# Pentagrammic retroprism

Pentagrammic retroprism Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymStarp
Coxeter diagrams2s10/3o (       )
Elements
Faces10 triangles, 2 pentagrams
Edges10+10
Vertices10
Vertex figureCrossed isosceles trapezoid, edge lengths 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius$\sqrt{\frac{5-\sqrt5}8} ≈ 0.58779$ Volume$\frac{5-2\sqrt5}{6} ≈ 0.087977$ Dihedral angles3–3: $\arccos\left(\frac{\sqrt5}3\right) ≈ 41.81031°$ 5/2–3: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°$ Height$\sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573$ Central density3
Number of external pieces72
Level of complexity24
Related polytopes
ArmyPap
RegimentStarp
DualPentagrammic concave antitegum
ConjugatePentagonal antiprism
Convex corePentagonal antitegum
Abstract & topological properties
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryI2(10)×A1/2, order 20
ConvexNo
NatureTame

The pentagrammic retroprism, or starp, also called the pentagrammic crossed antiprism, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is a crossed antiprism based on a pentagram, seen as a 5/3-gon rather than 5/2. This makes it the simplest uniform crossed antiprism.

Similar to how the pentagonal antiprism can be edge-inscribed into the regular icosahedron, the pentagrammic retroprism can be edge-inscribed into a great icosahedron. It can be constructed by diminishing two opposite vertices of the great icosahedron.

## Vertex coordinates

A pentagrammic retroprism of edge length 1 has vertex coordinates given by:

• $\left(±\frac{\sqrt5-1}2,\,±\frac12,\,0\right),$ • $\left(0,\,±\frac{\sqrt5-1}2,\,±\frac12\right),$ • $\left(\frac12,\,0,\,\frac{\sqrt5-1}2\right),$ • $\left(-\frac12,\,0,\,-\frac{\sqrt5-1}2\right).$ These coordinates are obtained by removing two opposite vertices from a great icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points along with their central inversions:

• $\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$ • $\left(±\frac{\sqrt5-1}4,\,\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$ • $\left(0,\,-\sqrt{\frac{5-\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{40}}\right).$ ## Related polyhedra

Two non-prismatic uniform polyhedron compounds are composed of pentagrammic retroprisms:

There are also an infinite amount of prismatic uniform compounds that are the crossed antiprisms of compounds of pentagrams.