SymSim - Symmetric Simulations on Orbifolds

Vladimir Bulatov

Introduction

The motivation of this work is the development of algorithms to create animated seamless patterns with discrete symmetry in various geometries.

It is relatively easy to create a seamless pattern with symmetry generated by pure reflections. The basic kaleidoscope is constructed in such a way. Reflections are continuous functions and the resulting patterns are continuous. Animation of such kaleidoscope keeps the pattern continuous but discontinuity of derivatives along reflection lines generates obvious visible artifacts.

In case of more general symmetries it is rather difficult to make the pattern even continuous. A way to make symmetric patterns with arbitrary symmetry is the rubber stamp approach used by M.C.Escher in his tessellation work. This requires very difficult manual fitting of the tiles. No general way to animate such patterns exists.

Another approach is to construct a function with fine tuned symmetric properties and use the function as a tool for domain coloring. It is non-trivial and hard to control process.

We are offering a general approach to make seamless patterns using dynamic (ODE or PDE) on an orbifold. An orbifold can be cut and flattened. The flattened orbifold will tile the whole space. If the pattern is seamless on the orbifold the tiled pattern will be seamless. So the problem of making a seamless symmetric pattern is reduced to creating a seamless pattern on the orbifold. This problem can be successfully solved using appropriate boundary conditions. Such patterns can be animated in real time in web browsers using javascript and WebGL.

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