Octagonal antiprismatic prism

Octagonal antiprismatic prism
Rank	4
Type	Uniform
Notation
Bowers style acronym	Oappip
Coxeter diagram	x2s2s16o ()
Elements
Cells	16 triangular prisms, 2 octagonal prisms, 2 octagonal antiprisms
Faces	32 triangles, 16+16 squares, 4 octagons
Edges	16+32+32
Vertices	32
Vertex figure	Isosceles trapezoidal pyramid, edge lengths 1, 1, 1, √2+√2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {8+2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}\approx 1.46360$
Hypervolume	${\frac {2{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}}{3}}\approx 4.26796$
Dichoral angles	Trip–4–trip: $\arccos \left({\frac {1-2{\sqrt {2+{\sqrt {2}}}}}{3}}\right)\approx 153.96238^{\circ }$
	Trip–4–op: $\arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 96.59451^{\circ }$
	Oap–8–op: 90°
	Oap–3–trip: 90°
Heights	Oap atop oap: 1
	Op atop gyro op: ${\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{2}}}\approx 0.86030$
Central density	1
Number of external pieces	20
Level of complexity	16
Related polytopes
Army	Oappip
Regiment	Oappip
Dual	Octagonal antitegmatic tegum
Conjugates	Octagrammic antiprismatic prism, Octagrammic retroprismatic prism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(16)×A1)+×A1, order 64
Convex	Yes
Nature	Tame

The octagonal antiprismatic prism or oappip is a prismatic uniform polychoron that consists of 2 octagonal antiprisms, 2 octagonal prisms, and 16 triangular prisms. Each vertex joins 1 octagonal antiprism, 1 octagonal prism, and 3 triangular prisms. As the name suggests, it is a prism based on an octagonal antiprism. It is also a CRF segmentochoron designated K-4.65 on Richard Klitzing's list.

Vertex coordinates

The vertices of an octagonal antiprismatic prism of edge length 1 are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,H,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,H,\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,-H,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0,\,-H,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,-H,\,\pm {\frac {1}{2}}\right),$

where $H={\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}$ is the distance between the antiprism's center and the center of one of its bases.

Representations

An octagonal antiprismatic prism has the following Coxeter diagrams:

• x2s2s16o (full symmetry)
• x2s2s8s ()
• xx xo8ox&#x (octagonal prism atop gyrated octagonal prism)

External links

• Bowers, Jonathan. "Category B: Antiduoprisms".

• Klitzing, Richard. "oappip".
• Wikipedia contributors. "Uniform antiprismatic prism".