If we look at the known cases of constructable polygons, the triangle, square and pentagon, each is constructable by a series of nested steps, in which a "knowable" magnitude is constructed, and then from that magnitude, another "knowable" magnitude is constructed, until the side of the polygon is found. For example, the triangle is constructed by first constructing the hexagon from the radius of the circle. Then the side of the triangle is constructed from the side of the hexagon. The square is constructed from one diameter and a second diameter is constructed perpendicular to it. The pentagon is constructed by first constructing the golden mean, and then the side of the pentagon is derived from the golden mean.

In each of the above examples, each magnitude in the chain is constructed from the its predecessor by simple circular action. Consequently, such magnitudes are commensurate with the type of magnitudes associated with doubling of the square, i.e. second degree magnitudes, which are generated by simple circular action. As distinguished from third degree magnitudes that are associated with doubling the cube, which as was seen in the construction of Archytus, require the complex action of rotation and extension.

Therefore, those polygons, whose constructions could be reduced to a nested chain of second degree magnitudes are, in principle, constructable. All others are not.

The crucial insight of Gauss was to recognize that each polygon ("planetary system") could be constructed as a chain of "orbital periods" and "sub-periods". The character of the magnitudes associated with these periods and sub-periods, is determined by the number-theoretic characteristics of the prime number, or more specifically, the prime number minus 1.

Herein lies the "profound connection" between the generation of transcendental magnitudes and higher Arithmetic. The arithmetical characteristics determine the geometry, while the geometry, in turn determines the arithmetical characteristics. Unlike formalists such as Euler, Lagrange and D'Alembert, Gauss saw no distinction between the geometrical and the arithmetical characteristics. The same physical principle that governed the circle, ruled number. What the circle concealed, number revealed. One need only be able, as Plato said, "to see the nature of number with the mind only." (Remember that the Greek word from which "arithmetic" is derived has the same root as the Greek word, "harmonia".)

For Gauss, the circle is not simply an object in visible space, but rather an artifact of an action in the complex domain. Successive divisions of the circle reflect a succession of different types of actions corresponding to the hierarchy of powers. The vertices of an "n" sided regular polygon are the "n" roots of 1. Inversely, as was shown last week, these vertices can be generated as a succession of powers.

Ironically, the principles of this so-called "imaginary" domain determine what is possible in the visible domain. Gauss showed that the deeper principle of their generation becomes known under examination of, what he called the "residues of powers" in his "Disquisitiones Arithmeticae".

Each prime number modulus has a characteristic period of residues with respect to a series of powers. For example, the modulus 5 produces the period of residues {1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3,etc.}, with respect to the powers of 2, and the period of residues {1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 2, etc, }, with respect to the powers of 3. (See Riemann for Anti-Dummies Parts 20-23.)

(Since the powers of 2 and 3 yield complete, albeit different, periods, they are called "primitive roots" of 5. Compare this result to the periods generated from the residues of the powers of 2 and 3 relative to modulus 7. In the case of 7, 3 is a primitive root, whereas 2 is not.)

These periods are completed periods and are not altered when all the elements are multiplied by any number. For example, multiply {1, 2, 4, 3} by any number, and take the residues relative to modulus 5. The resulting period will be the same as the one you started with. Similarly, for the period {1, 3, 4, 2}. (The reader is strongly encouraged to perform these experiments.)

Each complete period also has the two sub-periods. For the case of modulus 5, those sub- periods are {1, 4} and {2, 3}, which "orbit" each other. When either sub-period is multiplied by 2 or 3, they are transformed into the other. When multiplied by 1 or 4, they remain unchanged.

Similarly, the modulus 7 produces the period of residues, {1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5} with respect to the powers of 3. It contains 2 sub-periods of 3 elements each, {1, 2, 4} and {3, 6, 5 } and 3 sub-periods of 2 elements each, {1,6}, {3, 4}, and {2, 5}. (Much will be gained if the reader tries multiplying the elements of each sub-period to see what transformations occur.)

Modulus 17 produces the period of residues, {1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6} with respect to the powers of 3. It contains 2 sub-periods of 8, {1, 9, 13, 15, 16, 8, 4, 2} and {3, 10, 5, 11, 14, 7, 12, 6}; 4 sub-sub-periods of 4, {1, 13, 16, 4}; {9, 15, 8, 2}; {3, 5, 14, 12}; and {10, 11, 7, 6}. And, finally, 8 sub-sub-sub-periods of 2, {1, 16}, {3, 14}, {9, 8}, {10, 7}, {13, 4}, {5, 12}, {15, 2}, {11, 6}.

Notice that in all cases, the sum of the numbers of a period or sub-period is always congruent to 0 relative to the modulus, and that the lengths of all periods are always the modulus minus 1 or a factor of the modulus minus 1.

The Determination of the Polygon's Orbits

Being an artifact of an action in the complex domain, these individual vertices each corresponds to a complex number. The "n" complex numbers, corresponding to the "n" vertices, of an "n" sided polygon, comprise a complete period of "n" roots. The problem Gauss confronted was how to determine the positions of the individual vertices ("orbits") of a polygon ?

Gauss' discovery was to show that each of these "orbits" was completely determined by the harmonic nature of the whole. That harmonic principle is reflected in the nested chain of periods and sub-periods of the residues of powers. Gauss worked by inversion. Like Kepler with planetary orbits, Gauss understood that the harmonic principle determined the individual positions, so he developed a method to work from the top down, that is, from the harmonies to the notes, so to speak, showing how to "read" this chain of periods and sub-periods to determine the positions of the vertices, ("orbits") of the polygon.

For pedagogical purposes it is most efficient to illustrate by continuing with the example of the pentagon.

The first step in determining the vertices of the pentagon is organize the vertices into a "harmonic" period. As was shown last week, all the vertices can be generated as a series of powers from any one of them. Therefore, Gauss began with one of the vertices and generated all the others as a series of powers. But, to bring out the "harmonic" characteristic, they had to be ordered according to the principle exhibited by the residues of the primitive root. Continuing from the example of the pentagon from last week, that would mean generating the period from the powers of a^{2^0}, a^{2^1}, a^{2^2} and a^{2^3}. Taking the residues of these periods relative to modulus 5, these vertices will now be in the order, {1, 2, 4, 3}.

This period of can be divided into the two sub-periods {1, 4} and {2, 3}, that define the first set of magnitudes required to construct the pentagon. To determine the value of these magnitudes, Gauss considered them as "roots", and since there are two of them, they must be "roots" of a quadratic equation. Call the value of {1, 4}= r1 and the value of {2, 3} = r2.

Here again Gauss worked by inversion. Even without knowing what the values for r¹ and r² are, except that they are "roots" of some quadratic equation, Gauss could work backwards from the harmonic relationship between them, to determine what must produce them.

To solve this problem, Gauss relied on the relationship between the roots and coefficients of algebraic equations (introduced without demonstration). That relationship is that if a quadratic equation is in the form x² + Ax + B = 0, the sum of the roots equals -A and the product of the roots equals B.

Back to our example. Even without knowing the values of the individual vertices, we can know the sum and the products of them. The sum of the sub-periods {1, 4} and {2, 3} is {1 + 2 + 4 + 3}. This means adding together the complex numbers that correspond to the vertices 1, 2, 4, 3. Each complex number denotes a complex quantity of combined rotation and extension. To add complex numbers, you carry out the rotation and extension in series. In this example, you first carry out the rotation and extension that produces vertex 1. Then from the endpoint of vertex 1, carry out the rotation and extension that corresponds to vertex 2, and so forth. Geometrically, this turns the "inside-out" pentagon, "inside in". (See figure.) From this it can be seen that the sum of 1 + 2 + 3 + 4 = -1.

Similarly, we can also determine the product of the sub-periods, even without knowing the values of the individual vertices. The product of the sub-periods {1, 4} and {2, 3} is {(1 + 2) + (1 + 3) + (4 + 2) + (4 + 3)}. Taking the residues relative to modulus 5 this equals {3 + 4 + 1 + 2} which also equals -1. (See figure.) ( This is also evident from the fact that 1 x 4 x 2 x 3 = 24 which is congruent to -1 mod 5.)

Therefore {1, 4} and {2, 3} are the "roots" of the quadratic equation where A = 1 and B = -1, or, x² + x - 1 = 0. That means the {1, 4} = r1 = (-1+√5)/2 and {2, 3} = r2 = (-1-√5)/2.

The final step for the construction of the pentagon is to find the two vertices from the just discovered values of each sub-period. For example, the vertices 1 and 4are the "roots" of the sub- period {1, 4}, and the vertices 2 and 3, are the "roots" of the sub-period {2, 3}.

In sum, the action that generates the pentagon is a nested chain of second degree actions, and therefore, "knowable" geometrically.

What Gauss has demonstrated in general, is that any polygon is generated by a nested series of actions determined by the periods and sub-periods formed by the residues of powers. Since the number and length of these periods and sub-periods is determined by the factors of the modulus minus 1, the degree (or power) of each action will be determined by these factors.

For example, the construction of the heptagon will be determined by one cubic and one quadratic action. The 11-gon will be determined by one fifth power and one quadratic action; the 13-gon by one cubic and two quadratic actions; the 19-gon by two cubics and one quadratic action.

On the other hand, the 17-gon, 257-gon, the 65, 537-gon are all generated by a chain of quadratic powers, and are therefore geometrically "knowable"

Anyone who makes the effort to re-live this discovery of the 18 year old Gauss, will discover a corresponding increase in their own cognitive power.