# Gieseking manifold

The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of Epstein-Penner. Moreover, the angle made by the faces is $\pi/3$. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.