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. Its 27 vertices can be given coordinates in : for (λ, μ = 0,1,2). (0,ωλ,−ωμ) (−ωμ,0,ωλ) (ωλ,−ωμ,0) where . Its symmetry is given by 3[3]3[3]3 or , order 648. The configuration matrix for 3{3}3{3}3 is: The number of k-face elements (f-vectors) can be read down the diagonal. The number of elements of each k-face are in rows below the diagonal. The number of elements of each k-figure are in rows above the diagonal. These are 8 symmetric orthographic projections, some with overlapping vertices, shown by colors. Here the 72 triangular edges are drawn as 3-separate edges. The Hessian polyhedron can be seen as an alternation of , = . This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72 faces. Its vertices represent the union of the vertices and its dual . Its complex reflection group is 3[3]3[4]2, or , order 1296. It has 54 copies of , order 24, at each vertex. It has 24 order-3 reflections and 9 order-2 reflections. Its coxeter number is 18, with degrees of the fundamental invariants 6, 12, and 18 which can be seen in projective symmetry of the polytopes.