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 dicover 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.
This is a visualization of a tiling of the hyperbolic space. The
tiling is generated by reflections in the faces of Lambert cube
(Coxeter polyhedron) which becomes the fundamental polyhedron of
the symmetry group of the tiling. Only 4 out of 6 sides are used,
which results in sub-tiling (subgroup) filling only part of the
space. It let us see the internal structure of the tiling.
We use cylinder model of the hyperbolic space - a 3D
generalization of 2D band model. In this model the Poincare ball
is stretched into infinite cylinder. Cylinder's axis becomes one
of hyperbolic geodesics.
The tiling is oriented to make one it's plane to be orthogonal to
the cylinder's axis to have a feet to stand on.
The cylinder's axis is close to the axis of a loxodromic
transformation of the group, which gives the pieces its spiral
twist. The sharp boundary of the piece corresponds to the limit
set of the group. The limit set is fractal
Jordan curve at the infinity of the hyperbolic space.