2₂₁ polypeton

The 2₂₁ polypeton, also known as the 27-72-peton, long Greek icosi­hepta­hebdomeconta­di­peton, OBSA jak, is a uniform polypeton composed of 27 triacontaditera and 72 hexatera. Its vertex figure is a demipenteract. It has a circumradius of √63 ≈ 0.816497. It has the unique property of being the only regular facetted uniform polytope (excluding polygons and simplices) to have an odd number of vertices.

It appears as an exon of the 2₃₁ polyexon as well as as the vertex figure of the 3₂₁ polyexon.

Structure

Beginning from a triacontaditeron, half of its pentachoral tera join with 16 hexatera while the other half joins with 16 other triacontaditera, both of which touch the aequator. This leaves gaps at its edges, filled by 40 hexatera which touch the aequator, and at its vertices, filled by 10 other triacontaditera that span across the whole height of the shape. 16 hexatera are then placed after the first set of triacontaditera, completing the opposite demipenteractic vertex. From this perspective, the shape has a height of √62 ≈ 1.224745.

Beginning from a hexateron, all its pentachoral tera are joined by 6 triacontaditera which touch the aequator. This leaves gaps in the faces, filled by 20 other hexatera which touch the aequator, in the edges, filled by 15 aequatorial triacontaditera, and in the vertices, filled askew by 30 other aequatorial hexatera. The structure mirrors past this point, with 6 triacontaditera placed after the first set and 20 hexatera placed at tetrahedral junctions corresponding to the first set of hexatera. This then only leaves the opposite hexateron in the same orientation as the base, completing the shape. From this perspective, the shape has a height of 1.

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

Source: Incidence matrices — jak