# Demihexeract

(Redirected from 6-demicube)
Demihexeract
Rank6
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHax
Coxeter diagramx3o3o *b3o3o3o ()
Elements
Peta32 hexatera, 12 demipenteracts
Cells160+480 tetrahedra
Faces640 triangles
Edges240
Vertices32
Vertex figureRectified hexateron, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Hypervolume${\displaystyle \frac{43}{360} ≈ 0.11944}$
Dipetal anglesHin–pen–hix: ${\displaystyle \arccos\left(-\frac{\sqrt6}{6}\right) ≈ 114.09484°}$
Hin–hex–hin: 90°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Number of pieces44
Level of complexity4
Related polytopes
ArmyHax
RegimentHax
DualSemistellated hexacontatetrapeton
ConjugateNone
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryD6, order 23040
ConvexYes
NatureTame

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:

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

## 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)

## 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.

o3o3o *b3o3o3o truncations
Name OBSA CD diagram Picture
Demihexeract hax
Birectified hexeract brox
Birectified hexacontatetrapeton brag
Rectified hexacontatetrapeton rag
Hexacontatetrapeton gee
Truncated demihexeract thax
Hexeractihexacontatetrapeton xog
Bitruncated hexacontatetrapeton botag
Truncated hexacontatetrapeton tag
Rectified hexeract rax
Small rhombidemihexeract sirhax
Small birhombated hexacontatetrapeton siborg
Small rhombated hexacontatetrapeton srog
Bitruncated hexeract botox
Great rhombidemihexeract girhax
Great birhombated hexacontatetrapeton gaborg
Great rhombated hexacontatetrapeton grog
Small birhombated hexeract saborx
Great birhombated hexeract gaborx
Small prismatodemihexeract sophax
Small prismated hexacontatetrapeton spog
Prismatotruncatodemihexeract pithax
Prismatorhombated hexacontatetrapeton prog
Prismatorhombidemihexeract prohax
Prismatotruncated hexacontatetrapeton potag
Great prismatodemihexeract gophax
Great prismated hexacontatetrapeton gopog
Small biprismated hexeract sobpoxog
Biprismatorhombated hexacontatetrapeton boprag
Biprismatorhombated hexeract boprax
Great biprismated hexeract gobpoxog
Small cellidemihexeract sochax
Cellitruncatodemihexeract cathix
Cellirhombidemihexeract crohax
Celligreatorhombidemihexeract cagrohax
Celliprismatodemihexeract cophix
Celliprismatotruncatodemihexeract capthix
Celliprismatorhombidemihexeract caprohax
Great cellidemihexeract gochax
Small cellated hexacontatetrapeton scag
Celliprismated hexacontatetrapeton copog
Cellirhombated hexacontatetrapeton crag
Celliprismatorhombated hexacontatetrapeton coprag
Cellitruncated hexacontatetrapeton catog
Celliprismatotruncated hexacontatetrapeton captog
Celligreatorhombated hexacontatetrapeton cagorg
Great cellated hexacontatetrapeton gocog