Gauss and the regular heptodecagon.

Abstract. -- This note outlines the proof and provides a set of diagrams for the division of a circle into 17 equal arcs.

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Karl Friedrich Gauss (1777-1855) at the age of 18 gave a ruler-compass construction of dividing a circle into 17 equal arcs. This note consists of an outline of his proof and a set of diagrams which systematically give the construction.

Let

a[x.sup.2] + bx + c = 0, a [not equal to] 0 (1)

be a quadratic equation with real coefficients. Consider a rectangular coordinate system (Fig. 1). Let A and B, on the x - axis, correspond to [x.sub.1] and [x.sub.2], respectively, where [x.sub.1] and [x.sub.2] are the roots of (1). Let OU = 1 and OD = c/a. Then

(OU) (OD) = c/a = [x.sub.1][x.sub.2] = (OA) (OB). (2)

This implies that the four points, U, D, A and B are on circle (1). Since M, the midpoint of AB, corresponds to

[1/2] ([x.sub.1] + [x.sub.2]) = [-b]/[2a], (3)

one can construct the circle which passes through U, D, A and B. Therefore the roots of (1) can be constructed (Amir-Moez 1987).

THE SEVENTEEN ROOTS OF UNITY

Consider the equation

[z.sup.17] - 1 = 0. (4)

[FIGURE 1 OMITTED]

One can easily factor (4) into

(z - 1)([z.sup.16] + [z.sup.15] + ... + [z.sup.2] + z + 1) = 0 (5)

Let [z.sub.0] = 1 and [z.sub.1] = z (Fig. 2). Observe that

[z.sub.k] = [z.sup.k], k = 1,..., 16 (6)

and

[z.sub.k] = [z*.sub.17 - k], k = 9,..., 16 (7)

[FIGURE 2 OMITTED]

where, for example, z* is conjugate of z. This implies that

[z.sub.h] + [z*.sub.17 - h] = 2 cos[[2([pi] + h[pi])]/17]. (8)

Then

[z.sub.h] = [x.sub.h] + i[y.sub.h] (9)

and

[z.sub.h] + [z*.sub.17 - h] = 2[x.sub.h], h = 9,..., 16. (10)

One can write (10) as:

[z.sub.k] + [z*.sub.17 - k] = 2[x.sub.h], k = 1,..., 8. (11)

(Dickson 1952).

[FIGURE 3 OMITTED]

THE RULER-COMPASS CONSTRUCTION

Gauss has considered

[l.sub.1] = [1/2] {(z + [z.sup.16]) + ([z.sup.2] + [z.sup.15]) + ([z.sup.4] + [z.sup.13]) + ([z.sup.8] + [z.sup.9])} = [x.sub.1] + [x.sub.2] + [x.sub.4] + [x.sub.8] (12)

and

[l.sub.2] = [1/2] {([z.sup.3] + [z.sup.14]) + ([z.sup.5] + [z.sup.12]) + ([z.sup.6] + [z.sup.11]) + ([z.sup.7] + [z.sup.10])} = [x.sub.3] + [x.sub.5] + [x.sub.6] + [x.sub.7] (13)

Note that

[l.sub.1] + [l.sub.2] = - [1/2], [l.sub.1][l.sub.2] = -1. (14)

[FIGURE 4 OMITTED]

Thus [l.sub.1] and [l.sub.2] satisfy the quadratic equation

[l.sup.2] + [1/2] l - 1 = 0. (15)

One can construct the roots of (15) (Fig. 3).

Next let

[m.sub.1] = [1/2] [(z + [z.sup.16]) + ([z.sup.4] + [z.sup.13])] = [x.sub.1] + [x.sub.4] > 0, (16)

and

[m.sub.2] = [1/2] [([z.sup.2] + [z.sup.15]) + ([z.sup.8] + [z.sup.9])] = [x.sub.2] + [x.sub.8], (17)

[m.sub.1] + [m.sub.2] = [x.sub.1] + [x.sub.2] + [x.sub.4] + [x.sub.8] = [l.sub.1], (18)

and

[m.sub.1][m.sub.2] = - [1/4]. (19)

[FIGURE 5 OMITTED]

So [m.sub.1] and [m.sub.2] satisfy the quadratic equation

[m.sup.2] - [l.sub.1]m - [1/4] = 0. (20)

By carrying [l.sub.1] from (Fig. 3), [m.sub.1] and [m.sub.2] can be constructed (Fig. 4). The reader may examine the figure against Figures 1 and 3.

Now

[n.sub.1] = [1/2] [([z.sup.3] + [z.sup.14]) + ([z.sup.5] + [z.sup.12])] = [x.sub.3] + [x.sub.5], (21)

and

[n.sub.2] = [1/2] [([z.sup.6] + [z.sup.11]) + ([z.sup.7] + [z.sup.10])] = [x.sub.6] + [x.sub.7]. (22)

Note that

[n.sub.1] + [n.sub.2] = [x.sub.3] + [x.sub.5] + [x.sub.6] + [x.sub.7] = [l.sub.2], and [n.sub.1][n.sub.2] = - [1/4]. (23)

[FIGURE 6 OMITTED]

Therefore [n.sub.1] and [n.sub.2] satisfy the quadratic equation

[n.sup.2] - [l.sub.2]n - [1/4] = 0. (24)

Constructing the roots of (24), one obtains (Fig. 5).

Finally note that

[x.sub.1] = [1/2] (z + [z.sup.16]) and [x.sub.4] = [1/2]([z.sup.4] + [z.sup.13]). (25)

So

[x.sub.1] + [x.sub.4] = [1/2] (z + [z.sup.4] + [z.sup.13] + [z.sup.16]) = [m.sub.1] (26)

and

[x.sub.1][x.sub.4] = [1/4]([z.sup.5] + [z.sup.14] + [z.sup.3] + [z.sup.12]) = [1/2][n.sub.1]. (27)

Therefore [x.sub.1] and [x.sub.4] satisfy the quadratic equation

[x.sup.2] - [m.sub.1]x + [1/4][n.sub.1] = 0. (28)

Constructing roots of (28) gives the final result (Fig. 6). Note that z and [z.sup.4] can be obtained from [x.sub.1] and [x.sub.4]. The rest of the construction is quite clear.

LITERATURE CITED

Amir-Moez, A. R. 1987. Gauss and the Regular Pentagon, School of Science and Mathematics 87(4), pp. 293-301.

Dickson, L. E. 1952. New First Course in the Theory of Equations, John Wiley and Son, Inc., New York, p. 163.

Ali R. Amir-Moez

Department of Mathematics

Texas Tech University, Box 4319

Lubbock, Texas 79409