11428

# 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 .
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 .
3. The line perpendicular to through meets at and meets the circle at .
4. The side length of the heptagon is and a compass can be used to measure out the other vertices of the heptagon.
Verification
, so
.
Therefore
.
Define
so that
.
Eliminating gives
,
which has as its only positive solution (see Details).

### DETAILS

The points of a regular heptagon inscribed in the circle of radius 1 are given by . Since is a solution, divide the polynomial by to get
.
If
then
.
Substitute to get the third-degree equation
with solutions
,
,
.
These solutions also follow from the trigonometric identity
.
Set and to get
,
which factors as
.
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998 p. 45.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.
 © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS
 Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX Download or upgrade to Mathematica Player 7EX I already have Mathematica Player or Mathematica 7+