# Great prismatodemipenteract

Great prismatodemipenteract
Rank5
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGiphin
Coxeter diagramx3x3x3x *b3o ()
Elements
Tera40 truncated tetrahedral prisms, 16 great rhombated pentachora, 10 tesseractihexadecachora, 16 great prismatodecachora
Cells160 triangular prisms, 80+160 hexagonal prisms, 80+80 truncated tetrahedra, 80+80 truncated octahedra
Faces320 triangles, 240+240+480 squares, 160+320+320 hexagons
Edges480+480+480+960
Vertices960
Vertex figureSphenoidal pyramid, edge lengths 1 (1), 2(4), and 3 (5)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{170}}{4} \approx 3.25960}$
Hypervolume${\displaystyle \frac{86647\sqrt2}{120} \approx 1021.14469}$
Diteral anglesGrip–trip–tuttip: ${\displaystyle \arccos\left(-\frac{\sqrt{15}}{5}\right) \approx 140.76848^\circ}$
Tah–tut–tuttip: 135°
Gippid–hip–tuttip: ${\displaystyle \arccos\left(-\frac{\sqrt{10}}{5}\right) \approx 129.23152^\circ}$
Grip–toe–gippid: ${\displaystyle \arccos\left(-\frac35\right) \approx 126.86990^\circ}$
Tah–toe–grip: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) \approx 116.56505^\circ}$
Tah–tut–grip: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) \approx 116.56505^\circ}$
Gippid–hip–gippid: ${\displaystyle \arccos\left(-\frac15\right) \approx 101.53696^\circ}$
Central density1
Number of external pieces82
Level of complexity60
Related polytopes
ArmyGiphin
RegimentGiphin
DualSphenoidal-pyramidal enneacosihexecontateron
ConjugateNone
Abstract & topological properties
Flag count115200
Euler characteristic2
OrientableYes
Properties
SymmetryD5, order 1920
ConvexYes
NatureTame

The great prismatodemipenteract, or giphin, also called the steriruncicantic 5-cube, is a convex uniform polyteron. It consists of 40 truncated tetrahedral prisms, 16 great rhombated pentachora, 10 tesseractihexadecachora, and 16 great prismatodecachora. 2 great prismatodecachora, 1 great rhombated pentachoron, 1 tesseractihexadecachoron, and 1 trunctaed tetrahedral prism join at each vertex. It can be formed from an alternated faceting of the celligreatorhombated triacontaditeron.

## Vertex coordinates

The vertices of a great prismatodemipenteract of edge length 1 are given by all permutations and even sign changes of:

• ${\displaystyle \left(\frac{7\sqrt2}{4},\,\frac{5\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right)}$.

## Representations

A great prismatodemipenteract has the following Coxeter diagrams:

• x3x3x3x *b3o () (full symmetry)
• s4o3x3x3x () (as alternated faceting)

## Related polytopes

o3o3o *b3o3o truncations
Name OBSA CD diagram Picture
Demipenteract hin
Truncated demipenteract thin
Rectified penteract rin
Small rhombidemipenteract sirhin