Demipenteract

Demipenteract
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hin
Coxeter diagram	x3o3o *b3o3o ()
Elements
Tera	16 pentachora, 10 hexadecachora
Cells	40+80 tetrahedra
Faces	160 triangles
Edges	80
Vertices	16
Vertex figure	Rectified pentachoron, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{10}}{4} ≈ 0.79057$
Hypervolume	$\frac{13\sqrt2}{120} ≈ 0.15321$
Diteral angles	Hex–tet–pen: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ$
	Hex–tet–hex: 90°
Height	$\frac{\sqrt2}{2} ≈ 0.70711$
Central density	1
Number of pieces	26
Level of complexity	3
Related polytopes
Army	Hin
Regiment	Hin
Dual	Semistellated triacontaditeron
Conjugate	None
Abstract properties
Flag count	5760
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	D5, order 1920
Convex	Yes
Nature	Tame

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:

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

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 demipenteract 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

• Bowers, Jonathan. "Category 1: Primary Polytera" (#7).

• Klitzing, Richard. "hin".

• Wikipedia Contributors. "5-demicube".