2₃₁ polyexon

The 2₃₁ polyexon, also known as the 56-576-exon, long Greek penteconta­hexa­pentacosi­hebdomeconta­hexa­exon, OBSA laq, is a uniform polyexon composed of 56 2₂₁ polypeta and 576 heptapeta. Its vertex figure is a demihexeract. It has a circumradius of 1.

It appears as a zetton of the 2₄₁ polyzetton.

Structure

Beginning on a 2₂₁ polypetal exon, the triacontaditeral peta join 27 other 2₂₁ polypeta that span the entire height of the shape, while the hexateral peta join 72 heptapeta. This leaves gaps in the edges, filled by 216 additional heptapeta, and in the vertices, filled by 27 additional 2₂₁ polypeta that span the entire height of the shape. This can be visualized with its demihexeractic vertex figure. The structure repeats past this point: 216 heptapeta are placed in pentachoral junctions corresponding to the base's pentachora, leaving only the opposite 2₂₁ polypeton, which is inverted in relation to the base. The far side of the near set of 216 pentachora is aligned with the opposite 2₂₁ polypeton's pentachora that are opposite an edge. From this perspective, the shape has a height of 2√33 ≈ 1.154701.

Region	Layer	hop	jak
Near side	1	—	1
	2	72	27
	3	216	—
Far side	1	216	—
	2	72	27
	3	—	1
Grand total		576	56
		632 exa

Beginning on a heptapeton, all hexateral peta join to 7 2₂₁ polypeta which touch the aequator. This leaves gaps in the faces, filled by 35 heptapeta, in the edges, filled by 21 2₂₁ polypeta which straddle the aequator, and in the vertices, filled by askew 105 heptahepta and 7 heptapeta, both touching the aequator. This can be understood from the hexateron-first demihexeractic vertex figure. 140 heptapeta of two types are then added to junctions corresponding to the base's triangular faces and tetrahedral cells. The structure repeats past this point: 7 2₂₁ polypeta are placed after the initial set, 7 heptapeta are placed after the previous set of 7 heptapeta, askew 105 heptapeta are placed in aequatorial edges corresponding to vertices of a stericated heptapeton in relation to the base, 21 2₂₁ polypeta are placed in pentachoral junctions corresponding to the base's pentachora and 35 heptapeta are placed in vertex junctions corresponding to the base's tetrahedra. This then only leaves room for the opposite heptapeton in inverted orientation, completing the shape. From this perspective, the shape has a height of 4√77 ≈ 1.511858.

Region	Layer	hop	jak
Near side	1	1	—
	2	—	7
	3	35	—
	4	105	21
	5	7	—
Aequator		140+140	—
Far side	1	7	—
	2	105	21
	3	35	—
	4	—	7
	5	1	—
Grand total		576	56
		632 exa

Beginning on a demihexeractic vertex, there are 12 2₁₂ polypeta that touch the aequator and 32 heptapeta. This leaves gaps for 160 heptapeta that touch the aequator to inserted into tetrahedral junctions joining 3 2₂₁ polypeta and 1 heptapeton. This still leaves gaps for 192 aequatorial heptapeta to be placed at vertices corresponding to a pentic hexeract in relation to the base. 32 aequatorial 2₂₁ polypeta are then placed after the near heptapeta. The structure then repeats: 160 heptapeta are then placed opposite the aequatorial triangular faces of the second set of heptapeta, and finally 12 2₂₁ polypeta are placed after the near set and 32 heptapeta are placed after the aequatorial 2₂₁ polypeta, completing the opposite vertex. From this perspective, the shape has a height of 2.

Region	Layer	hop	jak
Near side	1	32	12
	2	160	—
Aequator		192	32
Far side	1	160	—
	2	32	12
Grand total		576	56
		632 exa

As for global vertex structure, the shape is the convex hull of a hexadecaexon and a small petihexadecaexon.

Having the same symmetry as the 1₃₂ 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 — laq