Great rhombicosidodecahedron

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Great rhombicosidodecahedron
Great rhombicosidodecahedron.png
Bowers style acronymGrid
Coxeter diagramx5x3x (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Faces30 squares, 20 hexagons, 12 decagons
Vertex figureScalene triangle, edge lengths 2, 3, (5+5)/2
Great rhombicosidodecahedron vertfig.png
Measures (edge length 1)
Dihedral angles6–4:
Central density1
Number of external pieces62
Level of complexity6
Related polytopes
DualDisdyakis triacontahedron
ConjugateGreat quasitruncated icosidodecahedron
Abstract & topological properties
Flag count720
Euler characteristic2
SymmetryH3, order 120

The great rhombicosidodecahedron or grid, also commonly known as the truncated icosidodecahedron, is the most complex of the 13 Archimedean solids. It consists of 12 decagons, 20 hexagons, and 30 squares, with one of each type of face meeting per vertex. It can be obtained by cantitruncation of the dodecahedron or icosahedron, or equivalently by truncating the vertices of an icosidodecahedron and then adjusting the edge lengths to be all equal.

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

It can be alternated into the snub dodecahedron after edge lengths are equalized.

Vertex coordinates[edit | edit source]

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

along with all even permutations of:

Representations[edit | edit source]

A great rhombicosidodecahedron has the following Coxeter diagrams:

  • x5x3x (full symmetry)
  • xuxxuAxFVFxx5xxFVFxAuxxux&#xt (H2 axial, decagon-first)

Semi-uniform variant[edit | edit source]

The great rhombicosidodecahedron has a semi-uniform variant of the form x5y3z that maintains its full symmetry. This variant has 12 dipentagons, 20 ditrigons, and 30 rectangles as faces.

With edges of length a (dipentagon-rectangle), b (dipentagon-ditrigon), and c (ditrigon-rectangle), its circumradius is given by and its volume is given by .

Related polyhedra[edit | edit source]

o5o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Dodecahedron doe {5,3} x5o3o
Uniform polyhedron-53-t0.png
Truncated dodecahedron tid t{5,3} x5x3o
Uniform polyhedron-53-t01.png
Icosidodecahedron id r{5,3} o5x3o
Uniform polyhedron-53-t1.png
Truncated icosahedron ti t{3,5} o5x3x
Uniform polyhedron-53-t12.png
Icosahedron ike {3,5} o5o3x
Uniform polyhedron-53-t2.png
Small rhombicosidodecahedron srid rr{5,3} x5o3x
Uniform polyhedron-53-t02.png
Great rhombicosidodecahedron grid tr{5,3} x5x3x
Uniform polyhedron-53-t012.png
Snub dodecahedron snid sr{5,3} s5s3s
Uniform polyhedron-53-s012.png

External links[edit | edit source]