Rhombic dodecahedron

Rhombic dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Rad
Coxeter diagram	o4m3o
Elements
Faces	12 rhombi
Edges	24
Vertices	6+8
Vertex figure	6 squares, 8 triangles
Measures (edge length 1)
Inradius	$\frac{\sqrt6}{3} ≈ 0.81650$
Volume	$\frac{16\sqrt3}{9} ≈ 3.07920$
Dihedral angle	120°
Central density	1
Related polytopes
Army	Rad
Regiment	Rad
Dual	Cuboctahedron
Conjugate	None
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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 $\sqrt2 ≈ 1.41421$ 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 $\sqrt2 ≈ 1.41421$ times the shorter diagonal, with acute angle $\arccos\left(\frac13\right) ≈ 70.52878°$ and obtuse angle $\arccos\left(-\frac13\right) ≈ 109.47122°$.

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

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

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.

External links

• Klitzing, Richard. "Rad".

• Wikipedia Contributors. "Rhombic dodecahedron".
• McCooey, David. "Rhombic Dodecahedron"

• Quickfur. "The Rhombic Dodecahedron".