1₂₂ polypeton

The 1₂₂ polypeton, also known as the 54-peton, long Greek penteconta­tetra­peton, OBSA mo, is a uniform polypeton composed of 54 demipenteracts. Its vertex figure is a dodecateron. It has a circumradius of 1. Since it is isogonal and isotopal, it is considered noble. It is the simplest Gosset–Elte figure by facet count.

It appears as an exon of the 1₃₂ polyexon.

Structure

Beginning from a demipenteract, its hexadecachoral tera join to 10 other demipenteracts which touch the aequator, while its pentachoral tera join to 16 other demipenteracts which span the height of the shape. This leaves gaps at the vertices, filled by 16 demipenteracts which also span the height of the shape. The aequator is shaped like a rectified triacontaditeron, with the hexadecachora representing the demipenteracts in contact with the aequator and the rectified pentachora representing vertex slices of demipenteracts. 10 demipenteracts are then placed after the first set of demipenteracts, forming a cross-like shape when unfolded. This then only leaves the opposite demipenteract in inverted orientation, completing the shape. This opposite demipenteract has its pentachora joining the second set of 16 demipenteracts, with its vertices meeting the first set of 16 demipenteracts. It has a very hexagonal peton structure, with a cycle of opposite demipenteracts being a hexagon. From this perspective, the shape has a height of √62 ≈ 1.224745.

Beginning from a dodecateral vertex, there are 12 demipenteracts. 30 aequatorial demipenteracts are then inserted into tetrahedral junctions which join two hexadecachora. The aequatorial demipenteracts are positioned in the vertices of a small cellidodecateron. This then only leaves the opposite cap containing 12 demipenteracts, completing the shape. From this perspective, the shape has a height of 2.

As for global vertex structure, the shape is the convex hull of two fully mutually inverted (triple rotated 180°) triangular trioprisms and three mutually perpendicular hexagons (as one hexagonal triotegum).

Having the same symmetry as the 2₂₁ polypeton and the rectified 1₂₂ polypeton, both with E₆ symmetry, their components have neat correspondences:

Source: Incidence matrices — mo