Compact hyperbolic coxeter thin cubes

Abstract

Andreev’s Theorem provides a complete characterization of 3-dimensional compact hyperbolic combinatorial polytope having non-obtuse dihedral angles. Cube is one of such polytope. In this article, with the help of Andreev’s Theorem, a special type of compact hyperbolic coxeter cube, called thin cube has been studied. Using graph theory and combinatorics, it has been found that there are exactly 3 such cubes in hyperbolic space upto symmetry.