# Constructing a Regular Heptagon Using Lill's Method

Constructing a Regular Heptagon Using Lill's Method

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

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

7

z

z=1

7

z

z-1

6

z

5

z

4

z

3

z

2

z

If , then

z=cosϕ+isinϕ

v=z+=2cosϕ

1

z

Substituting

v=z+

1

z

leads to the cubic equation

3

v

2

v

It has solutions , , .

2cos(2π/7)

2cos(4π/7)

2cos(6π/7)

This follows from the trigonometric identity

7

(2cosα)

5

(2cosα)

3

(2cosα)

Set and to get

7α=2π

x=2cosα

7

x

5

x

3

x

which factors as

2

(+-2x-1)

3

x

2

x

There are solutions when the points and coincide.

L

4

L'

3