# Square-pentagonal duoprism

Square-pentagonal duoprism Rank4
TypeUniform
Notation
Bowers style acronymSquipdip
Coxeter diagramx4o x5o (       )
Elements
Cells5 cubes, 4 pentagonal prisms
Faces5+20 squares, 4 pentagons
Edges20+20
Vertices20
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1) and 2 (base 2 and sides)
Measures (edge length 1)
Circumradius${\sqrt {\frac {10+{\sqrt {5}}}{10}}}\approx 1.10617$ Hypervolume${\frac {\sqrt {25+10{\sqrt {5}}}}{4}}\approx 1.72047$ Dichoral anglesCube–4–cube: 108°
Pip–5–pip: 90°
Cube–4–pip: 90°
Height1
Central density1
Number of external pieces9
Level of complexity6
Related polytopes
ArmySquipdip
RegimentSquipdip
DualSquare-pentagonal duotegum
ConjugateSquare-pentagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2×H2, order 80
ConvexYes
NatureTame

The square-pentagonal duoprism or squipdip, also known as the 4-5 duoprism, is a uniform duoprism that consists of 4 pentagonal prisms and 5 cubes, with two of each joining at each vertex.

It is also a CRF segmentochoron, being a pentagonal prism atop pentagonal prism. It is designated K-4.42 on Richard Klitzing's list. It can thus be thought of as a prism based on the pentagonal prism.

## Vertex coordinates

Coordinates for the vertices of a square-pentagonal duoprism with edge length 1 are given by:

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

A suare-pentagonal duoprism has the following Coxeter diagrams:

• x4o x5o (full symmetry)
• x x x5o (H2×A1×A1 symmetry, square as rectangle)
• xx xx5oo#x (H2×A1 axial, pentagonal prism prism)
• ofx xxx4ooo&#xt (BC2×A1 axial, square-first)
• ofx xxx xxx&#xt (A1×A1×A1, as above with rectangle symmetry)
• oqo xxx5ooo&#xt (H2×A1 axial, pentagon-first)