Calendar 2015 - Sayagata Loop

Sayagata Loop

The classic Sayagata tiling is a Euclidean (flat) plane tesselation based on a square grid. For May's calendar image I thought it would be fun to make a hyperbolic variation, with 5 squares meeting at each vertex instead of 4. Furthermore, to wrap the hyperbolic plane into a repeating ring shape like what Bulatov did (warning, that page is slow loading but worth it in the end).

I implemented it as a GLSL fragment shader in Fragmentarium (download my source code) with a few constants for changing the tiling parameters (the hyperbolic symmetry group, the orientation across the band, number of repetitions, whether to draw a pretty pattern or just triangles). Here are a few variations:

The code has a few comments; the gist of the overall operation is to transform the coordinates from the ring back into the Poincaré disc model of hyperbolic geometry, then repeatedly apply hyperbolic symmetries towards a central fundamental region, which finally gets the Sayagata texture. Note that tweaking some of the values leads to ugly seams with the parts not lining up - still some bugs I guess! (WONTFIX DONTDOTHAT)