# Great quasirhombated icositetrachoron

Great quasirhombated icositetrachoron
Rank4
TypeUniform
Notation
Bowers style acronymGaqri
Coxeter diagramx3x4/3x3o ()
Elements
Cells96 triangular prisms, 24 quasitruncated hexahedra, 24 quasitruncated cuboctahedra
Faces192 triangles, 288 squares, 96 hexagons, 144 octagrams
Edges288+288+576
Vertices576
Vertex figureSphenoid edge lengths 1 (1), 2 (2), 3 (1), and 2–2 (2)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {10-6{\sqrt {2}}}}\approx 1.23074}$
Hypervolume${\displaystyle 3\left(337-233{\sqrt {2}}\right)\approx 22.46472}$
Dichoral anglesQuitco–6–quitco: 120°
Quitco–8/3–quith: 45°
Quitco–4–trip: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{3}}\right)\approx 35.26439^{\circ }}$
Quith–3–trip: 30°
Number of external pieces2856
Level of complexity105
Related polytopes
ArmySemi-uniform Prico, edge lengths ${\displaystyle {\sqrt {2}}-1}$ (small rhombicuboctahedra), ${\displaystyle 3-2{\sqrt {2}}}$ (bases of triangular prisms)
RegimentGaqri
ConjugateGreat rhombated icositetrachoron
Convex coreIcositetrachoron
Abstract & topological properties
Flag count13824
Euler characteristic0
OrientableYes
Properties
SymmetryF4, order 1152
ConvexNo
NatureTame

The great quasirhombated icositetrachoron, or gaqri, is a nonconvex uniform polychoron that consists of 96 triangular prisms, 24 quasitruncated hexahedra, and 24 quasitruncated cuboctahedra. 1 triangular prism, 1 quasitruncated hexahedron, and 2 quasitruncated cuboctahedra join at each vertex. As the name suggests, it can be obtained by quasicantitruncating the icositetrachoron.

## Vertex coordinates

The vertices of a great quasirhombated icositetrachoron of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm 3{\frac {{\sqrt {2}}-1}{2}},\,\pm {\frac {2{\sqrt {2}}-1}{2}},\,\pm {\frac {{\sqrt {2}}-1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}-2}{2}},\,\pm ({\sqrt {2}}-1),\,\pm {\frac {2-{\sqrt {2}}}{2}},\,0\right).}$

The quasicantitruncation of the dual icositetrachron has coordinates given by all permutations of:

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