## Frequently Asked Questions

#### How do you prove that constructing a heptagon is impossible?

**Theorem**

Constructing a regular heptagon using compass and straight edge is impossible.

**Proof**

Please note that this proof assumes the knowledge that an construction that can be shown to be algebraically equivalent to a cubic containing rational coefficients and having irrational roots is impossible, proved in the Impossible Constructions document.

The proof for the construction of a heptagon (7-gon) is a quite difficult to follow and makes use of complex numbers.

We begin by recognising that, by using complex numbers, the seventh root of unity yields seven solutions. That is, `z`^{7} = 1 ⇒ `z`^{7} − 1 = 0. By considering the angle between the real number axis and the first root we shall produce the required angle, ^{360}/_{7} degrees.

So we proceed by writing `z`^{7} − 1 = (`z` − 1)(`z`^{6} + `z`^{5} + `z`^{4} + `z`^{3} + `z`^{2} + `z` + 1) = 0. As `z` ≠ 1 does not provide the given angle, the root must be a solution of `z`^{6} + `z`^{5} + `z`^{4} + `z`^{3} + `z`^{2} + `z` + 1 = 0.

Dividing by `z`³ gives `z`³ + `z`² + `z` + ^{1}/_{z} + ^{1}/_{z²} + ^{1}/_{z³} = 0.

This can be shown to be equivalent to (`z` + ^{1}/_{z})³ + (`z` + ^{1}/_{z})³ − 2(`z` + ^{1}/_{z})² − 1 = 0.

Let `x` = `z` + ^{1}/_{z}, so `x`³ + `x`² − 2x − 1 = 0. To show that the construction is not possible all we need demonstrate is that no rational roots exist.

Assume that `x` = `a`/`b`, where `a` and `b` have no common factors, so

`a`^{3}/`b`^{3} + `a`^{2}/`b`^{2} − 2`a`/`b` − 1 = 0, leading to `a`^{3} + `a`^{2}`b` − 2`ab`^{2} – `b`^{3} = 0.

By writing `a`^{3} = `b`(`b`^{2} + 2`ab` − `a`^{2}) and `b`^{3} = `a`(`a`^{2} + `ab` − 2`b`^{2}), we see that `a`^{3} is a multiple of `b` and `b`^{3} is a multiple of `a`. As `a` and `b` have no common factors, each must be ±1. But clearly `x` = ±1 does not satisfy the cubic equation, so we conclude that the construction of a regular heptagon by the use of compass and straightedge is impossible.