and the

Tropical Zodiac

May 1, 1997

PAR 649

Instructor: Richard Tarnas

PCC Faculty et al.

**Note: This paper contains some as yet uncorrected errors
in the construction of the derivation of the twelve fold zodiac. In brief, the
correct approach would involve starting with the circle's most fundamental harmonic
symmetry system, the hexagon, along with the two perpendicular bipolar axes
given by the solstices and the equinoxes. These two axes define a four fold
system. Two different hexagonal six fold systems are required to align with
both of the two perpendicular axes given by the four fold system of the seasons.
The result yields twelve points. There is still enough useful thinking in this
paper that I have not pulled it from the site, but it should NOT be quoted or
referenced as it stands.**

This paper represents the first exposition of a conceptual model which
I believe offers a new theoretical foundation for understanding, and possibly
deriving, the Tropical Zodiac. This model is based on the same principles
which underlie harmonic theory and quantum mechanics. It involves an area
of mathematics known as Bessel functions, but I will attempt explain the
concept through geometric diagrams which do not require any mathematical
background to understand. Ironically, this perhaps hyper-modern thinking
was inspired, at least in part, by the time I spent in recent months with
Robert Hand and Robert Schmidt becoming familiar with some of their tentative
conclusions about the theory and practice of Classical Greek and Medieval
astrology. When confronted with the history of the evolution of Western
Astrology it is difficult to understand how astrology could possibly work
now. So many of the details are so at odds with so much of what we now regard
as "traditional" knowledge.1 And yet,
it is not as if the ancient theory, at least as currently understood and
articulated, is necessarily more self-consistent and therefore theoretically
sound. In trying to make sense of the situation now, as well as then, several
possible insights have emerged which seem to have the potential to be efficacious.
The first is may be a re-exploration of the Tropical Zodiac from first principles,
but in light of modern theories of wave form and resonance modes.

We, as moderns, are accustomed to assuming that our scientific understanding
of the world is based on cause and effect relationships. By this we mean
that we may assume that any event or change of state which we observe in
the material world is the direct result of some material interaction or
exchange of energy. This is the only type of cause and effect relationship
accepted in the modern implicit philosophy. Mechanistic causation was a
combination of just two of the four types of causation articulated by Aristotle.
When we say cause in the modern world we mean material and efficient cause
in the Aristotelian sense. Science seems to have committed a logical error
wherein it has been assumed that because material cause and effect appears
to be valid in the vast majority of cases, then only material and efficient
causes *can* *ever* be valid. Thus, the Aristotelian formal and
final causes were banished from the implicit metaphysics of the modern age.
Yet, at least formal causes appear to have crept back in though quantum
physics while everyone else has continued to pretend that they are still
invalid.

At the very core of modern science lies quantum mechanics, which along with
general relativity, might be described as one of the twin pillars of the
modern scientific world view. Ironically, many of the other sciences still
seem to have "physics envy," striving always to show themselves
to be as mathematically rigorous and therefore as "scientific"
as classical Newtonian physics, while at the same time quantum physics itself
has sped past to become as acausal and paradox ridden as any system of mysticism.
One philosophical implication of quantum mechanical theory could be paraphrased
as follows: If a symmetrical way to divide up space into regions of greater
and lesser electron probability density exists mathematically, then some
state of that system will fulfill that possibility in matter. In other words,
each element of matter will fulfill all of the mathematically valid symmetries
for deploying its electron probability density field. This theoretical model
is accepted virtually as if it were fact even though no one has been able
to observe the wave forms, and no mechanism for explaining how the electrons
could actually move continuously through all these locations has ever been
elaborated. The idea that one could ever demonstrate this is in fact prohibited
by the uncertainty principle in the theory itself. All that we do know is
that the probability of observing an electron in the locations predicted
by the theory is so consistent with what is actually observed that we accept
the validity of the theory.

It is important to understand that theoreticians place far more emphasis
on symmetry and mathematical *elegance* than they do on experiment
and observation. In the end, it is essential that the experimental data
does confirm the mathematical theory, but it is not at all unusual for a
good theoretical physicist to have more confidence in a mathematically coherent
theory than in contradictory data from experiments. Of course, if the great
preponderance of data consistently goes against a theory it must be revised
or abandoned, but the point is that the theories themselves are driven largely
by a faith in the symmetry and coherence of math in nature. Our very understanding
of the quantum realm is based on mathematical constructs of symmetrical
density probability patterns, bolstered by observations which appear confirm
them. Yet these patterns could never be explained by any internal mechanism.
They are essentially expressions of formal causes. We know them only because
of abstract mathematics which predicts that such patterns should exist based
on the geometric and energetic configuration of the system as a whole. Quantum
systems are understood in terms of the wave equation of the whole rather
than through attempting to trace the interactions of the parts. What is
more, when the solutions to the wave equation imply that a certain form
may exist, mathematically, we say that it does in fact exist as a probability
density in matter. Since the theory never says that anything does or does
not exist, only that a probability for it to exist exists, this is equivalent
to saying that if a form exists in mathematical potential, it will also
exist (some of the time) in matter. Thus we have essentially reintroduced
formal causation.

To understand the theoretical basis of the zodiac we must first start
with a review of the geometry of the rational division of circles. There
is nothing new in this as virtually all treatments of the subject have started
with some theory of number, usually involving Pythagorean and or Platonic
associations with the principal numbers in question. While this may be historically
valid in re-deriving the thinking of the ancient Greeks, it is not the line
of thinking I will elaborate here. My principle reason is that we, as moderns,
for the most part do not find such arguments compelling. While some may
take them as esoterically valid, or even philosophically elegant, they are
not accorded an ontological status commensurate with scientific models of
reality. What I hope to show instead is that if we start from a somewhat
different set of premises we may find that we arrive at an identical conclusion,
but based on an argument which we might find more scientifically plausible,
if still not mechanistically compelling.

In some sense the idea of the division of the circle of the seasons into
four is as ancient as human thought, stretching back for millennia into
our deepest psyche. Jung saw it as a sign of integration of the psyche when
both the square and the circle are integrated into a single mandala. Yet,
when it comes to the division of the circle of the heavens not just into
four, but into twelve we have reached an entirely new level of insight and
abstraction. It has been pointed out for millennia that twelve is a combination
of three and four, in modern parlance their product, and the first number
divisible by both. The ancients from at least the time of Pythagoras (and
I suspect from at least the time of the building of the Great Pyramid in
Egypt) had sophisticated doctrines of the philosophical meaning of numbers
based on a kind of associative mathematical thinking which modern science
seems to at once scoff at and fail to comprehend. Without entering into
the details of those doctrines which might have actually led to a twelve
part division of a circle it is sufficient to observe that there are three
possible systems of division of a circle in plane geometry using only a
string (in modern education we say a compass and protractor).

The first, is essentially division by two, into half, and half again into
quarters, and again into eighths, sixteenths, thirty seconds etc. This has
come down to us from ancient systems of measure through the English system
of ounces in a pound, and quarts, pints and cups, as well as sixteenths
of an inch. Even as we are in the process of banishing such unwieldy systems
in the triumph of the rational metric system, the same numbers have re-emerged
in the natural count of bits and bytes of memory in our computers. These
numbers have also long lurked in our own biology, counting our thirty two
teeth and vertebrae like the number of black and white squares on a chess
board. The iteration of two, we have again rediscovered, is so fundamental
that we must make it the base of our computing even if not our counting
numbers.

The second system of division of a circle is into three and six parts. This
system is most closely associated with the nature of circles, as the circle’s
own radius walks around its perimeter exactly six times. Thus, the hexagon
is the most perfect rational approximation of a circle. This is equivalent
to saying that three is the rational approximation of *pi*, and thus
six is the exact approximation of 2*pi*. We can see this symmetry in
a honey comb close-packing pattern of circles in a plane, and also in the
hexagonal patterns in quartz crystals, and in the benzene ring in chemistry.
This circular pattern of six carbon atoms again represents a harmonic series
of standing waves at the quantum level. It is not difficult to see how the
ancients, who associated the circle with the perfection of unity, would
confer a related association upon the hexagon and equilateral triangle.
I am not here pretending to explain Pythagorean numerical symbolism in its
own terms, but rather to point out from first principles, which even we
as moderns would recognize, how and why certain geometric figures and the
numbers they represent would almost inevitably have had certain inherent
associations. Knowing that the equilateral triangle and hexagon are geometrically
most closely associated with a circle, which itself represents unity, may
allow us to better understand why trinities were so often seen as restatements
of unity. The concept of harmony was not far behind, as the elements in
an equilateral triangle seem to be most naturally associated with a cyclic
flow around the perimeter, while those in a square configuration may represent
the poles of two crossed pairs of opposites.

The third possible system of division of a circle is into five, and its
double, ten parts. It is interesting to notice that while it is theoretically
possible to divide a circle into five equal segments, it is not immediately
obvious how to do so unless one knows a trick. Thus, the division of a circle
into five must have constituted an initiation, and almost certainly a secret
initiation when it was first discovered in certain cultures. The division
of a circle into five parts is inconsistent with systems of both three and
four parts, and yet it is so fundamental to biology that we quite literally
hold it in the ten "digits" on our two hands. We have counted
by ten in spite of ourselves for so long that we "based" our number
system on ten, and now in an attempt to rationalize our systems of measurement
have made ten supreme in the modern world. We have done this at the expense
of the dozen which reconciled two, three and four, but not five and ten.
Yet, it is ironic that even while we have embraced five and ten in the realm
of counting numbers, we still seem to no better understand the principles
of the pentagonal organic symmetry in nature which put five in our hands
in the first place. But, for the most part apparently neither did the ancients,
and so it is easy to see why twelve, which successfully reconciled two,
three and four might have been chosen over ten for everything from the Zodiac,
to the clock face, to the counting dozen.

Twelve is also affirmed by the fact that a string, tied in a loop, with
twelve evenly spaced knots in it, may be pulled taught to form a Pythagorean
right triangle with sides of length three, four and five. The length of
each one of these sides represents one of the three fundamental systems
of rational division of a circle in plane geometry. To create the measuring
string knots may be located simply by pulling the closed loop of string
taught into thirds and then into halves twice. Thus, the knowledge of how
to construct such a string might have most likely constituted the first
secret geometric initiation, allowing one to carry a portable means of making
a perfect right angle. (Such a string might also have incidentally constituted
a model of the heavens as understood though the zodiac.)

It is also interesting to note that if one wishes to reconcile all three
systems of symmetry, represented by the numbers three, four and five, the
first number which will do so is sixty. We can see this in the minutes and
seconds on a circular clock face. The first number which will reconcile
all three systems, while also providing for both rational division into
eight parts (quartering the four directions on the compass rose), and division
by nine, is 360. Thus, we find that the number of degrees in a circle is
the circle’s self nature, six, times sixty. Among the single digit
numbers only seven remains outside the system of degree measure and thus
it retains a special status, perhaps as the recapitulation of unity.

In the course of listening to Robert Hand and Robert Schmidt discuss
their findings in Project Hindsight two new and startling bits of information
caused me to again rethink the meaning of the Zodiac. First, Robert Hand’s
discussion of the meaning of "Zoidion". And second, the insight
that the Greeks apparently did not differentiate the Zodiac into twelve
distinct "Signs" as we do, but rather they divided the ecliptic
into six pairs of Zoidia.2 The idea that opposite
Signs have opposite natures is nothing new to modern astrologers, but the
idea that the Greeks understood the Zoidia primarily as six axes of paired
opposites rather than twelve individual Signs was a revelation to me. This
is also apparently difficult to reconcile with the understanding of Zoidion
which Robert Hand so eloquently expurgated as "little bit (quanta)
of (animal) aliveness."3 He further explained
that his current understanding of this apparently baffling term is that
it described a figure standing out against the ground of the general background
aliveness. The idea being that the Greeks saw everything as alive and differentiated
the twelve Zoidia with a diminutive word for animal which could also mean
face. Both of these ideas made a strong impression on me along with a third
implicitly related idea: The Greeks apparently thought of the Zoidia (Signs)
as discrete discontinuous entities. To the Greeks what we would call a Sign
cusp,4 or boundary, was actually a discontinuous
break between one Zoidion and another.5

All of this once again started me thinking about the problem of the Tropical
Zodiac. For some time I had been bothered by the realization that the only
apparent "physical" reality of the Tropical Signs is at their
cusps (boundaries) where they represent exact aspects to the Equinox and
Solstice points. This too is in some way consistent with the Greek understanding
of aspects wherein the aspect apparently existed as a point in space which
a planet would then conjoin. Thus, when two planets were in aspect to each
other it was really as if a third thing were also present. One planet conjoins
one end of the aspect and a second planet conjoins the other end of the
aspect, they are in aspect to each other only because they are each separately
conjoining a third thing, which is itself the relationship between them.6
From a modern perspective this might all sound like double talk or naive
semantics, until I suddenly thought of both problems in terms of our modern
quantum view of resonance, wave forms, and normal modes of a system.

In quantum theory it is as if the solution to the wave equation has a
reality of its own. Those solutions represent the normal modes possible
in a given system. The fact that a particular mode can exist, mathematically,
implies that it has validity materially, that it is in fact materially manifested.
Experiments have demonstrated the efficacy of the relationship so consistently
that physicists take it for granted that the relationship is true, even
though no continuous mechanism for describing how the system moves smoothly
from one mode to another has ever been demonstrated. Indeed the same theory
asserts that such a continuous causal relationship can never be demonstrated.
All that it is important to understand is which values provide solutions
to the equation. These solutions describe the most symmetrical organizations
of electron density in space, though the equations often describe regions
of nodes and anti-nodes of density arrayed in patterns which one would not
otherwise intuitively expect to exist.

This line of thinking might suggest a new, modern, postmodern or perhaps
hyper-modern, way of understanding the Tropical Zodiac. If the Greeks were
in some sense correct that one of the most important characteristics of
the system is that the transition from one sign to another is a discontinuous
shift from one state to another, then this might be thought of as analogous
to the point where a sign wave crosses its horizontal axis. At this point
the curve is at what in mathematics is called *a point of inflection*,
a discrete point where the curve changes its fundamental nature from concave
up, to concave down (or vice versa). If the sine wave describes the physical
vibration of a string, then this point is called a "node."

The string is an example of a harmonic system in one dimension, while the
atom is an example of a harmonic system in three dimensions, but in each
case the material reality can be interpreted as a manifestation of essentially
the same pure mathematical potential. But, if the string is in one dimension,
and the atom in three dimensions, what happens in two dimensions? A drum
head is essentially a two dimensional model, a flat surface. The various
normal modes of a drum head vibrating are described by mathematical expressions
known as Bessel functions after the Prussian mathematician (and astronomer!)
Friedrich Wilhelm Bessel. The solutions to this function for a circular
drum head correspond to the ways in which the surface may symmetrically
vibrate in whole rational numbers of sectors, just as the equations describing
the vibration of the string describe the ways it may vibrate in rational
whole numbers of segments. If we think of the sine curve as the vertical
profile of the edge of a circular disk, our drum head described by a Bessel
function, then the node, the point where the curve crosses the axis, is
analogous to the boundary between two adjacent sectors on the drum surface,
where one is up and the adjacent one is down. Thus, only certain whole numbers
of sectors will work on a vibrating circular surface.

Just as with the string, the surface of the drum may be divided in half,
with one side moving up while the other side moves down. This might be best
visualized as a sort of yin-yang pattern on the surface of the circular
drum. Unlike the string, however, the surface of the drum may not be divided
into three sectors, because when one sector is moving up the other two would
have to both be moving down. But, then these two sectors could not be differentiated
from each other. However, as we have seen, the circle is most closely associated
with hexagonal symmetry and thus with the number six. If we look at the
drum head vibrating in six sectors we see that they alternate nicely, with
one sector up, and the next down, all the way around the circle. We can
see a natural pattern associated with circles in which the perimeter of
the circle is divided into six equal sectors wherein three of the these
are up, or positive, and three of them are down, or negative, and the positive
and negative sectors alternate around the circumference to form two opposed
equilateral triangles, one up and one down. Thus, we do see an inherent
connection between hexagonal and triangular division of the circle.

In the same manner, the original division of the circle into two sectors
gives rise to a new division into four sectors. As with the division into
six sectors, four sectors may alternate around the perimeter with one up,
or positive, and the next down or negative. Thus a pattern may be formed
with two up and two down, each pair forming an axis, and the two axes forming
a cross. Once we have an axis we may map any bipolar duality onto it such
as: black and white, good and evil, up and down or with perhaps more insight,
yin and yang. But, when we have two crossed axes we, as a species, seem
most likely to first map north and south vs. east and west to arrive at
the four directions. This mapping is found in virtually all cultures throughout
the world. The four directions also correspond to the four nodal points
in the solar year, the two solstices vs. the two equinoxes. So, we see that
the division of the circle representing the seasons is closely identified
with a division of the circle into four sectors.

The seasons by themselves imply a division of a circular disk into four
sectors where the solstices and the equinoxes represent the *boundary*
conditions on the system. In the language of mathematical physics the boundary
conditions are simply the constraints or fixed perameters which determine
the form of a system. The simplest model of our solar system is a flat disk
represented by the drum head. At first glance it appears that the four seasons
may be represented as simply two of crossed axis. The sectors representing
both ends of one axis would be up, while those at both ends of the other
axis would be down. But, the two solstices, summer and winter, represent
opposite phenomena in nature. What if we want them to be opposites in our
model?

To accomplish this we would have to also divide the circle by a multiple
of an odd number of sectors to cause the signs of opposite sectors to be
inverted. The first odd number is three. Even though three cannot be used
as a division by itself, when combined with (multiplied by) two this gives
us six, the symmetry system most closely associated with the circle’s
own symmetry. Now each sector has the opposite sign from the one opposite
it across the center of the circle. But, six is not divisible by the four
fold symmetry of the seasons. So, we must multiply the first odd number,
three, by the four seasons to yield a system in which mid-summer is up and
mid-winter is down. Thus we find that twelve is the minimum and most compact
number of sectors which can describe the normal modes of such a system on
a circle. This approach not only explains why there must be twelve sectors,
Signs, or Zoidion, but also why the system would be said to represent six
pairs of opposites, why the Sign boundaries would represent discontinuous
points where the nature of the Zodiac inverts, and perhaps most importantly
why the whole system would be tied to the seasons rather than to the fixed
stars.

In this model, the Tropical Zodiac could be thought of as essentially a
standing wave set up by the annual motion of the Earth in relation to the
Sun. The Solstices and the Equinoxes represent four fixed nodes (the points
where the amplitude of the wave form is zero as it crosses the axis representing
the plane of the Earth’s rotation around the Sun). In order for the
wave form representing the relationship when the Earth enters summer to
have the opposite sense from that at the point where the Earth enters winter,
the normal mode must have a three fold (odd) symmetry as well as four fold
(even) symmetry. When these two boundary conditions are applied the minimum
number of sectors which will satisfy the parameters of the system is twelve
sectors with their divisions tied to the locations in space of solstices
and equinoxes. Incidentally, the existence of this fundamental normal mode
in no way precludes the existence of other harmonics as well. It simply
makes explicit why this particular solution to the wave form should be so
fundamental and prominent.

The modes of electron probability density symmetry in molecular systems
are like the harmonic standing waves on a string, only in three dimensions.
They are said to be constrained by the boundary conditions of the nucleus
and the number of electrons, just as the string is constrained by the boundary
condition of the distance between its two ends. Once one sets out the boundary
conditions, the normal, or natural, modes are determined by geometry and
mathematics. It is as if each possible mathematical solution will be expressed
in the overall apparent behavior of the electrons, just as they are expressed
in the harmonic frequencies at which the string vibrates. Depending upon
the boundary conditions the string will vibrate in a different set of frequencies.
In both the case of the string and in the case of the quantum model of the
atom, given a certain set of boundary conditions or constraints, all of
the mathematical possibilities will be determined. And all of those potentials
which the math describes will correspond to frequencies which do in fact
appear to be manifested in matter. If a state potential exists in theory,
it will also be observed to exist in practice.

I am essentially arguing that the relationship between the motion of the
Earth and the Sun must be the same. Just as quantum theory moved us away
from a mechanistic orbital model of the atom to one in which the normal
modes of oscillation are seen as inherently valid, the same type of mathematics,
describing the solutions to a Bessel function, may turn out to validate
and corroborate the Tropical Zodiac as a natural way of describing the most
prominent standing wave in the Earth-Sun relationship. Because this relationship
is essentially described by a two dimensional plane, the solutions may very
likely be similar to those describing the vibrational behavior of a drum
head, but with the additional boundary condition of four fold seasonal symmetry.

It is also interesting to notice that in the quantum world we are always
talking about a probabilistic model, of electron probability density, just
as in the astrological model of the zodiac we are talking about a probability
of the manifestation of an archetype. In both realms a concrete prediction
is inherently impossible, yet the regions of high probability density may
be mathematically defined based solely upon the pure harmonic symmetry of
the system.

1. For example, many astrologers swear that their house cusp techniques are efficacious. Yet, not only are there so many different systems of cusps that it is rationally impossible to understand how they could all work, but it now appears that the entire idea of dividing a chart up by a system of twelve house divisions, other than the Signs, was based on a mistranslation of one line in Ptolemy during the Renaissance. Thus, the entire inception and proliferation of the house cusps we use today may be based entirely on a mistake - if the rational for the basis of astrology is ancient tradition. One could take the view that the Divine creator somehow meant this mistake to happen, or that evolution itself, whether of a species or an idea is itself divine and therefore valid. But, once one goes down this path it is impossible to evaluate anything, as everything, once it occurs, would by its very nature be perfect.

2. Robert Schmidt, Project Hindsight Conclave, Ithaca, NY, June 1996

3. Robert Hand, Plenary Lecture, Cycles & Symbols Conference, San Francisco, CA, February 15, 1997

4. Robert Hand has also pointed out at that what has been translated from some ancient texts as the "cusp" of a Sign may in fact refer to the middle of the sign. If this is true, he goes on to assert, then the doctrines which hold that the "cusp" (misinterpreted as the beginning) of a Sign is most powerful would really have been intended to describe the middle of the Sign. This may be just one of a number of confusions arising from the mistranslation of Ptolemy regarding whole sign houses. If the Ascendant falls in the middle of a Zoidion, it is in the cusp of that Zoidion. That Zoidion, in its entirety, is the first house. But, if one misinterprets the passage in Ptolemy and takes the Ascendant as the boundary of the

5 . Robert Hand, Project Hindsight Conclave, Ithaca, NY, June 1996

6. Robert Schmidt, Project Hindsight Conclave, Ithaca, NY, June 1996