The Mathematics of Wallpaper

Tessellation

Tessellation is a repeated geometric design that covers a plane without gaps or overlaps, like wallpapers. Tessellations in principle do not end, they should be thought of as continuing infinitely. M.C. Escher was a famous artist who enjoyed twisting perceptions of reality. He was responsible for works such as Reptiles, Horseman and many more that incorporated the use of tessellations. Tessellations can be seen all around us, on the pavements, in our bathroom floor tiles, hardwood floors, just to name a few. Tesselations can have any of the four types of isometry as symmetry, namely reflectional symmetry, translational symmetry, rotational symmetry, and glide reflection symmetry. Students should be able to recognize each of these symmetries in wallpaper patterns. If they study a variety of tessellations, they should see that a tessellation can have rotational symmetry of order 1, 2, 3, 4 or 6, and only these. (This is the crystallographic restriction.) Among the patterns with a given order of rotational symmetry, it is possible to find some that have reflectional or glide reflectional symmetry, and some that have only rotational symmetry. This already creates 10 different types of wallpaper. Some of these 10 bins can be further refined, making 17 different types in all. See how many your students can distinguish!

Tessellation using Geometer's Sketchpad

This is the part of the unit where the students will begin to learn about the computer program called Geometer's Sketchpad. This program is a geometry software program for exploring mathematics and it is very user friendly. I will teach the students how to, first, open the program on their computers. The next step will be to show them how to draw on the sketchpad. The software comes with a toolbox that allows you to

1. select parts of your drawings,
2. point tool that helps with drawing points,
3. compass tool that are used for drawing circles,
4. straightedge tool for drawing straight lines (or edges), and
5. text and custom tool that helps with inputting text in your drawings and customizing your drawings respectively.

The software also comes with a taskbar full of functions that will help you with any geometry drawing. When they are able to draw (or sketch) any shape on the sketchpad, then I will show them how to do some very basic tessellation. Simple tessellation on the sketchpad can be achieved by maneuvering the shape (or object) in different ways. By different ways, I mean you could, flip the shape (or object), slide the object, or even turn the object repeatedly. You can also use a combination of these to achieve some very beautiful tessellations. For example, it is described above how to create a tessellation starting with any quadrilateral, using only repeated rotations of 180 ^o in the midpoints of its sides.

The idea of tessellation is what is incorporated in the use of various wallpaper designs. Students will be shown different types of wallpaper designs and see how the idea of tessellation is used in them. Students will have a final project creating their own wallpaper design. They will begin this project by first choosing an object of interest, sketching the object using Geometer's Sketchpad and using the idea of tessellation to create wallpaper design.