Compound of five truncated tetrahedra
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	UC55
Polyhedra	5 truncated tetrahedra Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
Faces	20 triangles, 20 hexagons
Edges	90
Vertices	60
Dual	Compound of five triakis tetrahedra
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...	chiral Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...  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...  (I)
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	chiral Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...  tetrahedral 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...  (T)

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 truncated tetrahedra

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...

, formed by truncating each of the tetrahedra in the compound of 5 tetrahedra

Compound of five tetrahedra

This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.-As a compound:It can be constructed by arranging five tetrahedra in rotational icosahedral symmetry , as colored in the upper right model...

. A far-enough truncation creates the Compound of five octahedra

Compound of five octahedra

This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.- As a stellation :It is the second stellation of the icosahedron, and given as Wenninger model index 23....

. Its convex hull is a nonuniform Snub dodecahedron

Snub dodecahedron

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....

.

Cartesian coordinates

Cartesian coordinates for the vertices of this compound are all the cyclic permutations of

(±1, ±1, ±3)

(±τ⁻¹, ±(−τ⁻²), ±2τ)

(±τ, ±(−2τ⁻¹), ±τ²)

(±τ², ±(−τ⁻²), ±2)

(±(2τ−1), ±1, ±(2τ−1))

with an even number of minuses in the choices for '±', 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).