Rhombic triacontahedron
Rhombic triacontahedron  

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  
Volume  
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  H_{3}, 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 order5 and 20 order3 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 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 times the shorter diagonal, with acute angle and obtuse angle .
Vertex coordinates[edit  edit source]
A rhombic triacontahedron of edge length 1 has vertex coordinates given by all permutations of:
 ,
Plus all even permutations of:
 ,
 .
Dissection[edit  edit source]
The rhombic triacontahedron can be dissected into 10 acute golden rhombohedra and 10 obtuse golden rhombohedra.^{[1]}^{[2]}^{[3]}
Related polyhedra[edit  edit source]
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.

Medial rhombic triacontahedron

Great rhombic triacontahedron

Rhombihedron

Rhombic hexecontahedron
External links[edit  edit source]
 Klitzing, Richard. "rhote".
 Wikipedia contributors. "Rhombic triacontahedron".
 McCooey, David. "Rhombic Triacontahedron"
 Quickfur. "The Rhombic Triacontahedron".
References[edit  edit source]
 ↑ Hart, George. "Dissection of the Rhombic Triacontahedron". Archived from the original on 20121012.
 ↑ Kowalewski, Gehard (1938). Der Keplersche Korper und andere Bauspiele (in German).
 ↑ Bardos, Laszlo. "Golden Rhombohedra".
 ↑ Messer, Peter (1995). "Stellations of the Rhombic Triacontahedron and Beyond". Structural Topology (21): 25–46.