Rhombic triacontahedron

Rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Rhote
Coxeter diagram	o5m3o ()
Conway notation	jD
Elements
Faces	30 golden rhombi
Edges	60
Vertices	12+20
Vertex figure	12 pentagons, 20 triangles
Measures (edge length 1)
Inradius	${\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\approx 1.37638$
Volume	$4{\sqrt {5+2{\sqrt {5}}}}\approx 12.31073$
Dihedral angle	144°
Central density	1
Number of external pieces	30
Level of complexity	2
Related polytopes
Army	Rhote
Regiment	Rhote
Dual	Icosidodecahedron
Conjugate	Great rhombic triacontahedron
Abstract & topological properties
Flag count	240
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Flag orbits	2
Convex	Yes
Nature	Tame

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

Vertex coordinates

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

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

Plus all even permutations of:

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

Dissection

The rhombic triacontahedron can be dissected into 10 acute golden rhombohedra and 10 obtuse golden rhombohedra.

Related polyhedra

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

• 

  Medial rhombic triacontahedron

• 

  Great rhombic triacontahedron

• 

  Rhombihedron

• 

  Final stellation of the rhombic triacontahedron

• 

  Rhombic hexecontahedron

• 

  Grünbaum's new rhombic hexecontahedron

External links

• Klitzing, Richard. "rhote".
• Wikipedia contributors. "Rhombic triacontahedron".
• McCooey, David. "Rhombic Triacontahedron"

• Quickfur. "The Rhombic Triacontahedron".