# Square-decagonal duoprism

Square-decagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx4o x10o ()
Elements
Cells10 cubes, 4 decagonal prisms
Faces10+40 squares, 4 decagons
Edges40+40
Vertices40
Vertex figureDigonal disphenoid, edge lengths (5+5)/2 (base 1) and 2 (base 2 and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {5}}}{2}}}\approx 1.76580}$
Hypervolume${\displaystyle 5{\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 7.69421}$
Dichoral anglesCube–4–cube: 144°
Dip–10–dip: 90°
Cube–4–dip: 90°
Height1
Central density1
Number of external pieces14
Level of complexity6
Related polytopes
DualSquare-decagonal duotegum
ConjugateSquare-decagrammic duoprism
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryB2×I2(10), order 160
Flag orbits6
ConvexYes
NatureTame

The square-decagonal duoprism or squadedip, also known as the 4-10 duoprism, is a uniform duoprism that consists of 4 decagonal prisms and 10 cubes, with two of each joining at each vertex.

It is a prism based on the decagonal prism. As such it is also a CRF segmentochoron, designated K-4.97 on Richard Klitzing's list.

This polychoron can be alternated into a digonal-pentagonal duoantiprism, although it cannot be made uniform.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A square-decagonal duoprism has the following Coxeter diagrams:

• x4o x10o () (full symmetry)
• x4o x5x () (B2×H2 symmetry, decagon as dipentagon)
• x x x10o () (I2(10)×K2 symmetry, square as rectangle)
• x x x5x () (H2×K2 symmetry, both above)
• xx xx10oo (I2(10)×A1 symmetry, decagon prismatic prism)
• xx xx5xx&#x (H2×A1 axial, as above)
• oqo xxx10ooo&#xt
• oqo xxx5xxx&#xt (H2×A1 axial, decagon-first)