# 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 really be appreciated by everyone.

Limit Set #170923

20 x 20 x 10 cm

3D print, nylon

2017

M.C.Escher Circle Limit woodcuts are based on symmetry groups generated by inversions in 3 circles. The limit set of those groups is a circle. Limit set of group generated by inversions in spheres may be much more complicated. 4 generators groups may have intricate limit set which is always contained in a sphere. 5 or more generators may produce truly three dimensional structure. The calculations of 3D limit set involve reverse mapping of each point in space into fundamental domain of the group. The inverse Jacobian of the transformation is used as a measure of the distance to the limit set. We present few non trivial limit sets for groups with 5 or more generators.

The modeling was done in ShapeJS and printed by Shapeways.

The modeling 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 2 orthogonal planes and 3 spheres arranged inside of the wedge formed by the planes.

Tetrahedral Limit Set

20 x 20 x 20 cm

3D print, Nylon

2014

Limit set of a group generated by inversions in 8 spheres centered at the vertices of two concentric tetrahedra of different size. The limit set is intricate composition of a simpler spherical limit sets of two subgroups. Each subgroup is generated by inversions in 4 spheres centered on vertices of one tetrahedron. This creates that interplay of strcuctures with tetrahedral symmetry.

Limitr Set #1

20 x 20 x 10 cm

3D print, nylon

2012

Limit set of a group generated by inversions in 8 spheres centered at the vertices of a cube and a chain 4 symmetrically placed intersecting spheres surrounding that cube. The limit set is intricate composition of simper limit sets of two subgroups based on the subset of generators: spherical limit set for the group generated by 8 cube based generators and circular limit set of the group generated by chain of 4 spheres.