# Rectified Hessian polyhedron

Rectified Hessian polyhedron
Rank3
TypeRegular
SpaceComplex
Notation
Coxeter diagram
Schläfli symbol${\displaystyle _{3}\{3\}_{3}\{4\}_{2}}$
Elements
Faces54 Möbius–Kantor polygons
Edges216 3-edges
Vertices72
Vertex figure${\displaystyle _{3}\{4\}_{2}}$
Related polytopes
DualDouble Hessian polyhedron
Abstract & topological properties
Flag count1296
Properties
Symmetry3[3]3[4]2, order 1296

The rectified Hessian polyhedron is a regular complex polyhedron.

## Coxeter diagrams

A rectified Hessian polyhedron can be represented by the following Coxeter diagrams:

• (full symmetry)
• (L3 symmetry)

## 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:

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