Gauss’ Division of the Circle
The pursuit of a discovery of a universal principle always requires the pursuer to follow the Socratic method of negation, or, as Cusa called it, “Learned Ignorance”. This is the method by which Kepler ascended from the tangle of observed motions of the planets on the inside of an imaginary sphere, to the “hypergeometric” function, whose characteristics we’ve been exploring throughout this series. While the cause of these observations is determined from the top down by that function, like the shadows on Plato’s Cave, we cannot know that function directly. Rather, we must look between the gaps in the observations, and discern the hypergeometry of which those observations are a function.
It is the nature of this method, that, as our knowledge of the underlying hypergeometry increases, new gaps appear, through which new characteristics of the hypergeometry become discernable. These new characteristics, in turn, recast the previous discoveries, in a new light. For example, the principle that light travels the shortest path, discovered by the Greeks as a characteristic of reflection, becomes a special case of the principle that light travels the path of least time, discovered by Fermat as a characteristic of refraction. In both cases, light acts according to a minimizing principle. But, in the former, that principle is with respect to a manifold of space, while the latter is with respect to a manifold of space-time. It is the manifold from which the characteristic of the path is determined, but it is the characteristics of the path, by which the manifold is discovered.
Another example may help illustrate this point. What is the meaning of the question, “When did human life begin?”? If your conception of the universe is a reductionist/Darwinian one, in which non-living processes give rise to living ones, which, in turn, give rise to cognitive processes, the answer to that question, is, a search for a mystical point in time, in which something, becomes what it’s not. (i.e., non-living processes become living, or living processes becomes cognitive.) However, from the standpoint of a multiply-connected universe, which, as LaRouche most recently wrote, is organized from the top down, cognitive, to living, to non- living, the answer to the question, as Plato, Philo, and Augustine have said, is “from the beginning.” (cognition created man) Under a conception of the universe organized from the bottom up, the trajectory of evolution is a line. (Whether that line is curved or straight, it is still a one-dimensional magnitude.) Whereas, under the top-down conception, the trajectory is of a quite different nature.
To grasp the nature of that trajectory requires the metaphors developed by Riemann, toward which this series is heading. To get there, we must first traverse the intervening steps. This is what brings us to Gauss’ division of the circle. What follows may be at times dense, as it brings together discoveries that span more than 2500 years of cognition. But roll up your sleeves and work it through.You will be well rewarded with learned ignorance.
Kepler, in the first book of the “Harmonies of the World”, showed that the divisions of the circle generate a hierarchy of types of magnitudes. Following the Greek tradition, these magnitudes are ranked according to “knowability”. Those magnitudes that are measurable directly by the diameter of the circle, have the first degree of knowability, those measurable by a part of the diameter, have the second degree, and those whose squares are measurable by the diameter, have the third. All others are incommensurable. It is these incommensurable magnitudes, on which non-uniform motion depends, as Kepler had already shown in his Mysterium Cosmographicum and The New Astronomy. Therefore, to grasp the principles of non-uniform motion, one had to grasp the principles of generation of these incommensurables.
Magnitudes of the first three degrees of knowability, are also called constructable, because they can be constructed from the circle itself, or to put it colloquially, by straight-edge and compass. (The straight-edge being the diameter and compass being the circumference of the circle.) It is in the investigation of what is constructable, that we discover a gap, an anomaly, through which we ascend to the idea, that, the seemingly uniform circle, is, in fact, not uniform at all, but a special case of non-uniform action!
Ancient Greek philosophers had fully investigated, that, while the circle is uniform in all its parts, it does not divide uniformly. The circle can be divided into two parts by folding in half. By folding in half again, and again, the circle can be divided into 2,4,8,16,etc. parts. But to divide it into three parts, that is, to inscribe a triangle in it, requires the generation of a magnitude of the second degree of knowability. (Half the diameter, is the side of a hexagon.). Once that magnitude is created, it too can be doubled, to produce a division by 6, 12, 24, etc. parts. But, to divide the circle into five parts, requires the generation of a magnitude of the third degree of knowability, specifically, the golden section. The golden section is not commensurable with the diameter or a part of the diameter, but its square is. Hence, it is constructable.
However, it had been believed for more than 2000 years, that all other divisions of the circle were non-constructable. In other words, those divisions of the circle based on prime numbers greater than five, depended on magnitudes that were beyond Kepler’s third degree of knowability. This boundary condition, suggests that something outside the circle, a higher principle, is governing. It is that principle, that Gauss sought.
To discover this principle, Gauss, in effect, inverted the problem all together. Instead of thinking of two different types of magnitudes, constructable and non-constructable, he investigated a general principle governing the generation of magnitudes, of which the constructable ones were a special case. (A suggestive example of this concept was supplied by LaRouche several years ago, when he posed that the number 5 associated with the hypotenuse of a 3-4-5 right triangle, should be considered a special type of irrational number.)
Additionally, Gauss, using a unique application of geometry of position, which, at first may seem obscure. He inverted the conception of the circle. Instead of beginning with the circle and trying to find those positions that divided it, he sought the functions that created a divided circle. Thus, the positions were produced by the division, not the division by the positions.
Gauss showed that both the above principles relied on two, seemingly unrelated, conceptions that were at the heart of Greek science, and, that had been extended by the work of Kepler, Leibniz, Bernoulli, and Fermat: the geometric generation of incommensurables, and the more shrouded principles governing the generation of prime numbers. And, spurred by Kaestner’s prodding, Gauss showed that these principles extended beyond the circle.
As Gauss described it in the opening of the final chapter of “Disquistiones Arithmeticae”, “Among the most splendid developments contributed by modern mathematicians, the theory of circular functions without doubt holds a most important place. We shall have occasion in a variety of contexts to refer to this remarkable type of quantity, and there is no part of general mathematics that does not depend on it in some fashion…. I will speak of the theory of trigonometric functions as related to arcs that are commensurable with the circumference, or of the theory of regular polygons….The reader might be surprised to find a discussion of this subject in the present work which deals with a discipline apparently so unrelated; but the treatment itself will make abundantly clear that there is an intimate connection between this subject and higher Arithmetic.
“The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions….”
The first principle of Greek geometry that Gauss re-worked, is described by Plato in the Meno, Theatetus and Timaeus dialogues, and concerns the generation of incommensurable magnitudes as a consequence of a change in dimension. The reader can construct a geometrical representation of this by drawing a square, then drawing the diagonal, then, drawing a new square using the diagonal for its side. As Plato demonstrated in the Meno, the second square will have twice the area of the first, but the side of the second square, will be incommensurable to the side of the first. (In Kepler’s terms, the diagonal will be the third degree of “knowability”.) If you continue this drawing, you will produce a spiral sequence of squares, whose sides are the diagonals of the previous squares, and whose areas are double, the previous squares. The Greeks called the diagonals, the “geometric mean” between the two squares.
However, something new develops if you try and replicate this process in 3 dimensions, as in the case of doubling a cube. The diagonal of the cube does not correspond to the side of a cube with double the volume. This is the famous problem, the Delian priests brought to Plato. Eratosthenes reports Plato’s famous rebuke, that the Gods had posed this problem to the Greeks, because they wanted to chide the Greeks into studying geometry in order to improve their thinking. Hippocrates of Chios had shown that the incommensurable associated with the doubling of the cube, was of a different species, than the incommensurable associated with doubling the square. In other words, a change in dimensionality, produced a different species of incommensurable. In the Timeaus, Plato reports Hippocrates discovery:
“But it is not possible that two things alone be joined without a third; for in between there must needs be some bond joining the two…. Now if the body of the All had had to come into being as a plane surface, having no depth, one mean would have sufficed to bind together both itself, and its fellow-terms; but now it is otherwise for it behoved it to be solid in shape, and what brings solids into harmony is never one mean, but always two.”
By inversion, if two or more geometric means are required to double a magnitude, the doubling of that magnitude, is an action, that originates in a dimensionality greater than two.
Gauss’ insight rested on these Platonic principles, with a crucial extension supplied by Bernoulli’s discovery of the equiangular spiral, which he called, “spiral mirabilus”. Bernoulli showed that this spiral was an exemplar of geometric growth. For example, a line extending from the center of the spiral outward, will be cut at different intervals by each spiral arm. These intervals will be the same proportion to one another, as the areas and diagonals of Plato’s squares. Similarly, lines emanating from the center of the spiral at equal angles from each other, will cut the spiral arms in geometric proportion. Thus, equal divisions of the spiral, cut the spiral into parts that are in geometric proportion.
Now, if we think of the circle as a special case of the spiral, then the division of the circle by lines (radii) emanating from the center at equal angles, cuts the circumference in arcs that are in geometric proportion. The intersection of these lines with the circumference correspond to the vertices of an inscribed polygon. Thus, to divide the circle into “n” parts, corresponds to finding “n-1” geometric means. Those divisions, that can be accomplished by finding one mean between two others, are constructable; and those requiring two or more means are not. Thus, the different divisions of the circle are actually projections, of action originating in manifolds, of higher dimensionality, than the seemingly two dimensions of the circle. Again, like the shadows on Plato’s Cave, or Learned Ignorance, we can only ascend to knowledge of those manifolds, from the anomalies embedded in their reflection.
Gauss showed that these anomalies can be discovered, if we think of the circle, not in a Euclidean/Cartesian plane, but in the complex domain. A simple example is, perhaps, the easiest way to illustrate the point. To divide a circle into 4 parts, first think of the circle in the complex domain. Pick a point on the circumference for the first vertex, and call it 1. To divide the circle into four parts we would mark off three other points that are 90 degrees from each other. According to what we said above from Bernoulli and Plato, these points are all in geometric proportion to each other. Expressed in numbers, these points form a cycle of geometric means from 1 to 1. Using the letter i to denote the square root of -1, that series is, 1, i, -1, –i. These four numbers produce a cycle, such that if you multiply each one by itself 4 times, you get 1.
Gauss’ insight was based on the following: Dividing the circle into “n” parts requires finding “n-1” geometric means between 1 and 1. Each “n” divisions requires a function with specific characteristics. Gauss discovered a general principle that governed the nature of those characteristics.
Again, it is easiest to demonstrate this by example. Put five dots on a page, roughly in a circle and number the dots counter-clockwise, 1, 2, 3, 4, 5. Now, connect dot 1-3, then 3-5, then 5- 2, then 2-4, then 4-1. In this action, you went around the circle twice, connecting 2 dots in each turn. You also drew a pentagonal star. Perform the same action, connecting 1-4; 4-2; 2-5; 5-3; 3- 1. This produces the same result as above. Try the same with 7 dots. Connect 1-3; 3-5; 5-7; 7-2; 2- 4; 4-6; 6-1. This action required 2 cycles, connecting 3 dots in each cycle (3, 5, 7) and (2, 4, 6). Notice the shape of the heptagonal star. Make another configuration of 7 dots, and connect them in the following sequence: 1-4; 4-7; 7-3; 3-6; 6-2; 2-5; 5-1. This action required 3 cycles, connecting 2 dots in each cycle. Notice that the shape of this heptagonal star is different than the previous one. If you experiment around, using different cycles, (e.g. 1-5; 5-2;, etc. ) the result will be identical to one of the two produced above. Now, continue these types of experiments with 7, 11, 13 and 17 dots. In each case, you will find that the combinations of cycles and dots, is constrained by the prime number factors of 10, 12, and 16 respectively, ( n-1).
Gauss showed that these combinations of cycles corresponds to Plato’s principle of means. Those divisions that can be resolved completely into cycles of 2, correspond to inserting 1 geometric mean between two others, as in doubling of the square, and are therefore constructable. Those prime number divisions that cannot be so resolved, correspond to inserting two or more geometric means, and, like the doubling of the cube, are not constructable. Thus, it is possible to construct figures of 2, 3, 5, 17, 257, 65,537 and any other prime number divisions of the form (22)n + 1. All other divisions cannot be constructed because they are reflections of actions of a dimensionality higher than two.
It is the thinking underlying the above discovery of Gauss, that is at the heart of Gauss’ and Riemann’s development of the theory of functions.