RHOMBIC TRIACONTAHEDRON

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

Contents

Uses of rhombic triacontahedra

See also

References

External links

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

See also

★ Truncated rhombic triacontahedron

References

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

External links

★

★ Virtual Reality Polyhedra – The Encyclopedia of Polyhedra

★ Stellations of Rhombic Triacontahedron