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# Introduction to Wallpaper Patterns

Figure 1

In the last section we learned about applying isometries to a motif to get one or more images. By experimenting with mirrors, we saw how to use reflection to reproduce a motif and create a pattern on a plane or sphere. In this section, we learn more about plane patterns whose symmetries are translations, glide reflections, and rotations.

Some sample patterns are shown below. Notice that some are finite, while some can be extended indefinitely, and that the "motif" sometimes consists of more than one picture. Also, sometimes half of the images are mirror images of the original motif and sometimes each image can be transformed into the next by "sliding" the pattern.

Figure 2
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Figure 4 Figure 5

Two more examples of plane symmetry groups are illustrated by the Geometer's Sketchpad Sketches sketches equilateral kaliedoscope [GSP Help] and 30-60-90 rotational symmetry [GSP Help].

Figure 6
Some of the patterns above, called rosettes are made up of finitely many (4, 2, 8, 4, and 1) images. One pattern consists of a narrow band of images -- this is called a frieze pattern. Some patterns, called wallpaper patterns because they make good wallpaper, could be extended to cover the entire plane. (Some teachers use old wallpaper sample books when teaching this subject!)

We have already used KaleidoTile to create wallpaper patterns by repeating a triangular motif to cover the plane. We even made similar patterns on the sphere and the hyperbolic plane! Now we will learn to use a piece of software called Kali which is specifically designed to generate wallpaper patterns.

Our ultimate goal is to learn a language and a classification system for the seventeen different types of wallpaper pattern. Our first step is to study the features of these patterns and set up a system of classification of the patterns.

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