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Other kinds of patterns

If we allow more exotic transformations than the familiar isometries of the plane, we get more exotic patterns than the wallpaper-like ones we've seen so far. What other sorts of regular pattern can arise?

In what sense are the following "regular patterns"? Be generous-- the concept is really rather loose! In what ways do they fail to be "regular"?

Diamond shaped dynamical systems fractal

(Fractal by Chia-Hsing Nien)

(Fractal Clouds)


(Henon Map)

(Antoine's Necklace)

What sorts of patterns can you find around you? Examples can be found in snowflakes, trees, the eddies on a river, the pores of a golf ball, or the venetian blinds on the windows! Which of these patterns can be described using isometries? Which can't?

Patterns like the ones in the pictures above are more difficult to describe than the wallpaper patterns we've been discussing. That doesn't mean we shouldn't study them. Often things we encounter in the real world don't conform to the rules we know, and we have to continuously modify old systems to include new ideas.

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Author: Heidi Burgiel
Comments to: webmaster@geom.umn.edu
Created: Dec 7 1995 --- Last modified: Nov 07 1996
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