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Constructible Polygon

PolygonConstruction

A constructible polygon, in this context, is a regular polygon that can be constructed with a compass and straightedge. Compass and straightedge geometric constructions dating back to Euclid (Heath 1956) were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, ..., sides. Gauss (1801, pp. 662-665) gave a sufficient condition for a regular n-gon to be constructible, which he also asserted to be necessary without supplying a proof, thus showing that regular n-gons were constructible for n=3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, ... (OEIS A003401; Gauss 1801, pp. 662-665).

For n>=3, a regular n-gon is constructible iff n=2^kp_1...p_r, where k and r are nonnegative integers and p_1, ..., p_r are distinct Fermat primes. The empty product is 1. Equivalently, phi(n) is a power of 2, where phi is the totient function. Wantzel (1837) proved that Gauss's condition is necessary.

A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry angles.

PolygonConstructionTri

Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the numbers of sides for the known constructible regular polygons with odd numbers of sides are given by rows 1 through 31 of the Sierpiński sieve interpreted as binary numbers, giving 3, 5, 15, 17, 51, 85, 255, ... (OEIS A045544; Conway and Guy 1996, p. 140). Including row 0 adds the empty product 1 (OEIS A004729), which is not a polygon side count. These 32 row values are the products of subsets of the five known Fermat primes. An additional Fermat prime, if one exists, would give further odd constructible side counts.

See also

Compass, Constructible Number, Cyclotomic Polynomial, Fermat Number, Gaussian Period, Geometric Construction, Geometrography, Heptadecagon, Hexagon, Octagon, Pentagon, Polygon, Sierpiński Sieve, Square, Straightedge, Triangle, Trigonometry Angles

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References

Bachmann, P. Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie. Leipzig, Germany: Teubner, 1872.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 94-96, 1987.Bold, B. "The Problem of Constructing Regular Polygons." Ch. 7 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 49-71, 1982.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 190-191, 1996.Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, 1996.De Temple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991.Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352-386, 1955.Dixon, R. "Compass Drawings." Ch. 1 in Mathographics. New York: Dover, pp. 1-78, 1991.Gardner, M. "Pascal's Triangle." Ch. 15 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 194-207, 1977.Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany: Fleischer, pp. 662-665, 1801. English translation by A. A. Clarke. New Haven, CT: Yale University Press, 1966.Heath, T. L. The Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. New York: Dover, 1956.Joyce, D. E. "Euclid's Elements." http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.Kazarinoff, N. D. "On Who First Proved the Impossibility of Constructing Certain Regular Polygons with Ruler and Compass Alone." Amer. Math. Monthly 75, 647-648, 1968.Klein, F. "The Division of the Circle into Equal Parts." Part I, Ch. 3 in "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 16-23, 1980.Krížek, M.; Luca, F.; and Somer, L. 17 Lectures on Fermat Numbers: From Number Theory to Geometry. New York: Springer-Verlag, 2001.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 137-138, 1990.Sloane, N. J. A. Sequences A003401/M0505, A004729, and A045544 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. "cos(2pi/257) à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott, M. "cos(2pi/257) à la Gauss." §1.10.2 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 312-321, 2006. https://www.mathematicaguidebooks.org/.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 2, 366-372, 1837. https://www.numdam.org/item/JMPA_1837_1_2__366_0/.

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Constructible Polygon

Cite this as:

Weisstein, Eric W. "Constructible Polygon." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConstructiblePolygon.html

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