Great rhombicuboctahedron

Great rhombicuboctahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Girco
Coxeter diagram	x4x3x ()
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
Volume
Dihedral angles	6–4:
	8–4: 135°
	8–6:
Central density	1
Number of pieces	26
Level of complexity	6
Related polytopes
Army	Girco
Regiment	Girco
Dual	Disdyakis dodecahedron
Conjugate	Quasitruncated cuboctahedron
Abstract properties
Flag count	288
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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[edit | edit source]

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[edit | edit source]

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

$\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

Representations[edit | edit source]

A great rhombicuboctahedron has the following Coxeter diagrams:

x4x3x (full symmetry)
xxwwxx4xuxxux&#xt (BC2 axial, octagon-first)
wx3xx3xw&#zx (A3 symmetry, as hull of two inverse great rhombitetratetrahedra)
Xwx xxw4xux&#zx (BC2×A1 symmetry)
xxuUxwwx3xwwxUuxx&#xt (A2 axial, hexagon-first)

Semi-uniform variant[edit | edit source]

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 $\frac{\sqrt{3a^2+4b^2+2c^2+4bc+(4ab+2ac)\sqrt2}}{2}$ and its volume is given by $a^3+6ab^2+12abc+3ac^2+(9a^2b+9a^2c+5b^3+12b^2c+6bc^2+c^3)\frac{\sqrt2}{3}$ .

It has coordinates given by all permutations of:

$\left(±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2}\right).$

Related polyhedra[edit | edit source]

o4o3o truncations
Name	OBSA	Schläfli symbol	CD diagram
Cube	cube	{4,3}	x4o3o
Truncated cube	tic	t{4,3}	x4x3o
Cuboctahedron	co	r{4,3}	o4x3o
Truncated octahedron	toe	t{3,4}	o4x3x
Octahedron	oct	{3,4}	o4o3x
Small rhombicuboctahedron	sirco	rr{4,3}	x4o3x
Great rhombicuboctahedron	girco	tr{4,3}	x4x3x
Snub cube	snic	sr{4,3}	s4s3s