1₃₂ polyexon

The 1₃₂ polyexon, also known as the 56-126-exon, long Greek penteconta­hexa­hecaton­icosi­hexa­exon, OBSA lin, is a uniform polyexon composed of 56 1₂₂ polypeta and 126 demihexeracts. Its vertex figure is a birectified heptapeton. It has a circumradius of √72 ≈ 1.322876.

It appears as a zetton of the 1₄₂ polyzetton.

Structure

Beginning on a 1₂₂ polypeton, half of its demipenteracts join with 27 demihexeracts while the other half joins with 27 other 1₂₂ polypeta that straddle the aequator. This leaves room at the vertices for 72 aequatorial demihexeracts that span the entire height of the shape. Then, 27 demihexeracts are placed after the previous 1₂₂ polypeta and likewise 27 1₂₂ polypeta are placed after the previous demihexeracts. This then only leaves the opposite 1₂₂ polypeton, completing the shape. From this perspective, the shape has a height of √3 ≈ 1.732051.

Region	Layer	hax	mo
Near side	1	—	1
	2	27	27
Aequator		72	—
Far side	1	27	27
	2	—	1
Grand total		126	56
		182 exa

Beginning on a demihexeract, its hexateral peta join with 32 other demihexeracts, while its demipenteractic peta join with 12 1₂₂ polypeta, both of which touch the aequator. This then leaves room for an additional 32 aequatorial 1₂₂ polypeta that span the entire height of the shape at every vertex. 60 aequatorial demihexeracts are then placed at hexadecachoral junctions corresponding to the base's hexadecachora. The structure mirrors past this point: 12 1₂₂ polypeta are placed after the first set of 1₂₂ polypeta and 32 demihexeracts are placed after the first set of demihexeracts. This then only leaves room for the opposite demihexeract, which is in the same orientation as the base, completing the shape. From this perspective, the shape has a height of 2.

Region	Layer	hax	mo
Near side	1	1	—
	2	32	12
Aequator		60	32
Far side	1	32	12
	2	1	—
Grand total		126	56
		182 exa

Due to missing sources, its vertex-first structure is unavailable, although it can be inferred it has a height of √7 ≈ 2.645751.

As for global vertex structure, the shape is the convex hull of two mutually inverted octaexa scaled by 2 and two mutually inverted stericated octaexa.

Having the same symmetry as the 2₃₁ polyexon and 3₂₁ polyexon, both with E₇ symmetry, their components have neat correspondences:

Component	132	231	321
E6	mo	jak	pt [inv. jak]
D6	hax	pt [inv. hax]	gee
A6	pt [bril]	hop	inv. hop

Source: Incidence matrices — lin