Compass and straightedge geometric constructions dating back to Euclid
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. In 1796
(when he was 19 years old), Gauss gave a sufficient
condition for a regular necessary , thus showing that regular
A003401 ).
A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry
angles .
Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with
odd numbers of sides are given by the first 32 rows
of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85, 255, ...
(OEIS A004729 , Conway and Guy 1996, p. 140).
In other words, every row is a product of distinct Fermat
primes , with terms given by binary counting.
See also Compass ,
Constructible Number ,
Cyclotomic Polynomial ,
Fermat
Number ,
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. Ball, W. W. R. and Coxeter, H. S. M.
Mathematical
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Problems of Geometry and How to Solve Them. Conway, J. H. and Guy, R. K. The
Book of Numbers. Courant,
R. and Robbins, H. What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. De Temple, D. W. "Carlyle
Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math.
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Drawings." Ch. 1 in Mathographics. Gardner, M. "Pascal's Triangle."
Ch. 15 in Mathematical
Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. Gauss, C. F. §365
and 366 in Disquisitiones
Arithmeticae. Heath, T. L. The
Thirteen Books of the Elements, 2nd ed., Vol. 2: Books III-IX. Joyce, D. E. "Euclid's Elements." https://wwn.ikan3.com/6wa440r81_5goyxwra6zxygidwpd/7hz~hgetij/4xjaiei/8jichcdcewk/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 ,
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The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the
Circle." In Famous
Problems and Other Monographs. Krížek,
M.; Luca, F.; and Somer, L. 17
Lectures on Fermat Numbers: From Number Theory to Geometry. Sloane, N. J. A. Sequence A003401 /M0505
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C. S. Excursions
in Geometry. Trott,
M. "Mathematica Educ. Res. 4 , 31-36, 1995. Trott,
M. "The
Mathematica GuideBook for Symbolics. https://wwn.ikan3.com/4xj7455_4tqfiqkpfiqobinloxpyttghtzn/ . 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. Referenced on Wolfram|Alpha Constructible Polygon
Cite this as:
Weisstein, Eric W. "Constructible Polygon."
From MathWorld https://wwn.ikan3.com/7hz2929k26_9nqeuhigsnjcgsjpnuemse/ConstructiblePolygon.html
Subject classifications
-gon to be constructible, which he also conjectured (but did
not prove) to be necessary, thus showing that regular
-gons
were constructible for 
, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,
48, 51, 60, 64, ... (OEIS A003401).
à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott,
M. "
à la Gauss." §1.10.2 in