This essay was first published in TORUS, the Journal of the Meru Foundation, Vol. 2 #3 (July 1993).
A version in German was published in Tattva Viveka, Issue #5, October 1996, Frankfurt, Germany.
A shorter version was published in Gnosis #28, Summer 1993.
The Noetic Journal published
the accompanying piece to this article, Man
Bites Dog, as an Appendix to The God of Abraham. This article is usually published
with copious illustrations. A few of those illustrations are reproduced
here; many more are available in other places on this website. In particular,
the reader should refer to the following Meru Foundation Presentation Posters:
Unity exists when the Flame
is wedded to the Coal: Maximal Asymmetry and Maximal Symmetry
THE GOD OF ABRAHAM: A Mathematician's
The "idols" of sophisticated people are not literal
stone effigies. Instead they are the cultural, social, and scientific paradigms
of the societies in which they live.
The Talmud, expanding on the Biblical account, tells us
that Abraham discovered the One-God while he lived in the house of his
father Terach. Terach, we are also told, was an educated professional knowledgeable
in the skills, philosophies and spiritual practices known in his homeland
in "Ur of the Chaldees," a civilized and sophisticated city-state in "Babylonia."
"It was into this pagan atmosphere that a most
unique individual was born. From his earliest childhood,32 Abraham transcended his pagan environment33 and recognized that the world was governed by one Supreme Being.34 As one of the greatest geniuses of his time,35 Abraham was able to use his keen
mind to see through the sham and falsehood of the values of his generation,and
understand the true purpose of creation."1
Abraham is at home. He is reflecting on the many idols of
metal and stone and wood that his father makes and sells. He has a realization
that these idols are dead and inert and he "discovers" the One-God. This
is a paraphrase of the traditional story. How are we to understand it?
If we, as our scholars, assume that this story is to be
taken literally, then Abraham realizes that the idols his father makes
and sells are not alive, and thus not able to have any effect in the world
no matter what or how they are prayed to and no matter what sacrifice is
offered to them. We are assuming that a man whom we are told is educated
and sophisticated would make and sell inert effigies as deities. If we
met an educated and sophisticated person today we would not find it plausible
that they would be in the business of selling idols. Why should we make
that presumption about Terach?
Consider instead the following scenario. Terach, as an
educated and worldly man, would know and appreciate the arts, sciences
and spiritual beliefs of as many different peoples and cultures as were
known in his world. We assume this of educated persons today. If this is
so, then the "idols of metal and stone and wood" referred to in the traditional
story are really the particular cultural embodiments of the arts, the sciences,
and the sacred as they are known in various different cultures.
The "idols" of a sophisticated person are not, literally,
stone effigies and statuettes. These "idols" are the cultural, political,
social, and scientific paradigms comprising the world-views of the societies
in which they (and we) live.
When we examine the spiritual beliefs and cosmologies
of many ancient and modern cultures we find that they all include excellent
models of certain essential qualities of life - albeit each in its own
cultural context with its own particular perspective, emphasis and physical
The ancient Chinese developed a cosmology and an original
ideographic alphabet based on the 28-mansions of the lunar zodiac.
The Greeks and the Persians modeled the cyclicity of life
by a pantheon of gods, goddesses and their familial relationships
based on the 12-houses of the solar zodiac.2
The Druids of northern Europe modeled the self-propagation
of life on the life cycles of trees and other growing things.
Each and every culture has made accurate and effective models
of the cyclic, self-propagating and self-referential nature of all life
in terms appropriate to its needs and experience. These different cultural
embodiments of the same universal principles underlying all life are referred
to as "idols of metal and stone and wood." These "godlet" cultural paradigms
are honored (and, literally, stone statues of these "idols" are worshipped)
by the society that makes use of them.
All cultures model the same processes of the same overall
unity of the natural world and each uses a different physical example to
Abraham, seeing through each example to a Singular archetype, DEFINED the One-God as the Unity underlying all of them.
Abraham, in this view, acts as a mathematician: he postulates
a meaningful and functional definition of Unity. The mathematician's
model makes use of none of the "garments" of the many different cultural
embodiments. Even though it is a mathematical model that must make use
of geometry and form (or formalism) to be expressed, it (the model, not
the sacred) MUST be understood as a complete abstraction without physical
embodiment. A good mathematician tolerates no unneeded embellishments.
This perspective suggests why the Abrahamic faiths absolutely
prohibit "graven images" of God. Any "graven image" would be a physical
representation of only one culture's iconography during one historical
period - it could never be a timeless model of a universal underlying Unity.
Once we understand this mathematician's idea of God as
a DEFINITION necessary for universality we can, perhaps for the first time,
see how and why it is possible that the Abrahamic faiths' insistence that
God is the ONLY-GOD could be literally true, and not just the chauvinistic
religious puffery of these faiths - and in a way that does not impugn the
validity of other religions. The definition of Unity is in no way prejudicial
to any other view.
Even before Moses was given the Torah, it is possible
that Abraham realized that this ultimate, Singular, definition of the universal
One-God is also identical with the personal meditational experience of
God. That the Immanence of All-There-Is and the Transcendent Singularity
in our experience of meditation are one and the same may be the basis of
Abraham's understanding of and belief in the One-God.
This suggests the truly extraordinary possibility that
our ancient sages also realized that there is one particular mathematical
definition of Unity that is also a model of the sequence of feelings, the
"Yoga" and the "Hero's Journey", that leads to the meditational experience
of Unity. The meditational experience may be the gnosis that personally
validates and empowers these spiritual tradition(s).
Further, although the particular details and depth of
understanding of the idea of an explicit definition of the Unity of God
may have been most fully developed by the Abrahamic faiths, the principle
was known and considered fundamental in other cultures as well. Terach
and Abraham did not live in a cultural vacuum.
This is from Anne Macaulay in Lindesfarne Letter No. 14, p. 109 (emphasis added):
"And finally let us return to Apollo.
It came as a shock to me to find that this god was derived from a geometric
figure. This is no deified hero nor an archetypal godlet, but a synthesis
of observed facts about the relation of the stars, the moon and sun to
earth; it is a timepiece and a calendar; it is a statement of the laws
of the heavens in terms of geometrical and mathematical elements which
themselves exhibit absolute laws; and these laws also apply to music: this
must surely be the music of the spheres. The concept is total and
the harmonic nature of the music demonstrates the great harmony of creation. Apollo
can thus be seen as the logos or in another
sense as the definition of the absolute god."3
From The Egyptian Miracle by R.A. Schwaller de Lubicz,
p.147 (emphasis added):
"In Pharaonic Egypt (as also in India, incidentally, although with dangerous complications),
Neter Netru, the one and only God who is unknowable, is an idea
derived by logical conclusion."4
It is evidence of considerable philosophical sophistication
on their part that the sages of the ancient world were apparently aware
of the necessity of identifying the One-God with an abstract definition
SINGULARITY AND THE ONE-GOD
As many musicians and electronics enthusiasts know, "...the
harmonic nature of...music demonstrates the great harmony of creation."
(See Macaulay quotation, above.) Every musical pulse is made up of the
sum of many pure sine-wave tones; an ordinary "square wave" is made up
of many odd harmonics, and, by extrapolation, a truly infinite pulse w
ould consist of a sum of all possible pure tones.
The way musicians examine the spectrum of musical harmonics
is exactly the same as the procedure mathematicians call a Fourier Transform:
a sharp loud pulse consists of a broad spectrum of pure tones. Likewise
an infinitely loud, short, sharp pulse - which we could compare
to a musical Singularity - would produce the harmonic spectrum of ALL tones
- which we could liken to ALL-THERE-IS.
The Fourier Transform of ALL-THERE-IS, is a single pulse of infinite intensity and infinitesimal duration at the start of
time - at creation. This suggests that the Big Bang unfolds the modern
physicist's model of creation from an exquisitely singular and intense
pulse that may be mathematically equivalent to Abraham's DEFINITION of
THE FOURIER TRANSFORM between
UNITY and WHOLENESS
Fourier Transform of ALL-THERE-IS <=> Fourier Transform of PULSE
The Singular Pulse at "Creation"
is the Fourier Transform of an eternity of ALL-THERE-IS;
The Spectrum of ALL Tones and Harmonics,
is the Fourier Transform of the
If the universe is limited in extent and duration,
then its ultimate source must be less than omnipotent. Thus the presumption
of ONE exquisitely Infinite source demands that the universe be infinite,
open and eternal. The Singular Pulse may be the Kabbalist's "line" (Qav)
that extends from the "creation-contraction" (ZimZum) into All-There-Is.
Abraham's definition of the One-God and our modern understanding of this
universe may well be based on the same fundamental principles.
Here is another view that extends the idea of our closest logical
singularity - the sun, the ostensible god of many ancient peoples -
into a suitable definition of the Exquisite, Transcendental Unity of the
From physicist Roger Penrose's, The Emperor's New Mind, page
318-19, italics sic; [square brackets] and bold-face added for emphasis:
"Where indeed does our own low entropy
come from? The organization in our bodies comes from the food that we eat
and the oxygen that we breathe.
"The question is how we keep ourselves alive throughout out normal (mainly adult) lives. For that, we do not need to add to our energy content.
"...To keep ourselves alive, we need to keep
lowering the entropy that is within ourselves.
"Where does this supply of low entropy come from?
"... green plants ... [provide] it by making
use of sunlight. The light from the sun brings energy to the earth
in a comparatively low-entropy form, namely in the photons of visible
light. The earth, including its inhabitants, does not retain this
energy, but (after some while) re-radiates it all back into space. However,
the re-radiated energy is in a high-entropy form, namely what is
called 'radiant heat' - which means infra-red photons. Contrary to a common
impression, the earth does not gain [net] energy from the sun! What
the earth does is to take energy in low-entropy form, and then spew it all back again into space, but in a high-entropy form. What the sun has
done for us is to supply us with a huge source of low entropy. We (via
the plant's cleverness), make use of this, ultimately extracting some tiny
part of this low entropy and converting it into the remarkable and intricately
organized structures that are ourselves.
"The energy the earth spews
back into space is spread over many more degrees of freedom than is the
energy that it receives from the sun. Since there are so many more degrees
of freedom involved when the energy is sent back out again, ... the entropy has gone up enormously. The green plants, by taking in energy
in a low-entropy form (comparatively few visible-light photons)
and re-radiating it a high-entropy form (comparatively many infra-red
photons) have been able to feed on this low entropy and provide us with
this oxygen-carbon separation that we need.
"All this is made possible by the fact that
the sun is a hot-spot in the sky."5
Professor Penrose is telling us that the "raw" information needed to sustain
the self-organization of life on earth is due to the "separation" -
the distinction - between oxygen and carbon and this, in turn, is due
to the distinction - the contrast - between the hot, bright sun and
the cool, dark sky.
This model suggests why in traditional societies (as well as today)
those who believe that life is entirely physical and that consciousness
is an epiphenomenon of complex physical organization alone believe that
the source of life is the Sun. Exclusive materialism is the equivalent
of solar worship. In effect, materialists are not atheists. In this context,
materialists can truly be said to worship the hot "metal and stone" idol
the Greeks called Apollo.
Roger Penrose's TWISTER
We can take
Professor Penrose's observations one step further. After all, life on Earth
reaches one further step of organization. Human beings (and perhaps a few
other creatures such as some primates, elephants and the cetacea) are not
only living animals - whose living neg-entropic organization comes from
the contrast of the bright physical sun in the dark sky - but we are also
conscious and aware of ourselves. Humans have a special hand with an opposable
thumb marking our self-reflexive awareness and - some say consequently
- we form and use language.
Following Penrose's argument, where does our "hyper" self-aware consciousness
get its organizing information? What "hand" "informs" our special, "higher"
Does there exist or can we hypothesize a "higher" "hyper" contrast than
between our star and the sky? Conceptually, at least, we can. The contrast
between our physical sun and the sky can be idealized and extended by considering,
at least in principle, an even higher, brighter source - a source of ALL-THERE-IS.
If the sun is a very bright, yet finite source, very far away, but not
infinitely far away, then we could investigate the consequences of an infinite
source at an infinite distance from us. We can define a model of
an exquisitely singular and omnipotent source.
As with the Fourier Transform - musical harmonics ("music of the spheres")
model, our extension of Professor Penrose's High Contrast leads
us to an infinite pulse. (It is interesting to note that Kabbalists understand
this principle of highest contrast as the paradox of God withdrawing or
contracting (tzimtzum) and then projecting.) In this case the pulse
is an infinitely energetic Singularity infinitely far away.
This then could be the "hyper" source of our human self-aware consciousness.
This conceptually farthest, brightest source is also the conceptually highest
information, lowest entropy source. It is the "hyper-neg-entropic" source
of universal consciousness that corresponds to the One-God. By (Abraham's)
definition this is the most "high-contrast" source.
MODELS OF WHOLENESS, SINGULARITY AND HIGHEST
How are we to model this ultimate, exquisite, unknowable UNITY representing
our unique definition of the Singularity of the One-Living-God? Is there
an entirely abstract - non-idolatrous - mathematical model that
incorporates Singularity, Uniqueness, Self-organization, Universality,
Infinitude, Elegance and Simplicity?
What is the most elegant and exquisite model of "highest contrast"?
Could this same model also represent the meditational process or the path
and goal of the "Hero's Journey?"
In The Laws of Form, mathematician G. Spencer-Brown proposes the "mark
of distinction" archetypally distinguishing INSIDE from OUTSIDE as a definition of maximal contrast. Mathematicians have shown that all
of formal logic can be derived from G. Spencer-Brown's "mark of distinction."
The following is from The Laws of Form, p. xxix (emphasis added):
"The theme of this book is that a universe
comes into being when a space is severed or taken apart. The skin of
a living organism cuts off an outside from an inside. So
does the circumference of a circle in a plane. By tracing the way we represent
such a severance, we can begin to reconstruct, with an accuracy and coverage
that appear almost uncanny, the basic forms underlying linguistic, mathematical,
physical, and biological science, and can begin to see how the familiar
laws of our own experience follow inexorably from the original act of severance.
"Although all forms, and thus
all universes, are possible, and any particular form is mutable, it becomes
evident that the laws relating such forms are the same in any universe. It is this sameness, the idea that we can find a reality which is independent
of how the universe actually appears, that lends such fascination to the
study of mathematics."6
BET - "house"
In Hebrew the letter that most
represents this "mark of distinction" between inside and outside is Bet,
the first letter of the Hebrew text of Genesis.
It appropriately establishes, by definition,
the first logical distinction possible.
The distinction between archetypal Symmetry and Asymmetry is also primary and of absolute contrast. Each co-defines the other. Without
a representative standard of asymmetry how could we unambiguously define
Archetypal symmetry can be represented by the most compact structural
forms (in any given dimension). The five Platonic solids (Tetrahedron,
Cube, Octahedron, Icosahedron, and Dodecahedron) and the Archimedian semi-regular
solids can define fundamental symmetries in 3-dimensions. (Mathematicians
and physicists derive the formal symmetry groups that they use from these
Sample Platonic polyhedra based
on drawings from Keith Critchlow's Order in Space, p. 19.7
Archetypal asymmetry can be represented by a dynamic form that
continuously breaks symmetries as it unfurls. Meru Foundation research
suggests that this form is a particular, explicit vortex, which we call "Naked
Recursion" ("naked" in the mathematical sense - unadorned, without
any other quality) and which has been traditionally associated with "the
flame of consciousness", the archetypal living process of a "fruit tree
yielding fruit whose seed is inside itself" (Genesis I.11.), and with
its highest human embodiment - our hands.
As we will see later, besides its traditional association with the heart
Chakra, what is most astonishing about the vortex-shaped model HAND is
its direct relationship to both our personal consciousness and to our cosmological
model of Unity, Singularity and Wholeness.
Traditionally mystics and Kabbalists have referred
to the asymmetry vs. symmetry model of highest contrast as "Flame and Coal"
(Absolute Unity can only exist when the "flame" is wedded to the "coal"),8 the "Light in the Meeting Tent," or, perhaps, in Plato as the "same (symmetric)
and the different (asymmetric)" (although many other interpretations have
been given for this phrase).
In Taoism "hyper-contrast" is modeled by the Yin-Yang symbol:
Many representations of ultimate contrast are variations of the Inside
and Outside model. These include the hypercube and hypersphere which are
modeled by placing a small cube (or sphere) inside of a larger one. The
In-Out direction models the fourth dimension.
Following Eddington, mathematical philosopher Arthur
M. Young in his The Reflexive Universe, points out that the surface of a 4-dimensional Hypersphere is the same as the volume of a solid
Torus in 3-dimensions.9
In living things the Seed is INside and the Fruit is OUTside,
the Sperm is INside and the Ovum is OUTside:
Flame and Coal
Light and Vessel
Seed and Fruit
Wave and Particle
Female and Male
Spirit and Matter
Mind and Body
Transcendance and Immanence
In these embodiments the asymmetrical dynamic element is conceptually
INside (Seed, Mind, Spirit) with its symmetrical projective complement
OUTside (Fruit, Body, Matter). See Note #2. There are very many other examples.
The modern physical concept of the complementarity of Wave and Particle
as two mutually exclusive (highest contrast) aspects of all fundamental
entities also models the Inside / Outside relationship.
Mathematicians suggest that Singularities - formal mathematical definitions
of Unity - can be modeled by Torus knots. Torus knots are simple "basket-weaving"
or wreathes wound on rings. [For
more complete illustration, see The
3,10 Torus Knot, Ring, Sphere, Tetrahelix and Hand.] The simplest knot is also the simplest Torus Knot. There are explicit vortex
models based on particular Torus knots. The torus itself also defines the
primary distinction between Inside and Outside.
TRANSITION of the 3,10 TORUS
KNOT from a RING onto a DIMPLED-SPHERE
Break TORUS Ring and pull apart
A, B, and C
illustration ofCONTINUOUS CREATION is an attempt to show how a range of spiritual metaphors from different
traditions can be reconciled in one geometric model of the High Contrast
of Wholeness and Singularity. It shows some of the names by which the
DIMPLED SPHERE Torus is known in different cultures and spiritual traditions.
In this model, Continuous Creation refers to the unbroken chain of life:
...Seed-Tree-Fruit / Seed-Tree-Fruit / Seed-Tree... It represents the reflexive
self-organizing process that is the natural transformation and unfoldment
of every "SEED" (Singularity, Tao, Sun) via its "TREE" (Unfoldment, Flame,
World Mountain) into new "FRUIT" (Wholeness, Hand, World) of its kind.
It represents the general principle of the projection of life from life,
endlessly. It is a model of one cycle of Singularity, Unfoldment, and Wholeness.
The IDEALIZED meditational "Fruit
tree yielding fruit whose seed is inside itself," in
the form of a DIMPLED-SPHERE Torus,
defined by pairing off the letters
in the first
verse of the Hebrew text of Genesis.
It is the minimal, most compact
and most elegant representation
of the sequence of letters in the
CONTINUOUS CREATION represents the
verse from Genesis (I,11) that is quoted in the introduction to the Sefer
Zohar: "Fruit tree yielding fruit whose seed is inside itself."
The first verse of the Hebrew text of Genesis represents Naked (mathematically
unadorned) Recursion (...acorn-oak-acorn-oak-acorn-oak....) in the form
of a Human Hand.
Perhaps the most startling quality of this HAND form of NAKED RECURSION
is that when it is placed on our hand like a glove different gestures display
all of the letters of the Hebrew Alphabet. The Arabic and Greek sacred
alphabets are produced by a similar form.
The historical samples of the Hebrew Letters
reproduced in our poster, The Hebrew Alphabet - FIRST HAND (below) are from Reuben Leaf, Hebrew Alphabets, 400 B.C.E. to Our Days.
For each letter, the single right column (next to the printed letter name)
shows the form of the letter from the Elephantine Papyrus, circa 300 BCE.
The four samples in the middle are Merubah ("square-form") rabbinic script
(often referred to as "Rashi" script) from Islamic Spain, circa. 1000-1300
The far left column of Hebrew letters shows photocopy reductions of
actual photographs of various views (shadowgrams) of the same physical
model hand. [For a full-size
rendition of this poster, see The
Hebrew Alphabet - FIRST HAND.]
Because we can feel the model HAND on our hand and because we can always
"see" our own hands in our mind's eye, we can also see the model HAND in
our mind's eye. Close your eyes and point to the ceiling. You will find
that you can tell how your hand is oriented and where your fingers are
pointing even with your eyes closed. Now open your eyes and confirm that
your inner vision of your hand was correct. This means that each gesture
that displays a particular letter in our hand also displays that letter
in our minds.
Since letters generated by hand gestures can represent directions of
conscious focus in our minds, a sequence of these letters can be used to
specify and reconstruct a meditation. This may explain why certain Scriptural
texts were canonized and why these letters are rightly called "sacred."
Thus the same letter-signs are elements of both cosmology and consciousness
at the same time.
Meru Foundation research has also shown that because
the natural meaning of hand gestures is universal for all humans,12 each letter displayed (and seen in the mind's eye) by each gesture has
explicit meaning. For example, the name of the Hebrew letter Pe (Greek
Pi, Arabic Fe, English "P") traditionally means "mouth" (or swallow, puff,
or engulf). It can only be seen in the model hand when it is worn
on our hand - within the limits of motion of our arms and wrists - when
our hand points to our mouth, thumbs in, fingers flared, in a puffing gesture.
Every other letter is similarly explicitly defined by the gesture that
displays it to us. It is usually possible to "decipher" the operational
meaning (not the idiomatic meaning) of root words in all languages (that
can be phonetically related to Hebrew, Greek and Arabic) by examining the
sequence of letters alone.
There are many other examples we could suggest from every culture and
field of study.
Finally, there is the Hebrew word for Unity ("one"): E-cHoD. It means
"archetypal acuteness or sharpness." (Aleph - "archetype of"; cHet - "encompassing";
Dalet - "dividing.") By its exquisite singularity, E-cHoD represents the
essence of "highest contrast." The so-called "credo of Judaism," the Sh'Ma
(in the daily prayer book) states that "..., God is E-cHoD (One, exquisitely
This ancient model of the Unity of physics and consciousness is also
compelling and potent in a modern sense. As physicist Nick Herbert notes
in his discussion at the end of his book Quantum Reality, Beyond the
"One of the greatest scientific achievements imaginable
would be the discovery of an explicit relationship between the waveform
alphabets of quantum physics and certain human states of consciousness."13
It appears that Abraham's model of the One-God and the Hebrew (Greek and
Arabic) alphabet(s) that flow from it satisfy Professor Herbert's quest.
The perspective presented above suggests that we should not lightly dismiss
the spiritual claims of our traditional religions. If we dismiss these
teachings because they appear to be self-serving apologia presented by
interested partisans, as has been suggested to me by friendly scholars, then
we are explicitly excluding the possibility that they have real meaning. Whether or not spiritual discussions are often "damned by the faint praise"
of faith, they must still be explored honestly. Those who have preserved
and protected these traditions may not be the same persons who can make
use of their content.
Although it is bound to be considered insulting (and thus to have no
validity) by those who do not understand geometry and who therefore adhere
to exclusively literal literary understandings of traditional spiritual
traditions, there may in fact be no possible way to describe Exquisite
Singularity without mathematics.
Although some sources insist that it is anecdotal, it is generally believed
that over the entrance to the Platonic Academy it was written that:
"Only he who is familiar with geometry shall be admitted here."
Our spiritual traditions are the vessels of our faiths. The function
of these vessels, like nut-shells or seed-husks, is to protect the kernel
with its germ of life, the flame, the hand of the transcendent, inside.
If these vessels had not survived, there would be no hope for rebirth of
the germ - the "light" - they carry.
Perhaps our presumption that our sages of the past were not really knowledgeable
or wise has blinded us from seeing their knowledge and wisdom. The Abrahamic
claim of an exclusive, Singular and Unique One-God only appears to be prideful
puffery when our sight is overly narrow and literal. We misinterpret our
partial view of the "vessel." Once we are aware of the distinction between
Inside and Outside, between "vessel" and "light," we can appreciate the
true significance of the use of a real model of Singularity for the definition
of the Oneness of God.
35. Abraham was thus the world's greatest astronomer;
Bava Bathra 15a. He is also the leading personage of his time; Kiddushim
32b; Rambam on Genesis 26:29, 40:14.
It is interesting to note that in most of these examples
the positions of the complements could be reversed. For example, in our
models the geometric structure (Coal, Vessel, Particle, Tetrahedron, etc.)
is considered to be "male" because these structural elements, represented
by polyhedra, are rigid, discrete, and logically precise, while the complementary
process (Flame, Light, Wave, Spirit, etc.) is considered to be female because
the vortex forms which represent the process are cyclic, continuous, and
dynamic (or "emotional"). But it could be the other way around:. The polyhedra
could be seen as vessels and outside (like a womb) and thus "female" and
the vortex forms could be seen as penetrating and inside (like a penis)
and thus "male." This is similar to how Yin and Yang continuously flow
into and mutually define each other as each becomes the other.
The ideas presented above are work-in-progress. There
is much ancillary work that is not presented here and there is much work
yet to be done. For further information, please contact us directly.- Stan
Tenen, Meru Foundation, San Anselmo CA, 1993
A. Moran & David H. Kelley, The Alphabet and the Ancient Calendar
Signs, (Palo Alto, CA, Daily Press, 1953,1969), pp. 13-31.
Macaulay, APOLLO: The Pythagorean Definition of God, (West Stockbridge,
MA, Lindesfarne Letter #14: Homage to Pythagoras, 1982), pp. 85-110, quote
Schwaller de Lubicz, The Egyptian Miracle, (New York, Inner Traditions
International, Ltd., 1985), p. 147
Penrose, The Emperor's New Mind, (Oxford, Oxford University Press,
1989), chapter 7: The Arrow of Time, pp. 302-347, especially Fig. 7.7;
Spencer-Brown, Laws of Form, (New York, E.P. Dutton, 1979), from
the Introduction: A Note on the Mathematical Approach, pp. xxix-xxx; chapters
1; 2, pp. 1-7.
Critchlow, Order in Space, (New York, The Viking Press, 1970). This
book catalogs and illustrates the regular and semiregular polyhedra and
their relationships to each other.
R. Blumenthal, Understanding Jewish Mysticism, (New York, Ktav Publishing
House, 1978), from Sefer Yetzirah, Chapter 1, Mishna 7, translation, p.
M. Young, The Reflexive Universe, Evolution of Consciousness, (Lake
Oswego, OR, Robert Briggs Assoc., 1988), pp. 265-7. Highly recommended
for understanding of the process of self-reference and its relationship
to the torus. For more information on the work of Arthur Young, see the
Arthur Young Website at http://www.arthuryoung.com
10Ivars Peterson, The
Mathematical Tourist, (New York, W.H. Freeman, 1988), Fig. 4.9, p.
Leaf, Hebrew Alphabets, 400 B.C.E. to Our Days, (New York, Bloch
Publishing Co., 1950, 1976). This book gives historical samples of the
McNeill, Hand and Mind, What Gestures Reveal about Thought, (University
of Chicago Press, 1992).
Herbert, Quantum Reality, Beyond the New Physics, (New York, Anchor
Press/Doubleday, 1987), p. 249.