# Rhombic triacontahedron

Rhombic triacontahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymRhote
Coxeter diagramo5m3o ()
Conway notationjD
Elements
Faces30 golden rhombi
Edges60
Vertices12+20
Vertex figure12 pentagons, 20 triangles
Measures (edge length 1)
Inradius${\displaystyle {\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\approx 1.37638}$
Volume${\displaystyle 4{\sqrt {5+2{\sqrt {5}}}}\approx 12.31073}$
Dihedral angle144°
Central density1
Number of external pieces30
Level of complexity2
Related polytopes
ArmyRhote
RegimentRhote
DualIcosidodecahedron
ConjugateGreat rhombic triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits2
ConvexYes
NatureTame

The rhombic triacontahedron 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.

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 ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$ 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 ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$ times the shorter diagonal, with acute angle ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$ and obtuse angle ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$.

## Vertex coordinates

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

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

Plus all even permutations of:

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

## 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.