Constructible polygon

Construction of a regular pentagon

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.

Conditions for constructibility

Construction of the regular 17-gon

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,[1]:p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.[1]:pp. 49-50 This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons 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:

A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none).

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.

Detailed results by Gauss' theory

Restating the Gauss-Wantzel theorem:

A regular n-gon is constructible with ruler and compass if and only if n = 2kp1p2...pt where k and t are non-negative integers, and the pi's (when t > 0) are distinct Fermat primes.

The known Fermat primes are:

F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in OEIS)

The next twenty-eight Fermat numbers, F5 through F32, are known to be composite.[2]

Thus an n-gon is constructible if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, … (sequence A003401 in OEIS),

while an n-gon is not constructible with compass and straightedge if

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, … (sequence A004169 in OEIS).

Connection to Pascal's triangle

Since there are 5 known Fermat primes, we know of 31 numbers that are multiples of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, …, 4294967295 (sequence A045544 in OEIS). 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. (Because of this, the 1s in such a list form an approximation to the Sierpiński triangle.) This pattern breaks down after there, as the 6th Fermat number is composite, so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and is therefore unknown how many odd-sided constructible polygons exist. In general, if there are x Fermat primes, then there are 2x−1 odd-sided constructible polygons.

General theory

In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations.[3] In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field that is a power of two.

In the specific case of a regular n-gon, the question reduces to the question of constructing a length

cos(2π/n).

This number lies in the n-th cyclotomic field — and in fact in its real subfield, which is a totally real field and a rational vector space of dimension

½φ(n),

where φ(n) is Euler's totient function. Wantzel's result comes down to a calculation showing that φ(n) is a power of 2 precisely in the cases specified.

As for the construction of Gauss, when the Galois group is 2-group it follows that it has a sequence of subgroups of orders

1, 2, 4, 8, ...

that are nested, each in the next (a composition series, in group theory terms), something simple to prove by induction in this case of an abelian group. Therefore there are subfields nested inside the cyclotomic field, each of degree 2 over the one before. Generators for each such field can be written down by Gaussian period theory. For example for n = 17 there is a period that is a sum of eight roots of unity, one that is a sum of four roots of unity, and one that is the sum of two, which is

cos(2π/17).

Each of those is a root of a quadratic equation in terms of the one before. Moreover these equations have real rather than imaginary roots, so in principle can be solved by geometric construction: this because the work all goes on inside a totally real field.

In this way the result of Gauss can be understood in current terms; for actual calculation of the equations to be solved, the periods can be squared and compared with the 'lower' periods, in a quite feasible algorithm.

Compass and straightedge constructions

Compass and straightedge constructions are known for all known constructible polygons. If n = p·q 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 relatively prime, there exists integers a,b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a p·q-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 AD 150). See pentagon.
• 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. See heptadecagon.
• The first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822)[4] and Friedrich Julius Richelot (1832).[5]
• 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.[6]

Gallery

From left to right, constructions of a 17-gon, 257-gon and 65537-gon. Only the first stage of the 65537-gon construction is shown: the constructions of the 17-gon and 257-gon are given complete.