the I Ching.
Miraculously, when the 47 sectors of all seven enfolded, regular polygons are
35
36
37
converted into 2ndorder tetractyses, the number of dots in them is 3360
(Fig. 41), that is, the number of dots in 336 tetractyses. If, therefore,
a yin or yang line is considered to symbolize not 0 or 1 but 10, the numerical value of
the 56 offdiagonal hexagrams is 3360 — precisely, the same as the number of dots
composing the seven polygons! This illustrates the remarkable mathematical unity
underlying symbols from the mystical traditions of diverse cultures. The scientific
significance of this equivalence is the fact that, because each of the ten whorls of the
UPA/superstring makes five revolutions as it oscillates 1680 times, all ten whorls make
16800 oscillations, that is, 3360 oscillations for each revolution. Assigning the number
10 to a yin and yang line in the I Ching table is therefore equivalent to taking into
account the tenfold nature of the superstring as the realization of the tenfold Tree
of Life in the subatomic world. Amazingly, the geometry of the inner form of the latter
encodes through the tetractys the very number of helical turns of the ten whorls that
define the vibrational state of the superstring constituent of up and down quarks.
If the number 10 is assigned to a hexagram, the 64 hexagrams of the
table add up to 640. This is the number value of Shemesh (“the solar light”),
the Hebrew name of the Mundane Chakra assigned to Tiphareth. Indeed, 64 is the number
value of Nogah (“glittering splendour”), the Mundane Chakra of Netzach, the
Sephirah following Tiphareth. Hod, the next Sephirah, has a Mundane Chakra
called Kokab with the number value 48, which is the number of yin/yang lines in
a generic row of eight hexagrams. Starting with the Heaven trigram composed of three
yang lines, a further 87 trigrams border two sides of the I Ching table with 15
hexagrams prescribed by the Godname Yah. This is the number value of Levanah
(“the lunar flame”), the Mundane Chakra of Yesod, which follows Hod in the Tree of Life.
Including Cholem Yesodoth (“breaker of the foundations”), which is the Mundane
Chakra of the last Sephirah, Malkuth, whose number value 168 is the number of yin/yang
lines of the 28 offdiagonal hexagrams, we see that five consecutive Sephiroth
of the Tree of Life have Mundane Chakras whose number values aptly and naturally
characterize properties of the I Ching table. Assigning 10 to hexagrams gives a sum of
280 to these 28 hexagrams. This is the number value of Sandalphon (“Manifest
Messiah”), the Archangel associated with Malkuth. The sum of the eight diagonal
hexagrams becomes 80, which is the number value of Yesod, the penultimate Sephirah. As
shown in Fig. 20, their 24 yin lines and 24 yang lines correspond to the 24
corners defining the shapes of each set of six polygons embodying the superstring
structural parameter 168. In the Klein Configuration, they are the 24 coloured triangles
and the 24 grey triangles in each of its seven slices shown in Fig. 37. Between the first hexagram composed of two Heaven trigrams and
the last hexagram made up of two Earth trigrams there are 62 hexagrams. This is the
number value of Tzadkiel (“Benevolence of God”), the Archangel of Chesed. It is
the number of corners of the two sets of seven enfolded polygons constituting the inner
form of the Tree of Life that do not coincide with the positions of the Sephiroth when
its outer form is projected onto the plane containing the polygons (see Fig. 24).
The presence of Kabbalistic numbers in the I Ching raises the question
of whether it is just chance that the Hebrew Godnames have number values (Fig. 42) that naturally quantify types of hexagrams in the
table. Any supposedly more rational explanation that replaces one set of miraculous,
numerical correlations by an equally extraordinary series of coincidences is hardly more
plausible. The only reason why there are too many Kabbalistic numbers characterizing a
5,000years old Chinese diagram to be due to chance is that the I Ching is the Chinese
counterpart of the Tree of Life blueprint determining the nature of Tao. The only
sensible rationale for the remarkable isomorphism between the I Ching table, the Tree of
Life and the Klein Configuration
38
revealed in this article is simple: they are different ways of
describing fundamental reality. Despite outward differences in its expression,
Divine Truth remains the same whatever the period or culture in which it was mystically
revealed.
References
^{1}Recently, a physicist
called Shahriar Afshar has conducted experiments that seem to contradict the Principle of
Complementarity by demonstrating that particles can behave as waves at the same time. His
findings await replication and scrutiny by other researchers (see “Quantum Rebel,” by
Marcus Chown, New Scientist, 24 July, 2004).
^{2} http://www.chinesefortunecalendar.com/FAQ.htm.
^{3} A useful commentary on the 64 hexagrams can be found at: http://11members.aol.com/tig550908/ihexagrams.htm.
^{4} Besant, Annie and Leadbeater, Charles W. Occult Chemistry,
1st ed., Theosophical Publishing House, Adyar, Chennai, India, 1908.
^{5} Phillips, Stephen M. Extrasensory Perception of Quarks,
Theosophical Publishing House, Wheaton, Illinois, 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.
^{6} Phillips, Stephen M. Article 1: “The Pythagorean Nature of Superstring
& Bosonic String Theories,” (WEB, PDF), p. 5.
^{7} Phillips, S.M.The Mathematical Connection between Religion and
Science (Antony Rowe, England, 2009).
^{8} Ibid.
^{9} Phillips, Stephen M. Article 17: “The Logarithmic Spiral Basis of the
TitiusBode Law,” (WEB, PDF), pp. 25–26.
^{10} Ibid, p. 26.
^{11} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its Meaning,
Generalisation and Connection to the Tree of Life,” (WEB, PDF).
^{12} Phillips, Stephen M. Article 15: “The Mathematical Connection between
Superstrings and Their Micropsi Description: a Pointer Towards Mtheory,” (WEB, PDF), pp. 10–12.
^{13} Ibid.
^{14} Ref. 5.
39
