Demihexeract

Demihexeract
(OFF file)
Rank	6
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hax
Coxeter diagram	x3o3o *b3o3o3o ()
Elements
Peta	32 hexatera, 12 demipenteracts
Tera	192 pentachora, 60 hexadecachora
Cells	160+480 tetrahedra
Faces	640 triangles
Edges	240
Vertices	32
Vertex figure	Rectified hexateron, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Hypervolume	$\frac{43}{360} ≈ 0.11944$
Dipetal angles	Hin–pen–hix: $\arccos\left(-\frac{\sqrt6}{6}\right) ≈ 114.09484°$
	Hin–hex–hin: 90°
Height	$\frac{\sqrt2}{2} ≈ 0.70711$
Central density	1
Number of pieces	44
Level of complexity	4
Related polytopes
Army	Hax
Regiment	Hax
Dual	Semistellated hexacontatetrapeton
Conjugate	None
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	D6, order 23040
Convex	Yes
Nature	Tame

The demihexeract, or hax, also called the hemihexeract or 6-demicube, is a convex uniform polypeton. It has 12 demipenteracts and 32 hexatera as facets, with 6 of each at a vertex forming a rectified hexateron as the vertex figure. It is the 6-dimensional demihypercube and is formed by alternating the hexeract.

It is a segmentopeton, as a demipenteractic antiprism. It is also a tetrahedral duoantiprism and digonal trioantiprism.

The demihexeract contains the vertices of a tetrahedral duoprism, in fact being the convex hull of 2 tetrahedral duoprisms, one of which has bases in dual orientation compared to the other.

Vertex coordinates

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

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

Representations

A demihexeract has the following Coxeter diagrams:

• x3o3o *b3o3o3o (full symmetry)
• s4o3o3o3o3o () (as alternated hexeract)
• xo3oo3ox *b3oo3oo&#x (D₅ axial, demipenteract antiprism)
• xoo3ooo3oxo3ooo3oox&#xt (A₅ axial, hexateron-first)
• oooo3oxoo3oooo3ooxo3oooo&#x (A₅ axial, vertex-first)
• oxo xox3ooo3ooo *c3oxo&#xt (D₄×A₁ axial, hexadecachoron-first)
• xo3oo3ox *b3oo xo ox&#zx (D₄×A₁×A₁ symmetry)
• oxoo3ooxo xoxo3oooo3oxox&#xt (A₃×A₂ axial, tetrahedron-first)
• xo3oo3ox xo3oo3ox&#zx (A₃×A₃ symmetry, hull of two tetrahedral duoprisms)

Gallery

• 

  BC₆ orthographic projection

• 

  D₅

• 

  D₄

• 

  D₃

• 

  A₅

• 

  A₃

Related polytopes

The regiment of the demihexeract contains a total of 145 members, 66 fissaries, and 18 compounds. of these, 15 of the main members and one fissary has full D6 symmetry, all the others were only found in 2019 and have various subsymmetries.

External links

• Bowers, Jonathan. "Category 1: Primary Polypeta" (#9).

• Klitzing, Richard. "hax".

• Wikipedia Contributors. "6-demicube".