# Pentagonal-pentagrammic duoprism

Pentagonal-pentagrammic duoprism
Rank4
TypeUniform
Notation
Bowers style acronymStarpedip
Coxeter diagramx5o x5/2o ()
Elements
Cells5 pentagonal prisms, 5 pentagrammic prisms
Faces25 squares, 5 pentagons, 5 pentagrams
Edges25+25
Vertices25
Vertex figureDigonal disphenoid, edge lengths (5+1)/2 (base 1), (5–1)/2 (base 2), 2 (sides)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {5{\sqrt {5}}}{16}}\approx 0.69877}$
Dichoral anglesStip–5/2–stip: 108°
Pip–4–stip: 90°
Pip–5–pip: 36°
Central density2
Number of external pieces15
Level of complexity12
Related polytopes
ArmySemi-uniform pedip
RegimentStarpedip
DualPentagonal-pentagrammic duotegum
ConjugatePentagonal-pentagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2×H2, order 100
ConvexNo
NatureTame

The pentagonal-pentagrammic duoprism, also known as starpedip or the 5-5/2 duoprism, is a uniform duoprism that consists of 5 pentagonal prisms and 5 pentagrammic prisms, with 2 of each at each vertex.

This is the only duoprism aside from the tesseract to have a circumradius equal to its edge length.

The pentagonal-pentagrammic duoprism can be edge-inscribed into the small stellated hecatonicosachoron. The small stellated hecatonicosachoron's regiment contains a uniform compound of 24 pentagonal-pentagrammic duoprisms.

## Vertex coordinates

The coordinates of a pentagonal-pentagrammic duoprism, centered at the origin and with unit edge length, are given by:

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