# Enneagonal-square antiprismatic duoprism

Enneagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEsquap
Coxeter diagramx9o s2s8o ()
Elements
Tera9 square antiprismatic prisms, 8 triangular-enneagonal duoprisms, 2 square-enneagonal duoprisms
Cells72 triangular prisms, 18 cubes, 9 square antiprisms, 8+8 enneagonal prisms
Faces72 triangles, 18+72+72 squares, 8 enneagons
Edges72+72+72
Vertices72
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {2}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 1.67748}$
Hypervolume${\displaystyle {\frac {3{\sqrt {4+3{\sqrt {2}}}}}{4\tan {\frac {\pi }{9}}}}\approx 5.91601}$
Diteral anglesSquappip–squap–squappip: 140°
Tedip–ep–tedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Tedip–ep–sendip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Tedip–trip–squappip: 90°
Sendip–cube–squappip: 90°
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces19
Level of complexity40
Related polytopes
ArmyEsquap
RegimentEsquap
DualEnneagonal-square antitegmatic duotegum
ConjugatesEnneagrammic-square antiprismatic duoprism, Great enneagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(9)×I2(8)×A1+, order 288
ConvexYes
NatureTame

The enneagonal-square antiprismatic duoprism or esquap is a convex uniform duoprism that consists of 9 square antiprismatic prisms, 2 square-enneagonal duoprisms, and 8 triangular-enneagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-enneagonal duoprisms, and 1 square-enneagonal duoprism.

## Vertex coordinates

The vertices of an enneagonal-square antiprismatic duoprism of edge length 2sin(π/9) are given by:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{9}}}{2}}\right),}$

where j = 2, 4, 8.

## Representations

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

• x9o s2s8o () (full symmetry; square antiprisms as alternated octagonal prisms)
• x9o s2s4s () (square antiprisms as alternated ditetragonal prisms)