Poincaré Hyperbolic Tiling : The area inside this circle represents a hyperbolic plane filled with “ideal triangles”. Notice that all the angles inside these triangles go to zero at the edge of the circle. This image was generated using a series of reflections called anti-homographies. I recently learned about homographies while participating at the Experimental Geometry Lab at the University of Maryland. The right animation shows how a single homography can transform the upper half plane into the Poincaré disk. See also my POV-Ray code, Mathematica code, homography test, and circle inversion.
Hyperbolic Links
- hyperbolic animations – by Jos Leys
- Hyperbolic Java applet – by Don Hatch
- HypEngine – 3D real-time hyperbolic maze by Bernie Freidin
- Hyperbolic surfaces in nature – leaf edges and torn plastic sheets
- Mathematica package – by Matthias Weber
- Mathematica code – for animated Poincaré grid, by Matthew Cook
- “Circle Limit III” and “Circle Limit IV” – M.C. Esher’s famous hyperbolic tessellations
- Reducing Lizards – upper half plane tessellation by M.C. Escher
- “Escher Fish” – Mathematica version by Silvio Levy
- Kangaroo Tiling – hyperbolic tessellation by Guy Cousineau, et. all
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