Tessellation Proof

Date: 6/11/96 at 09:36:49 
From: Anonymous
Subject: Tesselation Proof

Hello,

I am looking for the proof of the statement that only three regular 
polygons tessellate in the Euclidean plane. I have an idea how it is 
done but am not quite sure how to write out the proof in a concise 
manner for a paper I am doing on tessellations. I would appreciate 
any information on this matter.

Thank you, Catherine

Date: 6/11/96 19:17:14 
From: Dr. Tom 
Subject: Re: Tesselation Proof

Well, a regular polygon has 3 or 4 or 5 or ... sides and angles, all 
equal. For each of these, you can work out the interior measure of 
the angles. For a triangle, it's 60 degrees; for a square, it's 90 
degrees; for a pentagon, it's 108 degrees; for a hexagon, it's 120 
degrees, and for anything with more than 6 sides, it's more than 
120 degrees. 

Since the regular polygons in a tessellation must fill the plane at 
each vertex, the interior angle must be an exact divisor of 360 
degrees. This works for the triangle, square, and hexagon, and you 
can show working tessellations for these figures. For all the others, 
the interior angles are not exact divisors of 360 degrees, and 
therefore those figures cannot tile the plane.

-Doctor Tom, The Math Forum