# criterion for constructibility of regular polygon

###### Theorem 1.

Let $n$ be an integer with $n\geq 3$. Then a regular $n$-gon (http://planetmath.org/RegularPolygon) is constructible (http://planetmath.org/Constructible2) if and only if a primitive $n$th root of unity (http://planetmath.org/PrimitiveRootOfUnity) is a constructible number.

###### Proof.

First of all, note that a is a constructible number if and only if $\displaystyle\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number. See the entry on roots of unity for more details. Therefore, without loss of generality, only the constructibility of the number $\displaystyle\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ will be considered.

Sufficiency: If a regular $n$-gon is constructible, then so is the angle whose vertex (http://planetmath.org/Vertex5) is the center (http://planetmath.org/Center9) of the polygon and whose rays pass through adjacent vertices of the polygon. The measure (http://planetmath.org/AngleMeasure) of this angle is $\displaystyle\frac{2\pi}{n}$.

By the theorem on constructible angles, $\displaystyle\sin\left(\frac{2\pi}{n}\right)$ and $\displaystyle\cos\left(\frac{2\pi}{n}\right)$ are constructible numbers. Note that $i$ is also a constructible number. Thus, $\displaystyle\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number.

Necessity: If $\displaystyle\omega=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number, then so is $\omega^{m}$ for any integer $m$.

On the complex plane, for every integer $m$ with $0\leq m, construct the point corresponding to $\omega^{m}$. Use line segments to connect the points corresponding to $\omega^{m}$ and $\omega^{m+1}$ for every integer $m$ with $0\leq m. (Note that $\omega^{0}=1=\omega^{n}$.) This forms a regular $n$-gon. ∎

Title criterion for constructibility of regular polygon CriterionForConstructibilityOfRegularPolygon 2013-03-22 17:18:40 2013-03-22 17:18:40 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Theorem msc 51M15 msc 12D15 RegularPolygon RootOfUnity TheoremOnConstructibleAngles