THE POWERS OF ONE
On the morning of March 30, 1796, Carl Friedrich Gauss discovered that the way people had been thinking for more than 2000 years was wrong. That was the day, when, after an intensive period of concentration, he saw on a deeper level than anyone before, the "profound connection" between transcendental magnitudes and higher Arithmetic.
The first public announcement of his discovery was at the initiative of E.A.W. Zimmerman, a collaborator of Abraham Kaestner, who headed the Collegium Carolineum, the school for classical studies, where Gauss had received his preparatory education. The notice was carried in the April 1796 issue of Allgemeine Literaturzeitung:
"It is known to every beginner in geometry that various regular polygons, namely the triangle; tetragon; pentagon; 15-gon, and those which arise by the continued doubling of the number of sides of one of them, are geometrically constructable.
"One was already that far in the time of Euclid, and, it seems, it has generally been said since then that the field of elementary geometry extends no farther; at least I know of no successful attempt to extend its limits on this side.
"So much the more, methinks, does the discovery deserve attention, that in addition to those ordinary polygons there is still another group, for example the 17-gon, that are capable of geometric construction. This discovery is really only a special corollary to a theory of greater scope, not yet completed, and is to be presented to the public as soon as it has received its completion."
Carl Friedrich Gauss
Student of Mathematics at Goettingen
"It deserves mentioning, that Mr. Gauss is now in his 18th year, and devoted himself here in Brunswick with equal success to philosophy and classical literature as well as higher mathematics."
E.A.W. Zimmerman, Prof.
Gauss did not construct the 17-gon. As the announcement indicates, the constructability of the 17-gon is merely a corollary of a much deeper principle--the generation of magnitudes of higher powers, as that principle was understood by Plato, Cusa, Kepler, Fermat, Leibniz and the Bernoulli's. As with his contemporaneous work on the fundamental theorem of algebra, Gauss' approach was explicitly anti-deductive, discovering a common physical principle that underlay both geometry and number. It was also a direct confrontation with the failed Aristotelean methods of the likes of Euler and Lagrange who understood the circle as an object in visible space and numbers as abstract formalisms.
Today's pedagogical exercise is the first of two, intended to guide the reader through the relevant concepts of Gauss' method. It will require some "heavy lifting" and the reader is advised to work it through all the way to the end, no matter how arduous it seems along the way, and then look back, surveying what has been gained from the vantage point of the summit. The reader is also advised to review the preliminary work on Gauss' theory of the division of the circle that was the subject of the several past pedagogicals, as it was summarized in the Winter 2001-2002 edition of 21st Century Science and Techonlogy, and the pedagogical exercises on the residues of powers (Riemann for Anti-Dummies Parts 20-25.) (Reference will also be made to several figures)
Polygons As Powers
As Gauss' announcement indicates, by Euclid's time, geometers had succeeded in finding the magnitudes that divided a circle in certain ways. What was not so evident, was why those ways and not others? From the standpoint of sense certainty, the circle, like the line, appears uniform and everywhere the same. Why then, is it not, like the line, divisible into whatever number of parts one desires? What unseen principle is determining which divisions are possible, and which are not?
Yet, when the circle is considered as a unit of action in the complex domain, it becomes evident that the division of the circle is based on the principle that generates magnitudes of successively higher powers. Those who have worked through the pedagogical exercises on Gauss' 1799 doctoral dissertation are familiar with how this works. There we saw that algebraic powers are generated by a non-algebraic, physical principle, as expressed, for example, by the catenary. This principle belongs to the domain of functions that Leibniz called transcendentala, and is expressed mathematically by the equiangular spiral, or alternatively, the exponential (logarithmic) functions. Gauss showed that these transcendental functions were themselves part of a higher class of functions that could only be adequately known through images in the complex domain.
From this standpoint, the generation of magnitudes of any algebraic power correspond to an angular change within an equiangular spiral. "Squaring" is the action associated with doubling the angle within an equiangular spiral, "cubing" by tripling, fourth power by quadrupling, and so forth. These angular changes are, consequently, what generates magnitudes of succesively higher algebraic powers. When the circle is correctly understood as merely a special case of an equiangular spiral, the generation of algebraic powers is reflected as a mapping of one circle onto another. Squaring, for example, maps one circle onto another twice, cubing maps three times, and so on for the higher powers. (The reader is referred to the figures from the pedagogical discussions on the fundamental theorem of algebra.)
The regular divisions of the circle are simply the inversion of this action. Each rotation around the "squared" circle divides the original circle in half. Each rotation around the "cubed" circle divides the original circle into thirds, each rotation around the fourth power circle, divides the original circle into fourths, and so on. Consequently, the vertices of a regular polygon, are the points on the original circle, that correspond to the complete rotations around the "powered" circle and the number of vertices corresponds to the degree of the power. For example, the fifth power will produce, by inversion, the five vertices of the pentagon; the inversion of the seventh power, will produce the seven vertices of the heptagon, etc. All the vertices of a given polygon are generated, "all at once", so to speak, by one function, which is the inversion of the function that generates the corresponding power. (By Gauss' time, such inversions had come to be called "roots", not to be confused with the misapplication of that term by ignorant translators of Plato's word, "dunamis".) Herein lies the paradox. If the triangle, square and pentagon are inversions of the generation of third, fourth, and fifth powers respectively, how come they are constructable and other polygons are not? (Constructable is used here in the same sense as Kepler uses the term "knowable" in the first book of the Harmonies of the World. By "knowable", Kepler meant those magnitudes that were commensurate with the diameter of the circle, part of the diameter, or the square of the diameter or its part. These magnitudes are the only magnitudes, "constructable" from the circle and its diameter, or by straight-edge and compass. All such magnitudes correspond to "square roots" or magnitudes of the second power. Magnitudes of higher powers, are not "knowable" from the circle alone, as is evident from the problem of doubling the cube, or trisecting the angle.)
Prime Numbers are Ones
It was Gauss' insight to recognize that the solution to this paradox lay, not in the visible circle, but in the nature of prime numbers. To begin with, throw out the common formal definition of prime numbers, and consider a physical principle in which prime numbers arise. This can be most efficiently illustrated by example. Perform the following experiment: draw 10 dots, in a roughly circular configuration, and number them 0 to 9. Connect the 10 dots sequentially (0, 1, 2,...) and call that sequence 1. Now connect every other dot, (0, 2, 4, 6...) and call that action sequence 2. Then every third dot, (0, 3, 6, 9, ..., for sequence 3) then every fourth dot, (0, 4, 8, ..., sequence 4) and so on.
Notice, that some sequences succeeded in connecting all 10 dots, namely, sequences 1, 3, 7 and 9, while sequences 2, 4, 5, and 8 connected only some of the dots. In the case of the latter, sequences, 2 and 5 became completed actions within one rotation, whereas 4 and 8 did not become completed actions until after more than one rotation.
Numbers are not formal symbols (or objects), to be manipulated according to a set of formal rules, but are relationships arising from physical action. In the above example, the number 10 becomes a One, or, as Gauss called it, a modulus. The numbers 1 through 9 are types of actions, not collections of things. With respect to modulus 10, the numbers (actions) 1, 3, 7, and 9 are called relatively prime, because those actions do not divide the modulus. The numbers 2, and 5, are called factors of 10, because those actions do divide the modulus within one rotation. (The numbers 4 and 8, divide the modulus but not within one rotation because they are not factors themselves but they share a common factor (namely 2) with 10.)
These relationships, of factors and relative primeness, are determined only by the nature of the modulus. If you begin sequence 2 on dot #1 instead of dot #0, it still connects only 5 dots. Similarly, if you begin sequence 3 on dot #1, it will still be relatively prime to 10. Additionally, if you continue the experiment with sequences 11, 12, 13, etc., the results will be identical to the sequences 1, 2, 3, etc. except that one rotation will be added. Gauss called these numbers congruent relative to modulus 10.
Thus, the modulus defines certain relationships, relative to the entire universe of whole numbers, in which some numbers are factors, some numbers are relatively prime, and some numbers are not factors themselves, but contain factors of the modulus.
However, when one dot is added, and the same experiment is performed with respect to 11 dots, all the sequences connect all the dots. Thus, 11 has no factors and all numbers are relatively prime to it. The relationship of modulus 11 to the entire universe of whole numbers is quite different than the modulus 10.
The modulus is the One. Some moduli, such as 10, define some numbers as factors,and some numbers as relatively prime and are called "composite". Those moduli under which all numbers are prime, are known as prime numbers.
There is nothing absolute about the quality of primeness. Relatively prime numbers gain this characteristic relative to a one (modulus). Those numbers that are prime relative to the One, are absolutely prime. (Gauss, in his treatises on bi-quadratic residues, would later show that even this characteristic of absolute primeness is not really absolute but relative to a still higher principle.)
Polygons as Planetary Systems
This leads us back to the original paradox. If the prime numbers are irreducible Ones, how come some prime number divisions of the circle are constructable and others not?
Take another look at the image of a circle in the complex domain. The vertices of a regular polygon are the roots (inversions) of a corresponding power. This relationship of "roots" and "powers" produces a type of harmonic "planetary system" for each polygon in which only those "planetary orbits" that correspond to the "roots" of that "power" are possible, and, these "roots" have a unique harmonic relationship to each other, whose characteristics are determined by the number-theoretic characteristics of the prime number.
Illustrate this pedagogically by an example. The vertices of a regular pentagon are the five "roots" of 1 and each of these "roots" is a complex number that has the power to produce a fifth degree magnitude. Such complex numbers represented the combined action of rotation and extension. Since in a circle the extension is constant, the complex numbers are at the endpoints of equally spaced radii. To construct the polygon it is necessary to determine the positions of these radii. To do this Gauss used the method of inversion and determined the positions of the radii from the harmonic relations among them. Even without knowing the positions of the radii, the harmonic relations can be known because the radii are inversions (roots) of powers. In other words, the vertices of the polygon are the endpoints of equally spaced radii.
But don't look at the endpoints (visible objects). Look for why the radii are equally spaced. They are equally spaced because they are the roots of an algebraic power. To illustrate this use the pentagon as an example, draw a circle with five approximately equally spaced radii. This should look like an "inside out" pentagon. (Since we are investigating only the relationships among the radii at this point it is not necessary that the radii be exactly equally spaced.)
Label the endpoints of the radii 1, a, b, c, d, with "a" representing 1/5 of a rotation, "b", 2/5, "c", 3/5, "d", 4/5 and "1" being 1 full rotation. If any of these individual angular actions is repeated (multiplied) five times, the resulting action will end up at 1. In other words, a5, b5, c5 and d5 are all equal to 1. Furthermore, a0=1, a1=a, a2=b, a3=c, a4=d; b0=1, b1=b, b2=d, b3=a, b4=c; c0=1, c1=c, c2=a,c3=d, c4=b; d0=1, d1=d, d2=c, d3=b, d4=a. Thus, any vertex can generate all the others. (For the general case, each of the vertices corresponds to a complex number of the form a + b √-1, such that (a + b√-1)n =1 for all "n's" of an "n" sided polygon.)
In the example of the pentagon, five is the modulus, the One, which establishes a certain harmonic ordering under which there are five and only five "orbits". A different modulus would produce a different number of "orbits", but the relationship just illustrated will remain; only the number of "orbits" will have changed, and consequently, the nature of the harmonies. Notice the congruence of these actions with our earlier experiment with dots illustrating the physical principle from which primeness, relative primeness and factors arise. Notice the similarity between the power sequences generated from each complex root, and the different number sequences used to connect the dots. This congruence is not discovered by looking at the visible objects, but by a method Leibniz called, "Geometry of Position", or "analysis situs", or what Gauss called, "geometrica situs". It reflects a higher principle, independent of any particular number and begins to shed light on that "profound connection" Gauss discovered between the geometry of transcendental functions and higher Arithmetic.
Next week we'll look further into that connection.