# Great quasitruncated icosidodecahedron

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.

Great quasitruncated icosidodecahedron
Rank3
TypeUniform
SpaceSpherical
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${\displaystyle \frac{\sqrt{31-12\sqrt5}}{2} ≈ 1.02068}$
Volume${\displaystyle 5(10\sqrt5-19) ≈ 16.80340}$
Dihedral angles10/3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747^\circ}$
10/3–6: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 79.18768^\circ}$
6–4: ${\displaystyle \arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) ≈ 69.09484^\circ}$
Central density13
Number of external pieces1140
Level of complexity58
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle \frac{3-\sqrt5}{2}}$ (dipentagon-rectangle), ${\displaystyle \sqrt5-2}$ (dipentagon-ditrigon), ${\displaystyle \frac{7-3\sqrt5}{2}}$ (ditrigon-rectangle)
RegimentGaquatid
DualGreat disdyakis triacontahedron
ConjugateGreat rhombicosidodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

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

along with all even permutations of:

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

## Related polyhedra

o5/3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great stellated dodecahedron gissid {5/3,3} x5/3o3o (       )
Quasitruncated great stellated dodecahedron quit gissid t{5/3,3} x5/3x3o (       )
Great icosidodecahedron gid r{3,5/3} o5/3x3o (       )
Truncated great icosahedron tiggy t{3,5/3} o5/3x3x (       )
Great icosahedron gike {3,5/3} o5/3o3x (       )
Quasirhombicosidodecahedron qrid rr{3,5/3} x5/3o3x (       )
Great quasitruncated icosidodecahedron gaquatid tr{3,5/3} x5/3x3x (       )
Great inverted snub icosidodecahedron gisid sr{3,5/3} s5/3s3s (       )