# Pentagrammic antiprism

Pentagrammic antiprism Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymStap
Coxeter diagrams2s10/2o (       )
Elements
Faces10 triangles, 2 pentagrams
Edges10+10
Vertices10
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius$\sqrt{\frac{15+\sqrt5}{40}} ≈ 0.65643$ Volume$\frac{\sqrt{5\sqrt5}}6 ≈ 0.55728$ Dihedral angles5/2–3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}3}\right) ≈ 114.80110°$ 3–3: $\arccos\left(\frac{2-\sqrt5}3\right) ≈ 94.51323°$ Height$\sqrt{\frac{\sqrt5-1}2} ≈ 0.78615$ Central density2
Number of external pieces32
Level of complexity11
Related polytopes
ArmySemi-uniform Pip
RegimentStap
DualPentagrammic antitegum
ConjugatePentagonal retroprism
Convex corePentagonal bifrustum
Abstract & topological properties
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexNo
NatureTame

The pentagrammic antiprism, or stap, 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 an antiprism based on a pentagram. It is one of two pentagrammic antiprisms, the other one being the pentagrammic retroprism. In this case, the pentagrams are aligned with one another.

## Vertex coordinates

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

• $\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,±\sqrt{\frac{\sqrt5-1}8}\right),$ • $\left(±\frac{\sqrt5-1}4,\,\sqrt{\frac{5+\sqrt5}{40}},\,±\sqrt{\frac{\sqrt5-1}8}\right),$ • $\left(0,\,-\sqrt{\frac{5-\sqrt5}{10}},\,±\sqrt{\frac{\sqrt5-1}8}\right).$ ## Related polyhedra

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

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

## In vertex figures

Pentagrammic antiprisms appear as vertex figures of four uniform polychora: the small prismatohecatonicosachoron, pentagrammal antiprismatoverted hexacosihecatonicosachoron, small pentagrammal antiprismatoverted dishecatonicosachoron, and great pentagrammal antiprismatoverted dishecatonicosachoron.