5-demicube

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5-demicube
Rank5
TypeUniform
Notation
Bowers style acronymHin
Coxeter diagramx3o3o3o *b3o ()
Elements
Tera16 pentachora, 10 hexadecachora
Cells40+80 tetrahedra
Faces160 triangles
Edges80
Vertices16
Vertex figureRectified pentachoron, edge length 1
Measures (edge length 1)
Circumradius
Hypervolume
Diteral anglesHex–tet–pen:
 Hex–tet–hex: 90°
Height
Central density1
Number of external pieces26
Level of complexity3
Related polytopes
ArmyHin
RegimentHin
DualSemistellated triacontaditeron
ConjugateNone
Abstract & topological properties
Flag count5760
Euler characteristic2
OrientableYes
Properties
SymmetryD5, order 1920
ConvexYes
NatureTame

The 5-demicube, also called the hemipenteract, demipenteract or hin, is a convex uniform 5-polytope. 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 5-cube.

This is the lowest-dimensional 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[edit | edit source]

The vertices of a 5-demicube of edge length 1, centered at the origin, are given by all even sign changes of:

  • .

Representations[edit | edit source]

A 5-demicube has the following Coxeter diagrams:

  • x3o3o3o *b3o () (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[edit | edit source]

The 5-demicube is the colonel of a 6-member regiment. Its other members include the dishexadecateron, hexadecapenteract, retrodishexadecateron, retropenteractihexadecateron, and the noble hexadecateron.

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

External links[edit | edit source]