Octagonal-truncated octahedral duoprism

Octagonal-truncated octahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Otoe
Coxeter diagram	x8o o4x3x ()
Elements
Tera	6 square-octagonal duoprisms, 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms
Cells	48 cubes, 64 hexagonal prisms, 12+24 octagonal prisms, 8 truncated octahedra
Faces	48+96+192 squares, 64 hexagons, 24 octagons
Edges	96+192+192
Vertices	192
Vertex figure	Digonal disphenoidal pyramid, edge lengths √2, √3, √3 (base triangle), √2+√2 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7+{\sqrt {2}}}{2}}}\approx 2.05112$
Hypervolume	$32+16{\sqrt {2}}\approx 54.62742$
Diteral angles	Tope–toe–tope: 135°
	Sodip–op–hodip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Hodip–op–hodip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Sodip–cube–tope: 90°
	Hodip–hip–tope: 90°
Central density	1
Number of external pieces	22
Level of complexity	30
Related polytopes
Army	Otoe
Regiment	Otoe
Dual	Octagonal-tetrakis hexahedral duotegum
Conjugate	Octagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(8), order 768
Convex	Yes
Nature	Tame

The octagonal-truncated octahedral duoprism or otoe is a convex uniform duoprism that consists of 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms, and 6 square-octagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-octagonal duoprism, and 2 hexagonal-octagonal duoprisms.

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

Vertex coordinates

The vertices of an octagonal-truncated octahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

Representations

An octagonal-truncated octahedral duoprism has the following Coxeter diagrams:

• x8o o4x3x (full symmetry)
• x8o x3x3x ()
• x4x o4x3x () (octagons as ditetragons)
• x4x x3x3x ()

External links

• Klitzing, Richard. "otoe".