6demicube
6demicube  

Rank  6 
Type  Uniform 
Notation  
Bowers style acronym  Hax 
Coxeter diagram  x3o3o3o3o *b3o () 
Elements  
Peta  32 5simplices, 12 5demicubes 
Tera  192 pentachora, 60 hexadecachora 
Cells  160+480 tetrahedra 
Faces  640 triangles 
Edges  240 
Vertices  32 
Vertex figure  Rectified 5simplex, edge length 1 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dipetal angles  Hin–pen–hix: 
Hin–hex–hin: 90°  
Height  
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  D_{6}, order 23040 
Flag orbits  4 
Convex  Yes 
Nature  Tame 
The 6demicube, also called the hemihexeract, demihexeract, or hax, is a convex uniform 6polytope. It has 12 5demicubes and 32 5simplices as facets, with 6 of each at a vertex forming a rectified 5simplex as the vertex figure. It is the 6dimensional demihypercube and is formed by alternating the 6cube.
It is a segmentopeton, as a 5demicubic antiprism. It is also a tetrahedral duoantiprism and digonal trioantiprism.
The 6demicube 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[edit  edit source]
The vertices of a 6demicube of edge length 1, centered at the origin, are given by all even sign changes of:
Representations[edit  edit source]
A 6demicube has the following Coxeter diagrams:
 x3o3o3o3o *b3o () (full symmetry)
 s4o3o3o3o3o () (as alternated 6cube)
 xo3oo3ox *b3oo3oo&#x (D_{5} axial, 5demicube antiprism)
 xoo3ooo3oxo3ooo3oox&#xt (A_{5} axial, 5simplexfirst)
 oooo3oxoo3oooo3ooxo3oooo&#x (A_{5} axial, vertexfirst)
 oxo xox3ooo3ooo *c3oxo&#xt (D_{4}×A_{1} axial, hexadecachoronfirst)
 xo3oo3ox *b3oo xo ox&#zx (D_{4}×K_{2} symmetry)
 oxoo3ooxo xoxo3oooo3oxox&#xt (A_{3}×A_{2} axial, tetrahedronfirst)
 xo3oo3ox xo3oo3ox&#zx (A_{3}×A_{3} symmetry, hull of two tetrahedral duoprisms)
Gallery[edit  edit source]

B_{6} orthographic projection

D_{5}

D_{4}

D_{3}

A_{5}

A_{3}
Related polytopes[edit  edit source]
The regiment of the 6demicube contains a total of 145 members, 66 fissaries, and 18 compounds. of these, 15 of the main members and one fissary has full D_{6} symmetry, all the others were only found in 2019 and have various subsymmetries.
External links[edit  edit source]
 Bowers, Jonathan. "Category 1: Primary Polypeta" (#9).
 Klitzing, Richard. "hax".
 Wikipedia contributors. "6demicube".