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.
|Bowers style acronym||Rhote|
|Faces||30 golden rhombi|
|Vertex figure||12 pentagons, 20 triangles|
|Measures (edge length 1)|
|Conjugate||Great rhombic triacontahedron|
|Abstract & topological properties|
|Symmetry||H3, order 120|
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 .
A rhombic triacontahedron of edge length 1 has vertex coordinates given by all permutations of:
Plus all even permutations of:
The rhombic triacontahedron can be dissected into 10 acute golden rhombohedra and 10 obtuse golden rhombohedra.
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.
- Klitzing, Richard. "rhote".
- Wikipedia Contributors. "Rhombic triacontahedron".
- McCooey, David. "Rhombic Triacontahedron"
- ↑ Hart, George. "Dissection of the Rhombic Triacontahedron". Archived from the original on 2012-10-12.
- ↑ 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.