Demihexeract

The demihexeract, also known by its OBSA hax, Coxeter symbol 1₃₁, is the alternation of the hexeract, composed of 12 demipenteracts and 32 hexatera. Its vertex figure is a rectified hexateron. It has a circumradius of √32 ≈ 0.866025. Unlike the demipenteract, it has a simplicial facet opposite another simplicial facet and a vertex opposite another vertex, just like any other even demicube.

It appears as an exon of the demihepteract and the 1₃₂ polyexon as well as the vertex figure of the 2₃₁ polyexon.

Structure

Beginning from a demipenteract, its hexadecachoral tera join to 10 other aequatorial demipenteracts, while its pentachoral tera join to 16 hexatera. This leaves gaps in the vertices filled by 16 hexatera. This then only leaves the opposite demipenteract in inverse orientation, completing the shape. All the non-base peta span the entire height of the shape. This opposite demipenteract has its pentachoral tera joined to the second set of hexatera, while its vertices meet the first set. From this perspective, the shape has a height of √22 ≈ 0.707107.

Beginning from a hexateron, all its pentachoral tera join to 6 demipenteracts that span the entire height of the shape. This leaves gaps for 15 more hexatera to be inserted at its edges and 6 more demipenteracts that also span the height of the shape to be inserted at its vertices. 15 hexatera are then inserted in aequatorial tetrahedral junctions corresponding to the base's tetrahedral cells. This then only leaves the opposite hexateron in inverse orientation, completing the shape. From this perspective, the shape has a height of 2√33 ≈ 1.154701.

Beginning from a rectified hexateral vertex, 6 hexatera and 6 demipenteracts are joined. 20 aequatorial hexatera are then inserted into triangular junctions corresponding to the vertex's triangular faces between two rectified pentachora. This finally leaves room for 6 hexatera and 6 demipenteracts, forming an inverted form of the base vertex cap, completing the shape. From this perspective, the shape has a height of √3 ≈ 1.732051.

Source: Incidence matrices — hax