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### Tessellation Proof

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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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Associated Topics:
High School Symmetry/Tessellations

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