Great rhombicosidodecahedron

Great rhombicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGrid
Coxeter diagramx5x3x ()
Elements
Faces30 squares, 20 hexagons, 12 decagons
Edges60+60+60
Vertices120
Vertex figureScalene triangle, edge lengths 2, 3, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{31+12\sqrt5}}{2} ≈ 3.80239}$
Volume${\displaystyle 5(19+10\sqrt5) ≈ 206.80340}$
Dihedral angles6–4: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484^\circ}$
10–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253^\circ}$
10–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ}$
Central density1
Number of pieces62
Level of complexity6
Related polytopes
ArmyGrid
RegimentGrid
DualDisdyakis triacontahedron
ConjugateGreat quasitruncated icosidodecahedron
Abstract properties
Flag count720
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

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

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

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2}\right),}$

along with all even permutations of:

• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{4+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right).}$

Representations

A great rhombicosidodecahedron has the following Coxeter diagrams:

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

Semi-uniform variant

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 ${\displaystyle \sqrt{\frac{9a^2+12b^2+5c^2+16ab+8ac+12bc+(3a^2+4b^2+c^2+8ab+4ac+4bc)\sqrt5}{8}}}$ and its volume is given by ${\displaystyle \frac{15a^3+60a^2b+45a^2c+60ab^2+60abc+15ac^2+30b^3+60b^2c+30bc^2+5c^3}{4}+(21a^3+72a^2b+45a^2c+108ab^2+180abc+45ac^2+34b^3+60b^2c+30bc^2+5c^3)\frac{\sqrt5}{12}}$.

Related polyhedra

o5o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Dodecahedron doe {5,3} x5o3o
Truncated dodecahedron tid t{5,3} x5x3o
Icosidodecahedron id r{5,3} o5x3o
Truncated icosahedron ti t{3,5} o5x3x
Icosahedron ike {3,5} o5o3x
Small rhombicosidodecahedron srid rr{5,3} x5o3x
Great rhombicosidodecahedron grid tr{5,3} x5x3x
Snub dodecahedron snid sr{5,3} s5s3s