criterion for constructibility of regular polygon

Theorem 1.

Let n be an integer with n3. Then a regularPlanetmathPlanetmathPlanetmath n-gon ( is constructiblePlanetmathPlanetmath ( if and only if a primitive nth root of unity ( is a constructible number.


First of all, note that a is a constructible number if and only if cos(2πn)+isin(2πn) 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 cos(2πn)+isin(2πn) will be considered.

Sufficiency: If a regular n-gon is constructible, then so is the angle whose vertex ( is the center ( of the polygonMathworldPlanetmathPlanetmath and whose rays pass through adjacent verticesMathworldPlanetmath of the polygon. The measure ( of this angle is 2πn.

By the theorem on constructible angles, sin(2πn) and cos(2πn) are constructible numbers. Note that i is also a constructible number. Thus, cos(2πn)+isin(2πn) is a constructible number.

Necessity: If ω=cos(2πn)+isin(2πn) is a constructible number, then so is ωm for any integer m.

On the complex planeMathworldPlanetmath, for every integer m with 0m<n, construct the point corresponding to ωm. Use line segmentsMathworldPlanetmath to connect the points corresponding to ωm and ωm+1 for every integer m with 0m<n. (Note that ω0=1=ωn.) This forms a regular n-gon. ∎

Title criterion for constructibility of regular polygon
Canonical name CriterionForConstructibilityOfRegularPolygon
Date of creation 2013-03-22 17:18:40
Last modified on 2013-03-22 17:18:40
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 51M15
Classification msc 12D15
Related topic RegularPolygon
Related topic RootOfUnity
Related topic TheoremOnConstructibleAngles