# Rhombic triacontahedron

The rhombic triacontahedron, or rhote, is one of the 13 Catalan solids. It has 30 rhombi as faces, with 12 order-5 and 20 order-3 vertices. It is the dual of the uniform icosidodecahedron.

Rhombic triacontahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymRhote
Coxeter diagram
Elements
Faces30 golden rhombi
Edges60
Vertices12+20
Vertex figure12 pentagons, 20 triangles
Measures (edge length 1)
Inradius${\displaystyle \sqrt{\frac{5+2\sqrt5}{5}} ≈ 1.37638}$
Volume${\displaystyle 4\sqrt{5+2\sqrt5} ≈ 12.31073}$
Dihedral angle144°
Central density1
Related polytopes
ArmyRhote
RegimentRhote
DualIcosidodecahedron
ConjugateGreat rhombic triacontahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

It can also be obtained as the convex hull of a dodecahedron and an icosahedron scaled so that their edges are orthogonal. For this to happen, the icosahedron's edge length must be ${\displaystyle \frac{1+\sqrt5}{2} ≈ 1.61803}$ times that of the dodecahedron's edge length. Each edge of the dodecahedron or icosahedron corresponds to one of the diagonals of the faces.

Each face of this polyhedron is a rhombus with longer diagonal ${\displaystyle \frac{1+\sqrt5}{2} ≈ 1.61803}$ times the shorter diagonal, with acute angle ${\displaystyle \arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°}$ and obtuse angle ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°}$.

## Vertex coordinates

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

• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}$

Plus all even permutations of:

• ${\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5-\sqrt5}{10}},\,0\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5+\sqrt5}{10}},\,0\right).}$

## Dissection

The rhombic triacontahedron can be dissected into 10 acute golden rhombohedra and 10 obtuse golden rhombohedra.[1][2][3]

## Related polyhedra

The rhombic triacontahedron has many stellations, including 227 fully supported stellations.[4] Some notable stellations of the rhombic triacontahedron include the medial rhombic triacontahedron, great rhombic triacontahedron, rhombihedron, and rhombic hexecontahedron.