Arnold's Cat Map

Arnold's Cat Map is a transformation that can be applied to an image. The pixels of the image appear to be randomly rearranged, but when the transformation is repeated enough times, the original image will reappear!

The transformation is as follows:

1. Shear the image in the both the x- and y-directions by a factor of 1.
2. Wrap the image back onto itself i.e. treat it like a torus (doughnut shape). This is a way of treating a 2D grid like it has no edges.

This transformation demonstrates some of the principles of chaos i.e. underlying order to an apparently random evolution of a system. Arnold's Cat Map is named after the Russian mathematician Vladimir Arnold, who discovered it using an image of a cat.

Links

• arnold's cat map by Sean Whalen.
• Arnold's cat map on Wikipedia.
• Arnold's cat map on Wolfram Mathworld.

Cat image courtesy of miss pupik.

Thanks to Denis Lukianov for his comments and inspiring the buttons for stepping through iterations one at a time.

© Jason Davies 2012.