16-sided Regular Polygon

Date: 07/31/2001 at 18:18:06
From: Tiffany Travis
Subject: 16 sided regular polygon

I have tried to construct a 16-sided polygon, but my attempts have 
been unsuccessful, and I don't know how to construct one.  If you 
could help I would appreciate it.

Date: 08/01/2001 at 13:43:15
From: Doctor Jubal
Subject: Re: 16 sided regular polygon

Hi Tiffany,

Thanks for writing to Dr. Math.

There's a straightforward way to construct any regular polygon whose 
number of sides is a power of two (square, octagon, 16-gon, etc...)

We'll start with a pair of perpendicular lines. Let's call the point 
where they intersect point P. If you draw a circle centered at P, the 
four points where that circle intersects the two perpendicular lines 
are the corners of a square.

The two perpendicular lines form four 90-degree angles at P. If you 
bisect all those angles, you'll have eight 45-degree angles at P, and 
the eight points where the circle centered at P intersects the 
original lines and the angle bisectors are the vertices of a regular 
octagon.

You can repeat the process, bisecting the 45-degree angles to make 
sixteen 22.5-degree angles, and the sixteen points where the circle at 
P intersects the two original lines or any of the angle bisectors are 
the vertices of a regular 16-gon.

You can make 32-gons, 64-gons, or any regular 2^n-gon in the same 
fashion. All you need is a consruction to bisect an angle.

I searched the Dr. Math archives using the keywords

  construct angle bisector (all, any order)

and found the page

   Geometry Constructions with Compass and Straightedge
   http://mathforum.org/dr.math/problems/zaidi11.13.98.html   

It contains instructions for bisecting an angle with compass and 
straightedge, complete with illustrations. Using that method, you 
should be able to construct any regular 2^n-gon you like.

As a generalization, we started with a square inscribed in a circle 
and kept bisecting the angles at the center of the circle to make a 
regular octagon, 16-gon, etc. If you start with an equilateral 
triangle inscribed in a circle, you can use the same method to 
construct a regular hexagon, dodecagon, 24-gon, etc., and if you start 
with a pentagon inscribed in a circle, you can use it to construct a 
regular decagon, 20-gon, 40-gon, etc. Using this method, if you can 
construct a regular n-gon, you can construct a regular (2^k)*n-gon.

Does this help? Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/