|Bowers style acronym||Girco|
|Coxeter diagram||x4x3x ()|
|Faces||12 squares, 8 hexagons, 6 octagons|
|Vertex figure||Scalene triangle, edge lengths √, √, √ |
|Measures (edge length 1)|
|Number of external pieces||26|
|Level of complexity||6|
|Abstract & topological properties|
|Symmetry||B3, order 48|
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.
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:
Representations[edit | edit source]
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[edit | edit source]
With edges of length a (ditetragon-rectangle), b (ditetragon-ditrigon), and c (ditrigon-rectangle), its circumradius is given by and its volume is given by .
It has coordinates given by all permutations of:
[edit | edit source]
- Bowers, Jonathan. "Polyhedron Category 5: Omnitruncates" (#57).
- Klitzing, Richard. "Girco".
- Quickfur. "The Great Rhombicuboctahedron".
- Hi.gher.Space Wiki Contributors. "Stauropantohedron".