| Rhombic triacontahedron | |
|---|---|
(Click here for rotating model) |
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| Type | Catalan solid |
| Face type | rhombus |
| Faces | 30 |
| Edges | 60 |
| Vertices | 32 |
| Vertices by type | 20{3}+12{5} |
| Face configuration | V3.5.3.5 |
| Symmetry group | Ih, [5,3], *532 |
| Dihedral angle | 144° |
| Properties | convex, face-transitive edge-transitive, zonohedron |
Icosidodecahedron (dual polyhedron) |
Net |
In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron.
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The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1(1/φ) = tan−1(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.
Being the dual of an Archimedean polyhedron, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic triacontahedron is also somewhat special in being one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic dodecahedron.
The rhombic triacontahedron is also interesting in that it has all the vertices of an icosahedron, a dodecahedron, a hexahedron, and a tetrahedron.
Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. (IQ for "Interlocking Quadrilaterals")
Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron. The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.
Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron.
In some roleplaying games, and for elementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided dice.
This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.
| Polyhedra | Euclidean tiling | Hyperbolic tiling | |||
|---|---|---|---|---|---|
| [3,3] | [4,3] | [5,3] | [6,3] | [7,3] | [8,3] |
Cube |
Rhombic dodecahedron |
Rhombic triacontahedron |
Rhombille |
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The rhombic triacontahedron forms the (hull of) the projection of a 6-cube to 3 dimensions.
The 3D basis vectors [u,v,w] are:
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Shown with inner edges hidden |
| There are 64 vertices and 192 unit length edges forming pentagonal symmetry along specific axis (as well as hexagonal symmetries on other axis). | |
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