Vladimir Bulatov

3D graphics Researcher
Shapeways, Inc
Corvallis, Oregon, USA
My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes: my interpretations of distance, transformations, and space. In my opinion, mathematics is a way of thinking, a way of life. My images and sculptures are like photographs of interesting mathematical ideas, which I discover and bring to the real world. I have always been intrigued by the possibility of showing the intrinsic richness of the mathematical world, whose charm and harmony can be appreciated by everyone.
Limit Set #170923
20 x 20 x 10 cm
3D print, white nylon
2017
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.
Limit Set #170922
20 x 20 x 20 cm
3D print, black nylon
2017
Limit set of a group generated by inversions in two orthogonal planes and three spheres arranged inside of the wedge formed by the planes.
Tetrahedral Limit Set
20 x 20 x 20 cm
3D print, red nylon
2014
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 #1
20 x 20 x 10 cm
3D print, orange nylon
2012
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.