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(con)tinuity | by Bill Horne
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Tilings of the Poincaré Disk are of interest to both mathematicians and artists. M.C. Escher, inspired by the work of the mathematician H.S.M. Coxeter, created four wood carvings based on tilings of the Poincaré disk titled Circle Limit I-IV.


The mathematics of such tilings has been extensively studied. While several papers have described how to create graphical tilings of the Poincaré disk using computers, there are a number of reasons why those techniques present difficulties for tiling photographs. In particular, the techniques are primarily intended for vector graphics or low resolution images, and are extremely inefficient for rendering large, high-resolution images. Recently, van Gagern and Richter-Gebert published the first description of an algorithm to efficiently render large tilings using a technique they described as reverse pixel lookup. However, they also only considered graphic tilings and their approach can generate significant visual artifacts due to rounding errors and aliasing problems.


I've created a program that efficiently renders photographic tilings and deals with these rounding and aliasing problems. Check out the large version to see improved visual details.


I've made my code available if anybody would like to try to create some of these images. A package that includes all of the source code, a win32 executable, a technical document describing how these images are generated, and an example to get you started can be found here on SourceForge.


I also created a Flickr group for these images here. If you try out the program, please post your results in this group.

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Taken on August 3, 2009