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Regular polygon that can be constructed with compass and straightedge

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.

Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, This led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which n gons (that is, polygons with n edges) are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17 gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons. Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem: A regular n gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes. Here, a power of 2 is a number of the form , where m ≥ 0 is an integer. A Fermat prime is a prime number of the form , where m ≥ 0 is an integer. The number of Fermat primes involved can be 0, in which case n is a power of 2. In order to reduce a geometric problem to a problem of pure number theory, the proof uses the fact that a regular n gon is constructible if and only if the cosine is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Equivalently, a regular n gon is constructible if any root of the nth cyclotomic polynomial is constructible.

Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 31 constructible odd sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo 2 Pascal's triangle, minus the top row, which corresponds to a monogon. (Because of this, the 1s in such a list form an approximation to the Sierpiński triangle.) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd sided constructible regular polygons exist. In general, if there are q Fermat primes, then there are 2q−1 odd sided regular constructible polygons.

Compass and straightedge constructions are known for all known constructible polygons. If n = pq with p = 2 or p and q coprime, an n gon can be constructed from a p gon and a q gon. If p = 2, draw a q gon and bisect one of its central angles. From this, a 2q gon can be constructed. If p > 2, inscribe a p gon and a q gon in the same circle in such a way that they share a vertex. Because p and q are coprime, there exists integers a and b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a pq gon can be constructed. Thus one only has to find a compass and straightedge construction for n gons where n is a Fermat prime. The construction for an equilateral triangle is simple and has been known since antiquity; see Equilateral triangle. Constructions for the regular pentagon were described both by Euclid (Elements, ca. 300 BC), and by Ptolemy (Almagest, ca. 150 AD). Although Gauss proved that the regular 17 gon is constructible, he did not actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work. The first explicit constructions of a regular 257 gon were given by Magnus Georg Paucker (1822) and Friedrich Julius Richelot (1832). A construction for a regular 65537 gon was first given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200 page manuscript.

From left to right, constructions of a 15 gon, 17 gon, 257 gon and 65537 gon. Only the first stage of the 65537 gon construction is shown; the constructions of the 15 gon, 17 gon, and 257 gon are given completely.

The concept of constructibility as discussed in this article applies specifically to compass and straightedge constructions. More constructions become possible if other tools are allowed. The so called neusis constructions, for example, make use of a marked ruler. The constructions are a mathematical idealization and are assumed to be done exactly. A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if where r, s, k ≥ 0 and where the pi are distinct Pierpont primes greater than 3 (primes of the form These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons that can be constructed with paper folding. The first numbers of sides of these polygons are:
3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, 296,