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In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in C 3 {\displaystyle \mathbb {C} ^{3}}. 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 {\displaystyle \left} or , 9 points lying by threes on twelve lines, with four lines through each point.
Its complex reflection group is 333 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.