# Pentagrammic retroprism

Pentagrammic retroprism
Rank3
TypeUniform
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${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle {\frac {5-2{\sqrt {5}}}{6}}\approx 0.087977}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$
5/2–3: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Central density3
Number of external pieces72
Level of complexity24
Related polytopes
ArmyPap, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentStarp
DualPentagrammic concave antitegum
ConjugatePentagonal antiprism
Convex corePentagonal antitegum
Abstract & topological properties
Flag count80
Euler characteristic2
OrientableYes
Genus0
Properties
Symmetry(I2(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:

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {1}{2}},\,0,\,{\frac {{\sqrt {5}}-1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,0,\,-{\frac {{\sqrt {5}}-1}{2}}\right).}$

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:

• ${\displaystyle \pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \pm \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \pm \left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right).}$

## 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.