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" , 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, 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.
Panels:
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 and p4m, shown in small versions below, and
cm. These have made popular T-shirts. Try making your
own.