Vladimir Bulatov

M.C.Escher Circle Limit woodcuts are based on a symmetry groups generated by inversions in 3 circles. The limit set of such a group is simple circle. Limit set of a group generated by inversions in spheres may be much more complicated. 4 generators groups may have rather intricate limit set which however always lies on a sphere. Only groups with 5 or more generators may have truly three dimensional structure of the limit set. To find the limit set we map each point into corresponding point in the fundamental domain. The inverse of the stretching factor of that transformation measures the distance of the point to the limit set. The isosurface of the distance data is converted into triangle mesh and 3D printed in nylon.

Two limit sets presented here are based on a similar arrangement of generators. Two identical spheres are placed inside of a wedge (with 90 and 60 degree angle) formed by two planes, each sphere intersect one plane and other sphere, third sphere is centered on planes intersection line and intersects first two spheres. The structure of the limit set reflects the structure of the group. The fractal curve on the surface of the sphere is limit set of subgroup generated by planes and first two spheres. The circles are images of limit sets of triangle subgroup generated by 3 spheres. The whole structure can be thought of as a union of those circles twisted in space. It has no rigid elements and is rather flexible.