Rhombic triacontahedron

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Rhombic triacontahedron
(Click here for rotating model)
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	I_h
Dihedral angle	144°
Dual	Icosidodecahedron
Properties	convex, face-transitive edge-transitive, zonohedron
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.

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.

1 Uses of rhombic triacontahedra
2 See also
3 References
4 External links

[edit] Uses of rhombic triacontahedra

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.

In some roleplaying games, and for elementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided die.

[edit] See also

Truncated rhombic triacontahedron

[edit] References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)