In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the rhombidodecadodecahedron is a nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

, indexed as U₃₈. It is given a Schläfli symbol t_0,2{5/2,5}, and by the Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

 this polyhedron can also be named a cantellated

Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

 great dodecahedron.

Related polyhedra

It shares its vertex arrangement with the uniform compounds of 10

Compound of ten triangular prisms

This uniform polyhedron compound is a chiral symmetric arrangement of 10 triangular prisms, aligned with the axes of three-fold rotational symmetry of an icosahedron.- Relatde polyhedra :...

 or 20 triangular prisms

Compound of twenty triangular prisms

This uniform polyhedron compound is a symmetric arrangement of 20 triangular prisms, aligned in pairs with the axes of three-fold rotational symmetry of an icosahedron.It results from composing the two enantiomorphs of the compound of 10 triangular prisms...

. It additionally shares its edges with the icosidodecadodecahedron

Icosidodecadodecahedron

In geometry, the icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U44. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms...

 (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron

Rhombicosahedron

In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :...

 (having the square faces in common).

convex hull Convex hull In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....	Rhombidodecadodecahedron	Icosidodecadodecahedron Icosidodecadodecahedron In geometry, the icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U44. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms...
Rhombicosahedron Rhombicosahedron In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :...	Compound of ten triangular prisms Compound of ten triangular prisms This uniform polyhedron compound is a chiral symmetric arrangement of 10 triangular prisms, aligned with the axes of three-fold rotational symmetry of an icosahedron.- Relatde polyhedra :...	Compound of twenty triangular prisms Compound of twenty triangular prisms This uniform polyhedron compound is a symmetric arrangement of 20 triangular prisms, aligned in pairs with the axes of three-fold rotational symmetry of an icosahedron.It results from composing the two enantiomorphs of the compound of 10 triangular prisms...

Cartesian coordinates

Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of

(±1/τ², 0, ±τ²))

(±1, ±1, ±(2τ−1))

(±2, ±1/τ, ±τ)

where τ = (1+√5)/2 is the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

(sometimes written φ).

External links

• Uniform polyhedra and duals