criterion for constructibility of regular polygon
Theorem 1.
Let be an integer with . Then a regular -gon (http://planetmath.org/RegularPolygon) is constructible (http://planetmath.org/Constructible2) if and only if a primitive 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 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 will be considered.
Sufficiency: If a regular -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 .
By the theorem on constructible angles, and are constructible numbers. Note that is also a constructible number. Thus, is a constructible number.
Necessity: If is a constructible number, then so is for any integer .
On the complex plane, for every integer with , construct the point corresponding to . Use line segments to connect the points corresponding to and for every integer with . (Note that .) This forms a regular -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 |