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### Types of Tessellations

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Date: 11/15/98 at 19:42:23
From: Kathy Siegel
Subject: Tessellations

Dear Dr. Math,

I am writing a math report on tessellations, and I must explain the
mathematics involved with tessellations. However I am not exactly sure
what mathematics are involved. I understand that the node must equal
360 degrees and that there can be no more then 42 sides in a polygon
if it is going to be tessellated, and there can be no less then three.
I figured that in my report I would explain the reasons for why there
are those rules on sides but I do not know what else to write about.
Please explain to me the other mathematical issues involved with
tessellations. Thank you.

Sincerely,
Kathy Siegel
```

```
Date: 11/16/98 at 19:23:43
From: Doctor Dianna
Subject: Re: Tessellations

Dear Kathy,

Some things you may mention in your paper include discussing regular
versus semiregular tessellations. A "regular" tessellation is made up
of the same regular polygon. (Remember that a regular polygon means
all angles have the same degree measure and all sides are the same
length.) For example, a tessellation made with only squares is called
regular.

Contrast that with "semiregular" tessellations which are made up of at
least two different polygons. Oftentimes you will see bathroom tile
using octagons and squares. Since two shapes are used we call this
tessellation "semiregular."

An interesting fact is that there exist only 3 regular tessellations.
Regular triangles, squares, and hexagons will form tessellations of the
plane by themselves. Maybe in your paper you could discuss why this is
so. (The answer has to do with the fact that these angle measures are
perfect divisors of 360.)

M.C. Escher is an artist who makes wonderful pictures with
tessellations. Maybe you could check out his work and show examples in

Finally, there are tessellations that may not cover the plane, but they
cover other shapes. For example, the soccer ball is a tessellation
that covers a sphere. Can you think of others?

Alejandre, on tessellations:

http://mathforum.org/sum95/suzanne/tess.intro.html

- Doctor Dianna, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Geometry
High School Projects
High School Symmetry/Tessellations
Middle School Geometry

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