Rhombic triacontahedron

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	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.

One face of the rhombic triacontahedron. The diagonals' lengths are in the golden ratio.

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.

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.

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.

Related polyhedra

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

The rhombic triacontahedron forms the (hull of) the projection of a 6-cube to 3 dimensions.

The 3D basis vectors [u,v,w] are: u = (1, φ, 0, -1, φ, 0) v = (φ, 0, 1, φ, 0, -1) w = (0, 1, φ, 0, -1, φ)	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).

See also

• Truncated rhombic triacontahedron
• Rhombille tiling
• Golden rhombus

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)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208  (The thirteen semiregular convex polyhedra and their duals, Page 22, Rhombic triacontahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, Rhombic triacontahedron )

External links

• Eric W. Weisstein, Rhombic triacontahedron (Catalan solid) at MathWorld.
• Rhombic Triacontrahedron – Interactive Polyhedron Model
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
• Stellations of Rhombic Triacontahedron
• IQ-light–Danish designer Holger Strøm's lamp
• Make your own
• a wooden construction of a rhombic triacontahedron box – by woodworker Jane Kostick
• 120 Rhombic Triacontahedra, 30+12 Rhombic Triacontahedra, and 12 Rhombic Triacontahedra by Sándor Kabai, The Wolfram Demonstrations Project.