This page gives a quick overview of the various kinds of computed images I have produced over the years. The file sizes are small for easy viewing; the resolution is not very high. Short descriptions accompany each picture, with links to sites with more information.
The story about images like this one appeared in the September 2001 issue of Math Horizons. You are seeing noneuclidean wallpaper, or more technically, a view of a modular function on the Poincare Upper Halfplane, that is, a function invariant under the action of SL(2,Z). Here are two more examples:
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In the left-hand image, we see the interval [0,1/2] on the real axis. There is one of those fans based at each rational number and, according to noneuclidean geometry, they are all exactly the same size.
The images above use the method of domain-coloring for picturing complex functions of one variable.
Here is a function with two zeroes and two poles, depicted using the same method: the zeroes are white, the poles are black; the limit of this function as z goes to infinity is 1/2, which is colored a light red.
OK, so it's not a computed image, it's a photograph of a woven rope frieze by Nils Kristian Rossing, from our joint paper in the February 1999 issue of Mathematics Magazine. The equation for this shape came first, and the rope was woven afterward.
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A lot of these ideas started with my desire to make Euclidean wallpaper using smooth functions instead of tiles. On the left above is a function with pmg symmetry; on the right is a function that gradually morphs from one p4m pattern to another.
Once I learned how to make images like these, it was natural also to investigate functions with antisymmetries. For the full story, see my online article in Communications in Visual Mathematics. In Shubnikov's rational notation, these are p4'gm' and p4'm'm .
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My newest project (summer 2001) might be called Forbidden Symmetries: Relaxing the Crystallographic Restriction. It is known that in 2-dimensional wallpaper patterns, you can have centers of rotational symmetry of orders 2, 3, 4, and 6, but never 5. But in the image below, don't you see 5-pointed stars?
The trick is that this is not wallpaper: it does not have any translational symmetries. However, it does have approximate symmetries; if you translate the given image 110 units to the right, you will get an image that your eye cannot distinguish from this one. Stay tuned for further explanations.
Finally, here is one image not computed by me. Joe Kain, a student in a course I taught in 1997 called Computer Visualization in Geometry, used fractional linear transformations to give a hyperbolic view of Leonardo's Tondo.
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