Quantum Mechanics, Bessel Functions
and the
Tropical Zodiac

James L. Fournier
May 1, 1997

Towards a New World View
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.

Formal Cause in Quantum Mechanics

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.

Rational Division of Circles

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 2pi. 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.

Insights about Ancient Greek Astrology

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.

Vibration on a String and Quantum Mechanics

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 first house, it would be easy to also misinterpret the passage to mean that the Ascendant is the cusp (boundary) of the first house. Cusp might thus have come to be used as a term for describing the boundary of a house rather than for the middle of a Sign. NCGR Lecture, Fort Mason Center, San Francisco, CA, Feb. 13, 1997

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

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