# Constructing a Regular Heptagon Using Gleason's Method

Constructing a Regular Heptagon Using Gleason's Method

This Demonstration shows Gleason's method for constructing a regular heptagon, using the following steps:

1. Draw a line segment of length 2 with midpoint and a circle with center and radius . Let , and be points on the line segment such that and . Draw an equilateral triangle .

O

O

1

A

B

P

AO=BO=1/2

OP=1

ABC

2. Draw a point between and so that . Draw an arc with center and radius . Let . The ray through with angle to meets the arc at a point .

D

A

O

DO=1/6

D

DC

3ϕ=∠CDB

D

ϕ

DP

F

3. The line perpendicular to through meets at and meets the circle at .

OP

F

OP

G

Q

4. The side length of the heptagon is and a compass can be used to measure out the other vertices of the heptagon.

PQ

Verification

tan(∠CDO)=3

3

cos(∠CDO)=cos(3ϕ)=

1

2

7

Therefore

4(cosϕ)-3cosϕ=

3

1

2

7

Define

x

2

so that

cosϕ=+/=

1

6

x

2

2

7

6

1+3x

2

7

Eliminating gives

ϕ

x+x-2x-1=0

3

2

which has as its only positive solution (see Details).

x=2cos(2π/7)