# Pentagrammic antiprism

Pentagrammic antiprism
Rank3
TypeUniform
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${\displaystyle {\sqrt {\frac {15+{\sqrt {5}}}{40}}}\approx 0.65643}$
Volume${\displaystyle {\frac {\sqrt {5{\sqrt {5}}}}{6}}\approx 0.55728}$
Dihedral angles5/2–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{3}}}\right)\approx 114.80110^{\circ }}$
3–3: ${\displaystyle \arccos \left({\frac {2-{\sqrt {5}}}{3}}\right)\approx 94.51323^{\circ }}$
Height${\displaystyle {\sqrt {\frac {{\sqrt {5}}-1}{2}}}\approx 0.78615}$
Central density2
Number of external pieces32
Level of complexity11
Related polytopes
ArmySemi-uniform Pip, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (base), ${\displaystyle {\sqrt {\frac {{\sqrt {5}}-1}{2}}}}$ (sides)
RegimentStap
DualPentagrammic antitegum
ConjugatePentagonal retroprism
Convex corePentagonal bifrustum
Abstract & topological properties
Flag count80
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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right),}$
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1}{8}}}\right).}$

## 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.