Great rhombicuboctahedral prism

Great rhombicuboctahedral prism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Gircope
Coxeter diagram	x x4x3x ()
Elements
Cells	12 cubes, 8 hexagonal prisms, 6 octagonal prisms, 2 great rhombicuboctahedra
Faces	24+24+24+24 squares, 16 hexagons, 12 octagons
Edges	48+48+48+48
Vertices	96
Vertex figure	Irregular tetrahedron, edge lengths √2, √3, √2+√2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7+3{\sqrt {2}}}{2}}}\approx 2.37093$
Hypervolume	$2(11+7{\sqrt {2}})\approx 41.79899$
Dichoral angles	Cube–4–hip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Cube–4–op: 135°
	Hip–4–op: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Girco–8–op: 90°
	Girco–6–hip: 90°
	Girco–4–cube: 90°
Height	1
Central density	1
Number of external pieces	28
Level of complexity	24
Related polytopes
Army	Gircope
Regiment	Gircope
Dual	Disdyakis dodecahedral tegum
Conjugate	Quasitruncated cuboctahedral prism
Abstract & topological properties
Flag count	2304
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×A1, order 96
Convex	Yes
Nature	Tame

The great rhombicuboctahedral prism, or gircope, is a prismatic uniform polychoron that consists of 2 great rhombicuboctahedra, 6 octagonal prisms, 8 hexagonal prisms, and 12 cubes. Each vertex joins one of each type of cell. As the name suggests, it is a prism based on the great rhombicuboctahedron. As such it is also a convex segmentochoron (designated K-4.125 on Richard Klitzing's list).

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in rhombicuboctahedron-first orientation.

This polychoron can be alternated into a snub cubic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritosnub alterprism, which is also nonuniform.

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, girco atop girco

• 

  Net

Vertex coordinates

The vertices of a great rhombicuboctahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

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

Representations

The great rhombicuboctahedral prism has the following Coxeter diagrams:

• x x4x3x (full symmetry)
• xx4xx3xx&#x (bases considered separately)
• xxxxxx xuxxux4xxwwxx&#xt (BC₂×A₁ symmetry, octagonal prism-first)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#944).

• Klitzing, Richard. "Gircope".
• Wikipedia contributors. "Truncated cuboctahedral prism".

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