# The Császár Polyhedron Subdivided into Tetrahedra

The Császár Polyhedron Subdivided into Tetrahedra

The Császár polyhedron has four visibly different versions. This Demonstration illustrates that each of the four versions can be subdivided into seven tetrahedra with planes fitted to the vertices in only one way.

In each of the four versions, the triangles shared by adjacent tetrahedra form a Möbius band. Each Möbius band contains all seven vertices of the Császár polyhedron, but only 14 of its edges, and within this subgraph of seven vertices and 14 edges, there are exactly two Hamiltonian cycles: one cycle traces the border of the Moebius band, while the other forms all the interior edges of the Möbius band (i.e., those shared by the adjacent pairs of triangles). The third Hamiltonian cycle is not a subgraph of the Möbius band subgraph.