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