poincaréphotographic tiling of the poincaré disk
| 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. These images are remarkable for both their aesthetic
beauty as well as the mathematical skill Escher acquired to create
The mathematics of such tilings has been extensively studied. While several techniques have been developed 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, which I utilize in this project. However, they also only considered graphic tilings. For photographic tilings, their approach can generate visual artifacts due to rounding errors and aliasing problems.
This project is an open source implementation of photographic tiling of the Poincaré Disk using reverse pixel lookup and techniques to address rounding and aliasing.
For me, this project was all about the challenge of figuring out how to do this, and to do it efficiently. It was a great journey. I learned a lot and had a lot of fun. I hope others might be able to benefit from that work and have some fun with the outcome of that effort.