Vladimir Bulatov

Limit set of a group generated by inversions in eight spheres centered at the vertices of a cube as well as a chain of four symmetrically placed intersecting spheres surrounding that cube. The limit set is intricate composition of simper limit sets of a pair of subgroups. These are given by two subsets of generators: the eight cube based generators and then the chain of four spheres.

Limit set of a group generated by inversions in eight spheres centered at the vertices of two concentric tetrahedra of different size. The limit set is a intricate composition of a simpler spherical limit sets of a pair of subgroups. Each subgroup is generated by inversions in four spheres centered on the vertices of one tetrahedron. This creates an interplay of structures with tetrahedral symmetry.

Limit set of a group generated by inversions in two orthogonal planes and three spheres arranged inside of the wedge formed by the planes.

Escher's "Circle Limit" woodcuts are based on symmetry groups generated by inversions in three circles. The limit sets of these groups are always contained in a round circle. Group generated by inversions in spheres can be much more complicated. Such a group, with only four generators, has a perhaps highly intricate limit set contained in a round sphere. However, as shown here, five or more generators can produce limit sets with truly three-dimensional structure. To calculate these limit sets we used a "reverse mapping", taking each point in space into a fundamental domain for the group. We use the inverse Jacobian to measure the distance to the limit set. All modelling was done in ShapeJS and printed by Shapeways.