# Great rhombicosidodecahedron

(Redirected from Grid)
Great rhombicosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymGrid
Coxeter diagramx5x3x ()
Conway notationbD
Stewart notationK5
Elements
Faces
Edges60+60+60
Vertices120
Vertex figureScalene triangle, edge lengths 2, 3, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}}}{2}}\approx 3.80239}$
Volume${\displaystyle 5(19+10{\sqrt {5}})\approx 206.80340}$
Dihedral angles6–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
10–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
10–6: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Central density1
Number of external pieces62
Level of complexity6
Related polytopes
ArmyGrid
RegimentGrid
DualDisdyakis triacontahedron
ConjugateGreat quasitruncated icosidodecahedron
Abstract & topological properties
Flag count720
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits6
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 with a Coxeter diagram comprising entirely ringed nodes, 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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}$,

along with all even permutations of:

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

## Representations

A great rhombicosidodecahedron has the following Coxeter diagrams:

• x5x3x () (full symmetry)
• xuxxuAxFVFxx5xxFVFxAuxxux&#xt (H2 axial, decagon-first)
• xxu(xX)BFDUDFCxCx(AF)UFx3xFU(AF)xCxCFDUDFB(xX)uxx&#xt (A2 symmetry, triangle-first)
• xuxXFBYUD(HA)F(JCx)(JCx)F(HA)DUYBFXxux xFCUAFxDX(uB)H(xYJ)(xYJ)H(uB)XDxFAUCFx&#xt (K2 axial, square-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 ${\displaystyle {\sqrt {\frac {9a^{2}+12b^{2}+5c^{2}+16ab+8ac+12bc+(3a^{2}+4b^{2}+c^{2}+8ab+4ac+4bc){\sqrt {5}}}{8}}}}$ and its volume is given by ${\displaystyle {\frac {15a^{3}+60a^{2}b+45a^{2}c+60ab^{2}+60abc+15ac^{2}+30b^{3}+60b^{2}c+30bc^{2}+5c^{3}}{4}}+(21a^{3}+72a^{2}b+45a^{2}c+108ab^{2}+180abc+45ac^{2}+34b^{3}+60b^{2}c+30bc^{2}+5c^{3}){\frac {\sqrt {5}}{12}}}$.

It has coordinates given by all even permutations of:

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

where ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$.