Octagonal-great rhombicuboctahedral duoprism

Octagonal-great rhombicuboctahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Ogirco
Coxeter diagram	x8o x4x3x ()
Elements
Tera	12 square-octagonal duoprisms, 8 hexagonal-octagonal duoprisms, 6 octagonal duoprisms, 8 great rhombicuboctahedral prisms
Cells	96 cubes, 64 hexagonal prisms, 24+24+24+48 octagonal prisms, 8 great rhombicuboctahedra
Faces	96+192+192+192 squares, 64 hexagons, 48+48 octagons
Edges	192+192+192+384
Vertices	384
Vertex figure	Mirror-symmetric pentachoron, edge lengths √2, √3, √2+√2 (base triangle), √2+√2 (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {17+8{\sqrt {2}}}}{2}}\approx 2.66053$
Hypervolume	$4(25+18{\sqrt {2}})\approx 201.82338$
Diteral angles	Sodip–op–hodip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Sodip–op–odip: 135°
	Gircope–girco–gircope: 135°
	Hodip–op–odip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Sodip–cube–gircope: 90°
	Hodip–hip–gircope: 90°
	Odip–op–gircope: 90°
Central density	1
Number of external pieces	34
Level of complexity	60
Related polytopes
Army	Ogirco
Regiment	Ogirco
Dual	Octagonal-disdyakis dodecahedral duotegum
Conjugate	Octagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(8), order 768
Convex	Yes
Nature	Tame

The octagonal-great rhombicuboctahedral duoprism or ogirco is a convex uniform duoprism that consists of 8 great rhombicuboctahedral prisms, 6 octagonal duoprisms, 8 hexagonal-octagonal duoprisms, and 12 square-octagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-octagonal duoprism, 1 hexagonal-octagonal duoprism, and 1 octagonal duoprism.

The octagonal-great rhombicuboctahedral duoprism can be vertex-inscribed into the cellirhombated penteractitriacontaditeron.

This polyteron can be alternated into a square-snub cubic duoantiprism, although it cannot be made uniform. The octagons can also be edge-snubbed to create a snub cubic-square prismantiprismoid or the great rhombicuboctahedra to create a square-pyritohedral prismantiprismoid, which are also both nonuniform.

Vertex coordinates

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

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

Representations

An octagonal-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

• x8o x4x3x (full symmetry)
• x4x x4x3x () (BC₃×BC₂ symmetry, octagons as ditetragons)

External links

• Klitzing, Richard. "ogirco".