# Rhombic dodecahedron

Rhombic dodecahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramo4m3o
Elements
Faces12 rhombi
Edges24
Vertices6+8
Vertex figure6 squares, 8 triangles
Measures (edge length 1)
Inradius${\displaystyle \frac{\sqrt6}{3} ≈ 0.81650}$
Volume${\displaystyle \frac{16\sqrt3}{9} ≈ 3.07920}$
Dihedral angle120°
Central density1
Related polytopes
DualCuboctahedron
ConjugateNone
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

The rhombic dodecahedron, or rad, is one of the 13 Catalan solids. It has 12 rhombi as faces, with 6 order-4 and 8 order-3 vertices. It is the dual of the uniform cuboctahedron.

It can also be obtained as the convex hull of a cube and an octahedron scaled so that their edges are orthogonal. For this to happen, the octahedron's edge length must be ${\displaystyle \sqrt2 ≈ 1.41421}$ times that of the cube's edge length. Each edge of the cube or octahedron corresponds to one of the diagonals of the faces.

It can also be constructed by augmenting a cube with square pyramids of height 0.5.

Each face of this polyhedron is a rhombus with longer diagonal ${\displaystyle \sqrt2 ≈ 1.41421}$ times the shorter diagonal, with acute angle ${\displaystyle \arccos\left(\frac13\right) ≈ 70.52878°}$ and obtuse angle ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$.

The rhombic dodecahedron is the only Catalan solid that can tile 3D space by itself, forming the rhombic dodecahedral honeycomb.

## Vertex coordinates

A rhombic dodecahedron of edge length 1 has vertex coordinates given by all permutations of

• ${\displaystyle \left(±\frac{\sqrt3}{3},\,±\frac{\sqrt3}{3},\,±\frac{\sqrt3}{3}\right),}$
• ${\displaystyle \left(±\frac{2\sqrt3}{3},\,0,\,0\right).}$

## Variations

The rhombic dodecahedron can be varied to remain isohedral under tetrahedral symmetry. In the process the rhombic faces turn into kites, and the resulting polyhedron can be called a deltoidal dodecahedron.

A variant can be made with golden rhombi for faces. This variant can be dissected into 4 acute golden rhombohedra and 4 obtuse golden rhombohedra.[1]