Constructing a Regular Heptagon Using Lill's Method

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This Demonstration shows how to construct a regular heptagon using Lill's method for solving cubic equations.

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The points of a regular heptagon with vertices on a circle of radius 1 are given by . Since is a solution, if we divide the polynomial by , we get

.

If , then

.

Substituting

leads to the cubic equation

.

It has solutions , , .

This follows from the trigonometric identity

.

Set and to get

,

which factors as

.

There are solutions when the points and coincide.

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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA


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Reference

[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998 p. 45.



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