Hessian polyhedron
In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in . It has 27 vertices, 72 3{} edges, and 27 3{3}3 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the Hessian configuration or (94123), 9 points lying by threes on twelve lines, with four lines through each point. Its complex reflection group is 3[3]3[3]3 or , order 648, also called a Hessian group. It has 27 copies of , order 24, at each vertex. It has 24 order-3 reflections. Its Coxeter number is 12, with degrees of the fundamental invariants 3, 6, and 12, which can be seen in projective symmetry of the polytopes. The Witting polytope, 3{3}3{3}3{3}3, contains the Hessian polyhedron as cells and vertex figures. It has a real representation as the 221 polytope, , in 6-dimensional space, sharing the same 27 vertices. The 216 edges in 221 can be seen as the 72 3{} edges represented as 3 simple edges. (Wikipedia).
