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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, $z^7 = 1 \Rightarrow z^7 - 1 = 0$. By considering the angle between the real number axis and the first root we shall produce the required angle, $\frac{360}{7}$ degrees.
So we proceed by writing $z^7 - 1 = (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1) = 0$.
Clearly $z \ne 1$, as this does not provide the given angle, so 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 $z^3 + z^2 + z + 1 + \dfrac{1}{z} + \dfrac{1}{z^2} + \dfrac{1}{z^3} = 0$.
This can be shown to be equivalent to $\left(z + \dfrac{1}{z}\right)^3 + \left(z + \dfrac{1}{z}\right)^2 - 2\left(z + \dfrac{1}{z}\right) - 1 = 0$.
Let $x = z + \dfrac{1}{z}$, such that we get $x^3 + x^2 - 2x - 1 = 0$. To show that the construction is not possible all we need demonstrate is that no rational roots exist.
Assume that $x = \dfrac{a}{b}$, where $a$ and $b$ have no common factors.
Therefore $\dfrac{a^3}{b^3} + \dfrac{a^2}{b^2} - \dfrac{2a}{b} - 1 = 0$, leading to $a^3 + a^2b - 2ab^2 - b^3 = 0$.
By rearranging we get $a^3 = b(b^2 + 2ab - a^2)$ and $b^3 = a(a^2 + ab - 2b^2)$. We can 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 $\pm 1$. But clearly $x = \pm 1$ 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.