# 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 . Since is a solution, if we divide the polynomial by , we get

z=1

7

z=1

z-1

7

z-1

z+z+z+z+z+z+1=0

6

5

4

3

2

If , then

z=cosϕ+isinϕ

v=z+=2cosϕ

1

z

Substituting

v=z+

1

z

leads to the cubic equation

v+v-2v-1=0

3

2

It has solutions , , .

2cos(2π/7)

2cos(4π/7)

2cos(6π/7)

This follows from the trigonometric identity

(2cosα)-7(2cosα)+14(2cosα)-7(2cosα)-2cos7α=0

7

5

3

Set and to get

7α=2π

x=2cosα

x-7x+14x-7x-2=0

7

5

3

which factors as

(x+x-2x-1)(x-2)=0

3

2

2

There are solutions when the points and coincide.

L

4

L'

3