Global InformationHessian polyhedron information
| Hessian polyhedron | |
|---|---|
 Orthographic projection (triangular 3-edges outlined as black edges) | |
| Schläfli symbol | 3{3}3{3}3 |
| Coxeter diagram |   
|
| Faces | 27 3{3}3 
|
| Edges | 72 3{} 
|
| Vertices | 27 |
| Petrie polygon | Dodecagon |
| van Oss polygon | 12 3{4}2 
|
| Shephard group | L3 = 3[3]3[3]3, order 648 |
| Dual polyhedron | Self-dual |
| Properties | Regular |
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.[1]
![{\displaystyle \left[{\begin{smallmatrix}9&4\\3&12\end{smallmatrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02ab5ff205bfbd1db47ba3a08a739a44b6236f87)
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.
- ^ Coxeter, Complex Regular polytopes, p.123