Demihepteract

Demihepteract
(OFF file)
Rank	7
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hesa
Coxeter diagram	x3o3o *b3o3o3o3o ()
Elements
Exa	64 heptapeta, 14 demihexeracts
Peta	448 hexatera, 84 demipenteracts
Tera	1344 pentachora, 280 hexadecachora
Cells	560+2240 tetrahedra
Faces	2240 triangles
Edges	672
Vertices	64
Vertex figure	Rectified heptapeton, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{14}}{4} ≈ 0.93541$
Hypervolume	$\frac{311\sqrt2}{5040} ≈ 0.087266$
Diexal angles	Hax–hix–hop: $\arccos\left(-\frac{\sqrt7}{7}\right) ≈ 112.20765°$
	Hax–hin–hax: 90°
Height	$\frac{\sqrt2}{2} ≈ 0.70711$
Central density	1
Number of pieces	78
Level of complexity	5
Related polytopes
Army	Hesa
Regiment	Hesa
Dual	Semistellated hecatonicosoctaexon
Conjugate	None
Abstract properties
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	D7, order 322560
Convex	Yes
Nature	Tame

The demihepteract, or hesa, also called the hemihepteract or 7-demicube, is a convex uniform polyexon. It has 14 demihexeracts and 64 heptapeta as facets, with 7 of each at a vertex forming a rectified heptapeton as the vertex figure. It is the 7-dimensional demihypercube and is formed by alternating the hepteract. It is also a segmentoexon, as a demihexeractic antiprism.

The demihepteract contains the vertices of a tetrahedral-hexadecachoric duoprism.

Vertex coordinates

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

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

Representations

A demihepteract has the folowing Coxeter diagrams:

• x3o3o *b3o3o3o3o (full symmetry)
• s4o3o3o3o3o3o (as alternated hepteract)
• xo3oo3ox *b3oo3oo3oo&#x (D6 axial, demihexeract antiprism)
• oooo3oxoo3oooo3ooxo3oooo3ooox&#xt (A6 axial, vertex-first)
• oxoo3ooxo xoxo3oooo3oxox *d3oooo&#xt (D4×A2 symmetry, hexadecachoron-first)
• xo3oo3ox *b3oo xo3oo3ox&#zx (D4×A3 symmetry, hull of two tetrahedral-hexadecachoric duoprisms)

External links

• Klitzing, Richard. "hesa".

• Wikipedia Contributors. "7-demicube".