# Great quasitruncated icosidodecahedron

Great quasitruncated icosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymGaquatid
Coxeter diagramx5/3x3x ()
Elements
Faces30 squares, 20 hexagons, 12 decagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle, edge lengths 2, 3, (5–5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {31-12{\sqrt {5}}}}{2}}\approx 1.02068}$
Volume${\displaystyle 5(10{\sqrt {5}}-19)\approx 16.80340}$
Dihedral angles10/3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
10/3–6: ${\displaystyle \arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 79.18768^{\circ }}$
6–4: ${\displaystyle \arccos \left({\frac {{\sqrt {15}}-{\sqrt {3}}}{6}}\right)\approx 69.09484^{\circ }}$
Central density13
Number of external pieces1140
Level of complexity58
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (dipentagon-rectangle), ${\displaystyle {\sqrt {5}}-2}$ (dipentagon-ditrigon), ${\displaystyle {\frac {7-3{\sqrt {5}}}{2}}}$ (ditrigon-rectangle)
RegimentGaquatid
DualGreat disdyakis triacontahedron
ConjugateGreat rhombicosidodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits6
ConvexNo
NatureTame

The great quasitruncated icosidodecahedron or gaquatid, also called the great truncated icosidodecahedron, is a uniform polyhedron. It consists of 12 decagrams, 20 hexagons, and 30 squares, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the great stellated dodecahedron or great icosahedron, or equivalently by quasitruncating the vertices of a great icosidodecahedron and then adjusting the edge lengths to be all equal.

It can be alternated into the great inverted snub icosidodecahedron after equalizing edge lengths.

## Vertex coordinates

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

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

along with all even permutations of:

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