Coxeter named it after Ludwig Otto Hesse for sharing the Hessian configuration
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.
[2] The configuration matrix for 3{3}3{3}3 is:[3] The number of k-face elements (f-vectors) can be read down the diagonal.
These are 8 symmetric orthographic projections, some with overlapping vertices, shown by colors.
This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72 faces.
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 , , resemble the real tetrahedron (), cube (), and octahedron ().
[4] Its real representation 54 vertices are contained by two 221 polytopes in symmetric configurations: and .
The elements can be seen in a configuration matrix: The rectification, doubles in symmetry as a regular complex polyhedron with 72 vertices, 216 3{} edges, 54 3{3}3 faces.
The elements can be seen in two configuration matrices, a regular and quasiregular form.