Vladimir Bulatov

Corvallis, OR

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.

Limit Sets
Limit Sets
5 x 5 x 5 cm each
3D printed steel and bronze composite

The inversive reflection group is symmetry group formed by reflections in planes and inversions in spheres.

The limit set of a such group is set of accumulation of actions of the group on arbitrary initial point. If spheres or planes have kaleidoscopic intersection angles ($\pi/n$) the group is discrete and its limit set can be empty, very simple or very complex.

Simplest example - familiar group formed by reflections in 6 sides of a room with mirror walls floor and ceiling. The limit set of this group has one point - infinity.

Another example. Reflection group formed by 3 intersecting spheres that have 3 intersection angles which form hyperbolic triangle (sum of angles is less than $\pi$). This group is isomorphic to two dimensional hyperbolic triangle groups and its limit set is a circle.

Here are few less simple examples of limit sets of group formed by reflections few spheres and planes. The limit set in these cases has complex self similar fractal structure.