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Some Student Research Projects:

Carry out analysis like that present here to construct all possible frieze functions, that is, functions invariant (or negating) with respect to the 7 frieze groups.

Minimal pair examples:
In our section, "Telling the types apart" [Back], we gave a few examples of patterns with similar names. Figure out why the notation is "rational" by doing this for other pairs whose names are very similar.

Wallpapers in art and film:
Anthropologist Dorothy Washburn, has done wonderful work relating the mathematics of wallpaper to cultural studies. Much more can be done, wherever there is a source of repeat patterns. For instance, we are curious to have students classify all the wallpapers in Stanley Kubrick's The Shining and the recently re-released Umbrellas of Cherbourg.

Spectral analysis of wallpaper:
Now that you know every wallpaper function can be constructed as a superposition of wallpaper waves [Back], you can use something like MatLab to analyze a given wallpaper sample and reconstruct it. You could use a black and white example, or use color filters to analyze the red, blue and green parts of the image. You could even develop a concept of almost wallpaper functions, where you can filter out small imperfections that interfere with a pattern's being exactly symmetrical.

Inspired by M. C. Escher's Metamorphosis, we have made some images showing one pattern morphing into another. We have only scratched the surface in our images of morphing using the groups p4g [Image] and p4m [Image], shown in small versions below, and cm [Image]. These have made popular T-shirts. Try making your own.
[a p4g panel]
[a p4m panel]

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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 30, 2003 9:26:59 AM
Comments to: CVM@maa.org