# Octagonal-square antiprismatic duoprism

Octagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOsquap
Coxeter diagramx8o s2s8o
Elements
Tera8 square antiprismatic prisms, 8 triangular-octagonal duoprisms, 2 square-octagonal duoprisms
Cells64 triangular prisms, 16 cubes, 8 square antiprisms, 8+8 octagonal prisms
Faces64 triangles, 16+64+64 squares, 8 octagons
Edges64+64+64
Vertices64
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {12+5{\sqrt {2}}}{8}}}\approx 1.54398}$
Hypervolume${\displaystyle {\frac {2{\sqrt {24+17{\sqrt {2}}}}}{3}}\approx 4.62080}$
Diteral anglesSquappip–squap–squappip: 135°
Todip–op–todip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Todip–op–sodip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Todip–trip–squappip: 90°
Sodip–cube–squappip: 90°
${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces18
Level of complexity40
Related polytopes
ArmyOsquap
RegimentOsquap
DualOctagonal-square antitegmatic duotegum
ConjugateOctagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)×I2(8)×A1+, order 256
ConvexYes
NatureTame

The octagonal-square antiprismatic duoprism or osquap is a convex uniform duoprism that consists of 8 square antiprismatic prisms, 2 square-octagonal duoprisms, and 8 triangular-octagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-octagonal duoprisms, and 1 square-octagonal duoprism.

## Vertex coordinates

The vertices of an octagonal-square antiprismatic duoprism of edge length 1 are given by all permutations of the first two coordinates of:

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

## Representations

An octagonal-square antiprismatic duoprism has the following Coxeter diagrams:

• x8o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x8o s2s4s (square antiprisms as alternated ditetragonal prisms)
• x4x s2s8o (octagons as ditetragons)
• x4x s2s4s