Tessellations
While creating repeated patterns, students talk about shapes, colors, and drawing techniques.

You will need:

• For each student, a piece of paper 8.5"x11".
• For each student, a template made of construction paper. A variety of templates will make the activity more interesting.
• Crayons or colored pencils.
Suggested vocabulary:

(various colors and shapes)  paper  template  draw  trace  copy  reverse/rotate  slide  touch  pattern  color (verb)  edge

Instructions:

Click on the thumbnail image if you want to see a larger picture. Teachers who use a screen reader
should ignore the links to the larger images, as the screen reader will read the instructions in order.

Tessellations are patterns which join together to cover a surface, leaving no empty spaces on that surface. This form of art is widely known due to the works of M.C. Escher, and is often used in math classes as a way of demonstrating congruency within patterns.
Geometric tessellations are not difficult to create, although they too require a high degree of precision. Artistic ones require greater efforts. The tessellations introduced here are simple-- you may want to explore more complex ones by following the links at the bottom of this page.

1. This is a sample template. Print out the large image (use link from thumbnail), cut out the shape, copy to construction paper, and cut out.

2. Trace the template on the paper, with the template edge flush with the edge of the paper.

3. Slide the template along the edge of the paper, trace it again, touching the previous shape.

4. Slide the template along the edge of the paper, trace it again, touching the previous shape (with smaller templates, you will repeat this step more times).

5. Invert the template vertically, trace repeatedly in the same way (illustrated here in green).

6. Invert the template again and repeat until the whole paper is covered.
Color the paper. Smaller templates produce a much richer effect, but take longer to complete

Here is a fish template. With interlocking designs like these, a good strategy is to leave the template on the paper, put the work paper on top of the template, and trace it this way, thus minimizing distortions.

These are the tessellations created with the fish template.

• The Coolmath site
An excellent introduction to tessellations: what they are and how to make them interesting. Superb drawings that should inspire as well as teach.

Explore the world of tessellations with Dr. Robert Fathauer. The site includes pages on "About Tessellations," "Lessons and Solutions," "The Mathematical Art of Robert Fathauer," and "Really Cool Stuff." There are also tempting products to buy.

• Tessellations: Technology and Culture
Tessellations with Java graphics, some of which were designed by award-winning programmers, are delightful. Not to be missed are William's Periodic Wallpaper , William Chow's excellent references, and his pages on "The Mathematical Basis of Interlocking Shapes" and "Hints and shortcuts to make tessellations"

• William W. Chow's Web pages on Escher
A wonderful introduction to Escher accompanied by several links to other first-rate sites.

• Geometry in Tessellations
An excellent lesson plan for teaching tessellations. This is a great way to introduce tessellations to your students.

• People Doing Tessellations
Andrew Crompton's Home Page on which he displays a sampling a tessellations made by thirty artists. He also displays "the world's first lifelike tiling" and has links to outstanding sites.

• 21st Century Escher Inspired Graphics
Andrew Crompton displays a collection of tessellations (from his wonderful Web site) inspired by Escher, that were created by a number of different artists.

• The World of Escher's Tessellation Contest
Paul Schoefield, Webmaster at The World of Escher, has displayed the top 60 entries for Contest 11. After admiring these tessellations, be sure to check out the rest of "The Site for Everything Escher," and be prepared to be inspired .

• Tantalizing Tessellations
Vivian Archambault, Danielle Desjardins, and Terry Wood present a superb unit on integrating math with subjects such as language arts and arts education. They provide 10 very complete lesson plans, several excellent Internet sites, a thorough list of resources, and an evaluation assessment tool. Teachers planning to introduce their students to tessellations would be well advised to take advantage of this first-rate site.