# Great prismatodecachoron

Great prismatodecachoron
Rank4
TypeUniform
Notation
Bowers style acronymGippid
Coxeter diagramx3x3x3x ()
Elements
Cells20 hexagonal prisms, 10 truncated octahedra
Faces30+60 squares, 20+40 hexagons
Edges120+120
Vertices120
Vertex figurePhyllic disphenoid, edge lengths 2 (3) and 3 (3)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {5}}\approx 2.23607}$
Hypervolume${\displaystyle {\frac {125{\sqrt {5}}}{4}}\approx 69.87712}$
Dichoral anglesHip–4–hip: ${\displaystyle \arccos \left(-{\tfrac {2}{3}}\right)\approx 131.81032^{\circ }}$
Toe–6–hip: ${\displaystyle \arccos \left(-{\tfrac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }}$
Toe–4–hip: ${\displaystyle \arccos \left(-{\tfrac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }}$
Toe–6–toe: ${\displaystyle \arccos \left(-{\tfrac {1}{4}}\right)\approx 104.47751^{\circ }}$
Central density1
Number of external pieces30
Level of complexity12
Related polytopes
ArmyGippid
RegimentGippid
DualDisphenoidal hecatonicosachoron
ConjugateNone
Abstract & topological properties
Flag count2880
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexYes
NatureTame

The great prismatodecachoron, or gippid, also commonly called the omnitruncated 5-cell or omnitruncated pentachoron, is a convex uniform polychoron that consists of 20 hexagonal prisms and 10 truncated octahedra. 2 hexagonal prisms and 2 truncated octahedra join at each vertex. It is the omnitruncate of the A4 family of uniform polychora.

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

Like the omnitruncated simplex of any dimension, this polychoron can tile 4D space. The resulting tetracomb is called the omnitruncated cyclopentachoric tetracomb.

It is the 5th-order permutohedron.

## Vertex coordinates

The vertices of a great prismatodecachoron of edge length 1 are given by the following points:

• ${\displaystyle \pm \left(0,\,{\tfrac {\sqrt {6}}{3}},\,-{\tfrac {\sqrt {3}}{3}},\,\pm 2\right)}$,
• ${\displaystyle \pm \left(0,\,{\tfrac {\sqrt {6}}{3}},\,-{\tfrac {5{\sqrt {3}}}{6}},\,\pm {\tfrac {3}{2}}\right)}$,
• ${\displaystyle \pm \left(0,\,{\tfrac {\sqrt {6}}{3}},\,{\tfrac {7{\sqrt {3}}}{6}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left(0,\,{\tfrac {2{\sqrt {6}}}{3}},\,-{\tfrac {\sqrt {3}}{6}},\,\pm {\tfrac {3}{2}}\right)}$,
• ${\displaystyle \pm \left(0,\,{\tfrac {2{\sqrt {6}}}{3}},\,-{\tfrac {2{\sqrt {3}}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left(0,\,{\tfrac {2{\sqrt {6}}}{3}},\,{\tfrac {5{\sqrt {3}}}{6}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left(\pm {\tfrac {\sqrt {10}}{2}},\,{\tfrac {\sqrt {6}}{6}},\,-{\tfrac {\sqrt {3}}{6}},\,\pm {\tfrac {3}{2}}\right)}$,
• ${\displaystyle \pm \left(\pm {\tfrac {\sqrt {10}}{2}},\,{\tfrac {\sqrt {6}}{6}},\,-{\tfrac {2{\sqrt {3}}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left(\pm {\tfrac {\sqrt {10}}{2}},\,{\tfrac {\sqrt {6}}{6}},\,{\tfrac {5{\sqrt {3}}}{6}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left(\pm {\tfrac {\sqrt {10}}{2}},\,\pm {\tfrac {\sqrt {6}}{2}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \left(\pm {\tfrac {\sqrt {10}}{2}},\,\pm {\tfrac {\sqrt {6}}{2}},\,\pm {\tfrac {\sqrt {3}}{2}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,{\tfrac {\sqrt {6}}{12}},\,-{\tfrac {\sqrt {3}}{3}},\,\pm 2\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,{\tfrac {\sqrt {6}}{12}},\,-{\tfrac {5{\sqrt {3}}}{6}},\,\pm {\tfrac {3}{2}}\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,{\tfrac {\sqrt {6}}{12}},\,{\tfrac {7{\sqrt {3}}}{6}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,-{\tfrac {\sqrt {6}}{4}},\,0,\,\pm 2\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,-{\tfrac {\sqrt {6}}{4}},\,\pm {\sqrt {3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,-{\tfrac {7{\sqrt {6}}}{12}},\,-{\tfrac {\sqrt {3}}{6}},\,\pm {\tfrac {3}{2}}\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,-{\tfrac {7{\sqrt {6}}}{12}},\,-{\tfrac {2{\sqrt {3}}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,-{\tfrac {7{\sqrt {6}}}{12}},\,{\tfrac {5{\sqrt {3}}}{6}},\,\pm {\tfrac {1}{2}}\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,{\tfrac {3{\sqrt {6}}}{4}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\tfrac {\sqrt {10}}{4}},\,{\tfrac {3{\sqrt {6}}}{4}},\,\pm {\tfrac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$.

Much simpler coordinates can be given in five dimensions, as all permutations of:

• ${\displaystyle \left(2{\sqrt {2}},\,{\tfrac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\tfrac {\sqrt {2}}{2}},\,0\right)}$.

## Representations

A great prismatodecachoron has the following Coxeter diagrams:

• x3x3x3x () (full symmetry)
• xxxux3xxuxx3xuxxx&#xt (A3 axial, truncated octahedron-first)

## Semi-uniform variant

The great prismatodecachoron has a semi-uniform variant of the form x3y3y3x that maintains its full symmetry. This variant uses 10 great rhombitetratetrahedra of form x3y3y and 20 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 a3b3b3a), its circumradius is given by ${\displaystyle {\sqrt {a^{2}+2b^{2}+2ab}}}$.

If it has only single pentachoric symmetry, the variant is called a great disprismatopentapentachoron.

## Related polychora

The antifrustary prismatohexacosichoron is a uniform polychoron compound composed of 60 great prismatodecachora.