6-demicube

6-demicube
Rank	6
Type	Uniform
Notation
Bowers style acronym	Hax
Coxeter diagram	x3o3o3o3o *b3o ()
Elements
Peta	32 5-simplices, 12 5-demicubes
Tera	192 pentachora, 60 hexadecachora
Cells	160+480 tetrahedra
Faces	640 triangles
Edges	240
Vertices	32
Vertex figure	Rectified 5-simplex, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Hypervolume	${\frac {43}{360}}\approx 0.11944$
Dipetal angles	Hin–pen–hix: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Hin–hex–hin: 90°
Height	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	44
Level of complexity	4
Related polytopes
Army	Hax
Regiment	Hax
Dual	Semistellated hexacontatetrapeton
Conjugate	None
Abstract & topological properties
Flag count	92160
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D6, order 23040
Flag orbits	4
Convex	Yes
Nature	Tame

The 6-demicube, also called the hemihexeract, demihexeract, or hax, is a convex uniform 6-polytope. It has 12 5-demicubes and 32 5-simplices as facets, with 6 of each at a vertex forming a rectified 5-simplex as the vertex figure. It is the 6-dimensional demihypercube and is formed by alternating the 6-cube.

It is a segmentopeton, as a 5-demicubic antiprism. It is also a tetrahedral duoantiprism and digonal trioantiprism.

The 6-demicube 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 6-demicube of edge length 1, centered at the origin, are given by all even sign changes of:

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right).$

Representations

A 6-demicube has the following Coxeter diagrams:

• x3o3o3o3o *b3o () (full symmetry)
• s4o3o3o3o3o () (as alternated 6-cube)
• xo3oo3ox *b3oo3oo&#x (D₅ axial, 5-demicube antiprism)
• xoo3ooo3oxo3ooo3oox&#xt (A₅ axial, 5-simplex-first)
• oooo3oxoo3oooo3ooxo3oooo&#x (A₅ axial, vertex-first)
• oxo xox3ooo3ooo *c3oxo&#xt (D₄×A₁ axial, hexadecachoron-first)
• xo3oo3ox *b3oo xo ox&#zx (D₄×K₂ symmetry)
• oxoo3ooxo xoxo3oooo3oxox&#xt (A₃×A₂ axial, tetrahedron-first)
• xo3oo3ox xo3oo3ox&#zx (A₃×A₃ symmetry, hull of two tetrahedral duoprisms)

Gallery

• 
  B₆ orthographic projection
• 
  D₅
• 
  D₄
• 
  D₃
• 
  A₅
• 
  A₃

Related polytopes

The regiment of the 6-demicube 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".