# Demipenteract

(Redirected from 5-demicube)
Demipenteract
Rank5
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHin
Coxeter diagramx3o3o *b3o3o ()
Elements
Cells40+80 tetrahedra
Faces160 triangles
Edges80
Vertices16
Vertex figureRectified pentachoron, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{10}}{4} ≈ 0.79057}$
Hypervolume${\displaystyle \frac{13\sqrt2}{120} ≈ 0.15321}$
Diteral anglesHex–tet–pen: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ}$
Hex–tet–hex: 90°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Number of pieces26
Level of complexity3
Related polytopes
ArmyHin
RegimentHin
ConjugateNone
Abstract properties
Flag count5760
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryD5, order 1920
ConvexYes
NatureTame

The demipenteract, or hin, also called the hemipenteract or 5-demicube, is a convex uniform polyteron. It has 10 hexadecachora and 16 pentachora as facets, with 5 of each at a vertex forming a rectified pentachoron as the vertex figure. It is the 5-dimensional demihypercube and is formed by alternating the penteract.

This is the lowest-dimesional demihypercube that is not identical to another regular polytope of the same dimension.

It is also a convex segmentoteron, as a hexadecachoron atop rotated hexadecachoron, or hexadecachoric alterprism. It is also the digonal-tetrahedral duoantiprism.

## Vertex coordinates

The vertices of a demipenteract of edge length 1, centered at the origin, are given by all even sign changes of:

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

## Representations

A demipenteract has the following Coxeter diagrams:

• x3o3o *b3o3o (full symmetry)
• s4o3o3o3o (as alternated penteract)
• xo3oo3ox *b3oo&#x (D4 axial, demitesseractic alterprism)
• ooo3oxo3ooo3oox&#xt (A4 axial, vertex-first)
• xox3ooo3oxo oxo&#xt (A3×A1 axial, tetrahedron-first)
• xo3oo3ox xo ox&#zx (A3×A1×A1 symmetry)
• xoxo oxox oxoo3ooxo&#xt (A2×A1×A1 axial, edge-first)

## Related polytopes

The hollow stellated icositetrachoric antiprism can be constructed as a blend of 6 demipenteracts.

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