Vladimir Bulatov
Artists
Vladimir Bulatov
Mathematics and Arts Explorer
Corvallis, Oregon
Statement
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.
Artworks
Limit Sets
5 x 5 x 5 cm each
3D printed steel and bronze composite
2012
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.