# Octagonal-decagonal duoprism

Octagonal-decagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymOdedip
Coxeter diagramx8o x10o ()
Elements
Cells10 octagonal prisms, 8 decagonal prisms
Faces80 squares, 10 octagons, 8 decagons
Edges80+80
Vertices80
Vertex figureDigonal disphenoid, edge lengths 2+2 (base 1), (5+5)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {2}}+{\sqrt {5}}}{2}}}\approx 2.07970}$
Hypervolume${\displaystyle 5{\sqrt {15+10{\sqrt {2}}+6{\sqrt {5}}+4{\sqrt {10}}}}\approx 37.15093}$
Dichoral anglesOp–8–op: 144°
Dip–10–dip: 135°
Dip–4–op: 90°
Central density1
Number of external pieces18
Level of complexity6
Related polytopes
ArmyOdedip
RegimentOdedip
DualOctagonal-decagonal duotegum
ConjugatesOctagonal-decagrammic duoprism, Octagrammic-decagonal duoprism, Octagrammic-decagrammic duoprism
Abstract & topological properties
Flag count1920
Euler characteristic0
OrientableYes
Properties
SymmetryI2(8)×I2(10), order 320
ConvexYes
NatureTame

The octagonal-decagonal duoprism or odedip, also known as the 8-10 duoprism, is a uniform duoprism that consists of 8 decagonal prisms and 10 octagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a square-pentagonal duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pentagonal-square prismantiprismoid, which is also nonuniform.

## Vertex coordinates

The vertices of an octagonal-decagonal duoprism of edge length 1, centered at the origin, are given by:

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

## Representations

An octagonal-decagonal duoprism has the following Coxeter diagrams:

• x8o x10o () (full symmetry)
• x5x x10o () (octagons as ditetragons0
• x5x x8o () (decagons as dipentagons)
• x4x x5x () (both of these applied)