Demihexeract

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)

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.