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
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