Great rhombicuboctahedron

Great rhombicuboctahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Girco
Coxeter diagram	x4x3x ()
Conway notation	bC
Stewart notation	K4
Elements
Faces	12 squares, 8 hexagons, 6 octagons
Edges	24+24+24
Vertices	48
Vertex figure	Scalene triangle, edge lengths √2, √3, √2+√2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {13+6{\sqrt {2}}}}{2}}\approx 2.31761$
Volume	$2(11+7{\sqrt {2}})\approx 41.79899$
Dihedral angles	6–4: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	8–4: 135°
	8–6: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
Central density	1
Number of external pieces	26
Level of complexity	6
Related polytopes
Army	Girco
Regiment	Girco
Dual	Disdyakis dodecahedron
Conjugate	Quasitruncated cuboctahedron
Abstract & topological properties
Flag count	288
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

For the nonconvex conjugate of the small rhombicuboctahedron, see quasirhombicuboctahedron.

The great rhombicuboctahedron or girco, also commonly known as the truncated cuboctahedron, is one of the 13 Archimedean solids. It consists of 12 squares, 8 hexagons, and 6 octagons, with one of each type of face meeting per vertex. It can be obtained by cantitruncation of the cube or octahedron, or equivalently by truncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

This is one of three Wythoffian non-prismatic polyhedra whose Coxeter diagram has all ringed nodes, the other two being the great rhombitetratetrahedron and the great rhombicosidodecahedron.

It can be alternated into the snub cube after equalizing edge lengths.

Naming

Rhombi refers to the twelve square faces on the axis of the rhombic dodecahedron, cub(e) refers to the six faces on the axis of the cube, and octahedron for the eight hexagons on the axis of the octahedron.

Alternate names include:

• Truncated cuboctahedron (a translation of Kepler's Latin name), because it can be derived by truncating the cuboctahedron. However, it is not a true truncation of the rhombicuboctahedron, as a true truncation would result in rectangles rather than squares. Kepler used a different word for this sense of "truncated", but it was lost in translation.
• Rhombitruncated cuboctahedron (similar to the above, but also refers to the planes of the truncated faces).
• Great rhombcuboctahedron (no "i") - alternate spelling.

Vertex coordinates

A great rhombicuboctahedron of edge length 1 has vertex coordinates given by all permutations of:

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

Representations

A great rhombicuboctahedron has the following Coxeter diagrams:

• x4x3x (full symmetry)
• xxwwxx4xuxxux&#xt (B₂ axial, octagon-first)
• wx3xx3xw&#zx (A₃ symmetry, as hull of two inverse great rhombitetratetrahedra)
• Xwx xxw4xux&#zx (B₂×A₁ symmetry)
• xxuUxwwx3xwwxUuxx&#xt (A₂ axial, hexagon-first)

Semi-uniform variant

The great rhombicuboctahedron has a semi-uniform variant of the form x4y3z that maintains its full symmetry. This variant has 6 ditetragons, 8 ditrigons, and 12 rectangles as faces.

With edges of length a (ditetragon-rectangle), b (ditetragon-ditrigon), and c (ditrigon-rectangle), its circumradius is given by ${\frac {\sqrt {3a^{2}+4b^{2}+2c^{2}+4bc+(4ab+2ac){\sqrt {2}}}}{2}}$ and its volume is given by $a^{3}+6ab^{2}+12abc+3ac^{2}+(9a^{2}b+9a^{2}c+5b^{3}+12b^{2}c+6bc^{2}+c^{3}){\frac {\sqrt {2}}{3}}$.

It has coordinates given by all permutations of:

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

External links

• Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#57).

• Klitzing, Richard. "Girco".

• Quickfur. "The Great Rhombicuboctahedron".

• Wikipedia contributors. "Truncated cuboctahedron".
• McCooey, David. "Truncated Cuboctahedron"

• Hi.gher.Space Wiki Contributors. "Stauropantohedron".