Types of Tessellations

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 
your paper.

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?

For more information, here is a good introductory source, by Suzanne 
Alejandre, on tessellations:

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

Good luck with your paper!

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