Hessian polyhedron

Its 27 vertices can be given coordinates in $\backslash mathbb\{C\}^3$: for (λ, μ = 0,1,2).

:(0,ω^λ,−ω^μ)

: (−ω^μ,0,ω^λ)

: (ω^λ,−ω^μ,0)

where $\backslash omega\; =\; \backslash tfrac\{-1+i\backslash sqrt3\}\{2\}$.

Its symmetry is given by ₃[3]₃[3]₃ or {{CDD|3node|3|3node|3|3node}}, order 648.Coxeter Regular Convex Polytopes, 12.5 The Witting polytope

The configuration matrix for ₃{3}₃{3}₃ is:Coxeter, Complex Regular polytopes, p.132

:

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 {{CDD|node_1|4|3node|3|3node}}, {{CDD|node_h|4|3node|3|3node}} = {{CDD|label-33|nodes_10ru|split2|node|label3}}. This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72 {{CDD|node_1|4|3node}} faces. Its vertices represent the union of the vertices {{CDD|3node_1|3|3node|3|3node}} and its dual {{CDD|3node|3|3node|3|3node_1}}.

Its complex reflection group is ₃[3]₃[4]₂, or {{CDD|3node|3|3node|4|node}}, order 1296. It has 54 copies of {{CDD|3node|3|3node}}, 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.

Coxeter noted that the three complex polytopes {{CDD|3node_1|3|3node|3||3node}}, {{CDD|node_1|4|3node|3||3node}}, {{CDD|3node_1|3|3node|4|node}} resemble the real tetrahedron ({{CDD|node_1|3|node|3|node}}), cube ({{CDD|node_1|4|node|3|node}}), and octahedron ({{CDD|node_1|3|node|4|node}}). The Hessian is analogous to the tetrahedron, like the cube is a double tetrahedron, and the octahedron as a rectified tetrahedron. In both sets the vertices of the first belong to two dual pairs of the second, and the vertices of the third are at the center of the edges of the second.Coxeter, Complex Regular Polytopes, p.127

Its real representation 54 vertices are contained by two 2₂₁ polytopes in symmetric configurations: {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}} and {{CDD|nodes_01r|3ab|nodes|split2|node|3|node}}. Its vertices can also be seen in the dual polytope of 1₂₂.

The elements can be seen in a configuration matrix:

{{-}}

The rectification, {{CDD|3node|3|3node_1|3|3node}} doubles in symmetry as a regular complex polyhedron {{CDD|3node_1|3|3node|4|node}} with 72 vertices, 216 ₃{} edges, 54 ₃{3}₃ faces. Its vertex figure is ₃{4}₂, and van oss polygon ₃{4}₃. It is dual to the double Hessian polyhedron.Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47

It has a real representation as the 1₂₂ polytope, {{CDD|nodes|3ab|nodes|split2|node|3|node_1}}, sharing the 72 vertices. Its 216 3-edges can be drawn as 648 simple edges, which is 72 less than 1₂₂'s 720 edges.

The elements can be seen in two configuration matrices, a regular and quasiregular form.

{{Reflist}}

• Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
• Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,

Category:Polytopes

Category:Complex analysis