Modern Decomposition Problems

Eventhough Hilbert's challenge problem about decomposing tetrahedra didn't take very long to solve, there are still many unsolved problems about tetrahedral decompositions of other Platonic Solids besides the tetrahedron.

Earlier in this article we posed the problem of finding the largest and smallest number of tetrahedra that can be used in a decomposition. The series of pictures below show the most economical way of decomposing a regular dodecahedron into tetrahedra.

Here is an animation showing the minimal decomposition for the dodecahedron.

A similar problem is to find the decomposition with the largest number of tetrahedra. The pictures below show the decomposition with the maximal number of tetrahedra for the icosahedron.

For mathematicians, who love generality and abstraction, the most interesting problem of all is coming up with a general method of find the largest or the smallest tetrahedralizations for a polyhedron. This is complicated question -- researchers have tried for many years just to understand the minimal decompositions of higher dimensional analogues of a cube, which actually has practical applications.

Recently, however, researchers at the Geometry Center of the University of Minnesota developed general techniques to compute such minimal decompositions. The fundamental insight is that the collection of all tetrahedralizations can in fact be thought of as a large polyhedron, called Universal Polytope, that lives in high dimensions. Studying this strange, abstract, other-worldy object leads to a way of finding optimal tetrahedralizations.