Compound of six pentagrammic antiprisms
Type	Uniform compound
Index	UC44
Polyhedra	6 pentagrammic antiprisms
Faces	60 triangles, 12 pentagrams
Edges	120
Vertices	60
Symmetry group	chiral icosahedral (I)
Subgroup restricting to one constituent	5-fold dihedral (D5)

This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic antiprisms, aligned with the axes of five-fold rotational symmetry of a dodecahedron. A Uniform polyhedron compound is a Polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement In Geometry, the pentagrammic antiprism is one in an infinite set of nonconvex Antiprisms formed by triangle sides and two regular Star polygon caps in A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is A regular Icosahedron has 60 rotational (or orientation-preserving symmetries and a total of 120 symmetries including transformations that combine a reflection and a rotation In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of This article deals with three infinite series of Point groups in three dimensions which have a Symmetry group which as abstract group is a Dihedral group Dih A Uniform polyhedron compound is a Polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement In Geometry, the pentagrammic antiprism is one in an infinite set of nonconvex Antiprisms formed by triangle sides and two regular Star polygon caps in A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal

Cartesian coordinates

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

(±(τ+√τ⁻¹), ±τ⁻¹, ±(−1+√τ))

(±√τ⁻¹, ±2, ±√τ)

(±(−τ+√τ⁻¹), ±τ⁻¹, ±(1+√τ))

(±(−1+√τ⁻¹), ±(−τ), ±(τ⁻¹+√τ))

(±(1+√τ⁻¹), ±(−τ), ±(−τ⁻¹+√τ))

with an even number of minuses in the '±' choices, where τ = (1+√5)/2 is the golden ratio (sometimes written φ). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the

References

• John Skilling, Uniform Compounds of Uniform Polyhedra, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 79, pp. 447-457, 1976.

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