Constructible polygon

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

Some regular polygons are easy to construct with compass and straightedge; others are not. 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.

Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in 1837. It seems very unlikely that Gauss had a correct proof, because by taking n = 9, one can immediately deduce the impossibility of trisecting an angle of 120°, a fact of which Gauss was certainly aware.

Detailed results in terms of Fermat primes

Only five Fermat primes are known:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537

The next seven Fermat numbers, F₅ through F₁₁, are known to be composite.

(sequence A019434 in the OEIS).

Thus an n-gon is constructible if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ...

(sequence A003401 in the 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 the OEIS).

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. 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ぱい/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ふぁい(n),

where φふぁい(n) is Euler's totient function. Wantzel's result comes down to a calculation showing that φふぁい(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ぱい/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 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 are two vertices a central angle 360°/(p·q) apart. 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 didn't actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work. See heptadecagon.
The first explicit construction of a regular 257-gon was given by 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. (Conway has cast doubt on the validity of Hermes' construction, however.)

Other constructions

It should be stressed that the concept of constructibility as discussed in this article applies specifically to compass and straightedge construction. More constructions become possible if other tools are allowed. The so-called neusis constructions, for example, make use of a marked ruler. The construction of a regular heptagon is then easy, although most polygons remain inconstructible.

External links

Friedrich Julius Richelot (1832). "De resolutione algebraica aequationis x²⁵⁷ = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata". Journal für die reine und angewandte Mathematik. 9: 1–26, 146–161, 209–230, 337–358.
Duane W. DeTemple (1991). "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions". The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. MR1089454.
Christian Gottlieb (1999). "The Simple and Straightforward Construction of the Regular 257-gon". Mathematical Intelligencer. 21 (1): 31–37. MR1665155.
Regular Polygon Formulas, Ask Dr. Math FAQ.
Why Gauss could not have proved necessity of constructible regular polygons