# Vladimir Bulatov

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 try to discover and to 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.

Three dimensional tilings of hyperbolic 3-space are difficult to visualize. In the conformal models of hyperbolic space the tiles become very small as they approach the boundary of the hyperbolic space. Here we attempt to visualize the 3d tiling on the two dimension surface by making cross section of the tiling with the surface of a cyclide - surface of equal distance from a line. The cyclide is flattened into a plane using conformal mapping. The tiling is generated by reflections in 6 sides of Lambert cube - box shaped hyperbolic polyhedron which has three edges with acute dihedral angles (pi/12 in this case) and remaining 9 edges having right angles. The key properties of the visualization are its uniformity in scale and periodicity.

M.C.Escher inspired tiling on the horizon (Riemann sphere) of three dimensional hyperbolic space.

The tiling is generated by reflections in 5 sides of hyperbolic cuboid (box shaped polyhedron).

Using 5 out of 6 sides of the cuboid makes the tile to be infinite and extending to the horizon.

The two dimensional intersection of infinite tile with the horizon is filled with pattern and replicated over the whole Riemann sphere by reflections in 5 sides of the cuboid.