5.5: Ruler and compasses constructions for regular polygons
Euclid’s Elements include methods for constructing the regular polygons that are required for the construction of the regular polyhedra (see Section 5.6 ). In one sense, Euclid is thoroughly modern: he is reluctant to work with entities that cannot be constructed. And for him, geometrical construction means construction “using ruler and compasses” only.
For each regular polygon, there are two related (and sometimes very different) construction problems:
- given two points A and B, construct the regular n-gon with AB as an edge of the regular polygon;
- given two points O and A, construct the regular n-gon ABCD . . . inscribed in the circle with centre O and passing through A, that is with circumradius OA .
Before Problem 137 we saw how to construct an equilateral triangle ABC given the points A, B. And in Problem 36 we saw that every regular polygon has a circumcentre O .
Problem 182 Given points O, A, show how to construct the regular 3-gon ABC with circumcentre O.
Problem 183
(a) Given two points O , A , show how to construct a regular 4-gon ABCD with circumcentre O .
(b) Given points A , B , show how to construct a regular 4-gon ABCD.
Problem 184
(a)(i) Given two points O , A , show how to construct a regular 6-gon ABCDEF with circumcentre O .
(ii) Given two points O , A , show how to construct a regular 8-gon ABCDEFGH with circumcentre O .
(b)(i) Given points A , B , show how to construct a regular 6-gon ABCDEF .
(ii) Given points A , B , show how to construct a regular 8-gon ABCDEFGH .
Problem 185
(a) (i) Given two points O , A , show how to construct a regular 5-gon ABCDE with circumcentre O .
(ii) Given points O , A , show how to construct a regular 10-gon ABCDEFGHIJ with circumcentre O .
(b) (i) Given points A , B , show how to construct a regular 5-gon ABCDE .
(ii) Given points A , B , show how to construct a regular 10-gon ABCDEFGHIJ .
We shall not prove it here, but it is impossible to construct a regular 7-gon, or a regular 9-gon, or a regular 11-gon using ruler and compasses. All constructions with ruler and compasses come down to two moves:
- if a is a known length, then can be constructed (see Problem 173 (c));
- if an n -gon can be constructed, then the sides can be bisected to produce a 2n-gon.
Put slightly differently, all ruler and compasses constructions involve solving linear or quadratic equations, so the only new points, or lengths we can construct are those which involve iterated square roots of expressions or lengths which were previously known.
This iterated extraction of square roots is linked to a fact first proved by Gauss (1777-1855), namely that the only regular p-gons (where p is a prime) that can be constructed are those where p is a Fermat prime - that is, a prime of the form p = 2 k + 1 (in which case k has to be a power of 2: see Problem 118 ). Gauss proved (as a teenager, though it was first published in his book Disquisitiones arithmeticae in 1801):
a regular n-gon can be constructed with ruler and compasses if and only if n has the form
where p 1 ,p 2 ,p 3 ,... ,p k are distinct Fermat primes.
As we noted in Chapter 2 , the only known Fermat primes are the five discovered by Fermat himself, namely 3, 5, 17, 257, and 65 537.