Chapter 3:

Tessellations

You may have heard the word "tessellations" before, and perhaps you have even studied them in an art class or a mathematics class. Whether you are familiar with them or not, you will find tessellations are interesting, beautiful, and fun to create! In this chapter, you will learn what a tessellation is, and how to create them using a compass, pens and a ruler, or using the Geometer's Sketchpad software.

So, what is a tessellation? First, let's look at an example of a tessellation, and then the next paragraph below will give you the mathematical definition.

The tessellation above was created by Mr. Jamal Jamalov, an automotive engineer from Baku, Azerbaijan. You can see his tessellations, and tessellations created by other people, at the following amazing website:

http://www.cromp.com/tess/people/T37.shtml

The word "tessellation" in Mathematics is defined as "a tiling of the plane in which the tiles are congruent polygons".

Tessellations are a fascinating blend of mathematics and art. They are patterns formed by interlocking congruent figures (congruent means "the same size and the same shape"), At first glance, it may seem that it would be easy to create such patterns. But when you look more closely, you will realize that it takes a lot of thought and knowledge of geometry, to make congruent figures interlock with no space left over.

So how are tessellations created? The following page, from the Math Forum website, will give you an excellent introduction to Tessellations:

http://mathforum.com/sum95/suzanne/whattess.html

Test Question #2: What must be true about the interior angle of the regular polygons in a tessellation, according to the web page linked above, and why?

Some of my own students have created tessellations. We were honored to have these tessellations published on The Math Forum website:

http://forum.swarthmore.edu/sum95/suzanne/hawaii.html

Of all the regular polygons, there are only three that will fit together perfectly: the equilateral triangle, square, and regular hexagon, because theirs are the angles that divide evenly into 360 degrees.

This tessellation is based on squares:

There an infinite number of irregular polygons and figures that will interlock and fill the plane, and therefore can be used to create tessellations. The green and pink figures in this simple example of a geometric tessellation below are said to "tessellate". The tessellation below is based on parallelograms, as you can see by the second image which shows how it was created:

Some tessellations are made of abstract shapes, while others can be made to look like animals, birds, fish or other creatures. The tessellation above looks a little bit like a strange fish, and with eyes, a nose and scales, would look even more so!

We see examples of tessellations and other geometric patterns in fabric designs and in ornamental tile work. Some very beautiful examples of this can be seen in the tile work of the Alhambra palace in Spain.

Many people find tessellations fascinating, and like to create their own. One of the most famous "tessellators" of all time is Maurits Cornelis Escher. You will find an amazing amount of information about M.C. Escher at the ThinkQuest web site below.

http://library.thinkquest.org/16661/escher.html

http://library.thinkquest.org/16661/escher/biography.1.html

http://library.thinkquest.org/16661/escher/biography.3.html

http://library.thinkquest.org/16661/escher/tessellations.1.html

http://library.thinkquest.org/16661/escher/tessellations.2.html

http://library.thinkquest.org/16661/escher/tessellations.3.html

http://library.thinkquest.org/16661/escher/tessellations.4.html

http://library.thinkquest.org/16661/escher/tessellations.5.html

http://library.thinkquest.org/16661/escher/tessellations.8.html

http://library.thinkquest.org/16661/escher/tessellations.12.html

Besides tessellations, Escher did a number of other fascinating types of mathematical artwork. To see some of his unusual and fascinating geometric artwork, click on the link below. You may, of course, go on to see more of the pages in this amazing website if you like,by clicking on the "next" button at the bottom of each of the pages on the ThinkQuest website, but you are not required to for this class:

http://library.thinkquest.org/16661/escher/trends.1.html

ThinkQuest is an organization that sponsors contests for students who are interested in creating web pages. There are some truly wonderful web pages on this site, made by school students like yourself. You might want to take some time to explore the ThinkQuest site, and maybe become prize winner with them! To learn more about ThinkQuest, click on the link below, and explore!

http://www.thinkquest.org/competitions.html

If you find Escher interesting, as many people do, you will find many pages on the internet about his work. Just go to your favorite search engine ( I like Google, at http://www.google.com/, myself). Type in "Escher" and you will be amazed at what you find. I just tried it, and got 171,000 websites to choose from!

Here are some of my favorites, which you may visit if you like. You are not required to visit this particular group of pages, for this class:

From ThinkQuest - an unbelievable website on Escher and tessellations.

http://library.thinkquest.org/16661/escher.html

Check out the thumbnails from this same ThinkQuest site:

http://library.thinkquest.org/16661/gallery/sourcebook/4.html

A very cool web page with tessellations in motion (You need the Shockwave Plug-in to view these movies):

http://www.cromp.com/tess/apc/page09.html

Of course there are other people besides Escher who create tessellations. You can see wonderful tessellations by Andrew Crompton at these pages:

http://www.cromp.com/tess/apc/page01.html

http://www.cromp.com/tess/apc/page04.html

Tessellations can be found in many different applications: fabric patterns, friezes and tile work in architecture, and many types of graphic design. For example, look at the quilt below:

(from http://memory.loc.gov/ammem/qlthtml/qlthome.html)

Visit the following website to see tessellations in the repeating patterns of wallpaper:

http://mathmuse.sci.ibaraki.ac.jp/pattrn/PatternE.html

... And visit this educational website on the applications of mathematics in art and architecture:

http://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html

Test Question #4: On the web page you just visited, what is the definition of a regular polygon?

You can create tessellations using a pencil, ruler, and tracing paper, or you can create tessellations with The Geometer's Sketchpad.

To construct a tessellation using a pencil, ruler, and tracing paper, click on the link below:

http://mathforum.org/~sanders/geometry/GP08CreatingTessellations.html

To construct a tessellation using The Geometer's Sketchpad, first read the page linked above for some background information,then click on this link for step-by-step GSP instructions:

http://forum.swarthmore.edu/sum95/suzanne/tess.gsp.tutorial.html

Chapter 3 Project

Your assignment for this project is to create an artistic tessellation, using geometric shapes. First do some research on the Internet, and find some tessellations. Read some information about the tessellations, and explore any aspects that interest you. Then design and create a tessellation. If you would like to see some examples of graphics created by students like yourselves, please go to the Student Work section from the Table of Contents.

Go To Chapter 4: 3D Drawing