# Hessian polyhedron

Hessian polyhedron
Rank3
TypeRegular
SpaceComplex
Notation
Coxeter diagram
Schläfli symbol${\displaystyle _{3}\{3\}_{3}\{3\}_{3}}$
Elements
Faces27 Möbius–Kantor polygons
Edges72 3-edges
Vertices27
Vertex figureMöbius–Kantor polygon
Related polytopes
Real analog221 polytope
DualHessian polyhedron
Van Oss polygonγ 3
2

Abstract & topological properties
Flag count648
Properties
Symmetry3[3]3[3]3, order 648

The Hessian polyhedron is a regular complex polyhedron. It is self-dual.

## Related polytopes

The relationship between the three Platonic solids (left), and the analygous relationship between the three Hessian polyhedra (right)

The three regular complex polyhedra:

share analogous relationships to three Platonic solids:

1. the tetrahedron
2. the cube
3. the octahedron

Those relationships are:

• 1 is self-dual.
• 2 is dual to 3.
• 1 is the halving of 2.
• 3 is the rectification of 1.

If the vertices of the Hessian polyhedron are treated as vertices in ${\displaystyle \mathbb {R} ^{6}}$ rather than ${\displaystyle \mathbb {C} ^{3}}$, they are identical to those of the 221 polytope.