# Great prismatotetracontoctachoron

Great prismatotetracontoctachoron
Rank4
TypeUniform
Notation
Bowers style acronymGippic
Coxeter diagramx3x4x3x ()
Elements
Cells192 hexagonal prisms, 48 great rhombicuboctahedra
Faces288+576 squares, 384 hexagons, 144 octagons
Edges1152+1152
Vertices1152
Vertex figurePhyllic disphenoid, edge lengths 2 (3), 3 (2), and 2+2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {14+9{\sqrt {2}}}}\approx 5.16991}$
Hypervolume${\displaystyle 24(62+43{\sqrt {2}})\approx 2947.46840}$
Dichoral anglesHip–4–hip: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {2}}}{3}}\right)\approx 160.52878^{\circ }}$
Girco–6–hip: 150°
Girco–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Girco–8–girco: 135°
Central density1
Number of external pieces240
Level of complexity12
Related polytopes
ArmyGippic
RegimentGippic
DualDisphenoidal chiliahecatonpentaconta-dichoron
ConjugateGreat quasiprismatotetracontocta-choron
Abstract & topological properties
Flag count27648
Euler characteristic0
OrientableYes
Properties
SymmetryF4×2, order 2304
ConvexYes
NatureTame

The great prismatotetracontoctachoron, or gippic, also commonly called the omnitruncated 24-cell, is a convex uniform polychoron that consists of 192 hexagonal prisms and 48 great rhombicuboctahedra. 2 hexagonal prisms and 2 great rhombicuboctahedra join at each vertex. It is the omnitruncate of the F4 family of uniform polychora.

This polychoron can be alternated into a snub tetracontoctachoron, although it cannot be made uniform.

## Vertex coordinates

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

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

## Representations

A great prismatotetracontoctachoron has the following Coxeter diagrams:

• x3x4x3x () (full symmetry)
• xux4wxx3xxx3xwX&#zx (B4 symmetry)

## Semi-uniform variant

The great prismatotetracontoctachoron has a semi-uniform variant of the form x3y4y3x that maintains its full symmetry. This variant uses 48 great rhombicuboctahedra of form y4y3x and 192 ditrigonal prisms of form x x3y as cells, with 2 edge lengths.

With edges of length a and b (so that it is represented by a3b4b3a), its circumradius is given by ${\displaystyle {\sqrt {2a^{2}+6b^{2}+6ab+(a^{2}+4b^{2}+4ab){\sqrt {2}}}}}$.

If it has only single icositetrachoric symmetry, the variant is called a great disprismatoicositetricositetrachoron.