# Octagonal-decagonal duoprismatic prism

Octagonal-decagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymOddip
Coxeter diagramx x8o x10o ()
Elements
Tera10 square-octagonal duoprisms, 8 square-decagonal duoprisms, 2 octagonal-decagonal duoprisms
Cells80 cubes, 8+16 decagonal prisms, 10+20 octagonal prisms
Faces80+80+160 squares, 20 octagons, 16 decagons
Edges80+160+160
Vertices160
Vertex figureDigonal disphenoidal pyramid, edge lengths 2+2 (disphenoid base 1), (5+5)/2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+2{\sqrt {2}}+2{\sqrt {5}}}}{2}}\approx 2.13896}$
Hypervolume${\displaystyle 5{\sqrt {15+10{\sqrt {2}}+6{\sqrt {5}}+4{\sqrt {10}}}}\approx 37.15093}$
Diteral anglesSodip–op–sodip: 144°
Odedip–op–sodip: 90°
Height1
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyOddip
RegimentOddip
DualOctagonal-decagonal duotegmatic tegum
ConjugatesOctagonal-decagrammic duoprismatic prism, Octagrammic-decagonal duoprismatic prism, Octagrammic-decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)×I2(10)×A1, order 640
ConvexYes
NatureTame

The octagonal-decagonal duoprismatic prism or oddip, also known as the octagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 octagonal-decagonal duoprisms, 8 square-decagonal duoprisms, and 10 square-octagonal duoprisms. Each vertex joins 2 square-octagonal duoprisms, 2 square-decagonal duoprisms, and 1 octagonal-decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

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

## Vertex coordinates

The vertices of an octagonal-decagonal duoprismatic prism of edge length 1 are given by all permutations of the first two coordinates of:

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

## Representations

An octagonal-decagonal duoprismatic prism has the following Coxeter diagrams:

• x x8o x10o (full symmetry)
• x x4x x10o () (BC2×I2(10)×A1 symmetry, octagons as ditetragons)
• x x8o x5x () (H2×I2(8)×A1 symmetry, decagons as dipentagons)
• x x4x x5x () (BC2×H2×A1 symmetry)
• xx8oo xx10oo&#x (octagonal-decagonal duoprism atop octagonal-decagonal duoprism)
• xx4xx xx10oo&#x
• xx8oo xx5xx&#x
• xx4xx xx5xx&#x