# Pentagrammic retroprism

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

Similar to how the pentagonal antiprism can be vertex-inscribed into the regular icosahedron, the pentagrammic retroprism can be vertex-inscribed into a great icosahedron.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{\sqrt5-1}2,\,±\frac12,\,0\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt5-1}2,\,±\frac12\right),}$
• ${\displaystyle \left(\frac12,\,0,\,\frac{\sqrt5-1}2\right),}$
• ${\displaystyle \left(-\frac12,\,0,\,-\frac{\sqrt5-1}2\right).}$

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:

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

## 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.