Demihepteract

(Redirected from 7-demicube)
Demihepteract
Rank7
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHesa
Coxeter diagramx3o3o *b3o3o3o3o ()
Elements
Exa64 heptapeta, 14 demihexeracts
Peta448 hexatera, 84 demipenteracts
Cells560+2240 tetrahedra
Faces2240 triangles
Edges672
Vertices64
Vertex figureRectified heptapeton, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{14}}{4} ≈ 0.93541}$
Hypervolume${\displaystyle \frac{311\sqrt2}{5040} ≈ 0.087266}$
Diexal anglesHax–hix–hop: ${\displaystyle \arccos\left(-\frac{\sqrt7}{7}\right) ≈ 112.20765°}$
Hax–hin–hax: 90°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Number of pieces78
Level of complexity5
Related polytopes
ArmyHesa
RegimentHesa
DualSemistellated hecatonicosoctaexon
ConjugateNone
Abstract properties
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryD7, order 322560
ConvexYes
NatureTame

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:

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

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)