Compound of five cuboctahedra
Encyclopedia
| Compound of five cuboctahedra | |
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| Type | Uniform compound Uniform polyhedron compound A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling... |
| Index | UC59 |
| Polyhedra | 5 cuboctahedra Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... |
| Faces | 40 triangles, 30 squares Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... |
| Edges | 120 |
| Vertices | 60 |
| Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation... |
icosahedral Icosahedral symmetry A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation... (Ih) |
| Subgroup Subgroup In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H... restricting to one constituent |
pyritohedral Tetrahedral symmetry 150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group... (Th) |
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 ....
, this uniform polyhedron compound
Uniform polyhedron compound
A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...
is a composition of 5 cuboctahedra
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
. It has icosahedral
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
symmetry Ih.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of- (±2, 0, ±2)
- (±τ, ±τ−1, ±(2τ−1))
- (±1, ±τ−2, ±τ2)
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).