Constructible Polygon
Euclid 
were capable of inscribing regularpolygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, this listing is not a completeenumeration of ``constructible'' polygons. A regular 
-gon (
) can be constructed by Straightedge and Compass Iff
where
is in Integer 
and the 
are distinct Fermat Primes. FermatNumbers are of the form
where
is an Integer . The only known Primes of this form are 3, 5, 17, 257, and 65537. The fact thatthis condition was Gauß in 1796 when he was 19 years old. That thiscondition was also Gauß, and the first proof of this fact iscredited to Wantzel (1836).See also Compass, Constructible Number, Geometric Construction, Geometrography,Heptadecagon, Hexagon, Octagon, Pentagon, Polygon, Square, Straightedge,Triangle
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 94-96, 1987. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 119, 1996. De Temple, D. W. ``Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions.'' Amer. Math. Monthly 98, 97-108, 1991. Dixon, R. ``Compass Drawings.'' Ch. 1 in Mathographics. New York: Dover, pp. 1-78, 1991. Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. Translated by A. A. Clarke. New Haven, CT: Yale University Press, 1965. 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. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 137-138, 1990. 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. 1, 366-372, 1836.
- Handlebody
- Hankel Function
- Hankel Function of the First Kind
- Hankel Function of the Second Kind
- Hankel Matrix
- Hankel's Integral
- Hankel Transform
- Hann Function
- Hanning Function
- Hanoi Graph
- Hanoi Towers
- Hansen-Bessel Formula
- Hansen Chain
- Hansen Number
- Hansen's Problem
- Happy Number
- Harada-Norton Group
- Harary Graph
- Hard Hexagon Entropy Constant
- Hard Square Entropy Constant
- Hardy-Littlewood Conjectures
- Hardy-Littlewood Constants
- Hardy-Littlewood k-Tuple Conjecture
- Hardy-Littlewood Tauberian Theorem
- Hardy-Ramanujan Number