3₂₁ polyexon

The 3₂₁ polyexon, also known as the 126-576-exon, long Greek hecaton­icosi­hexa­pentacosi­hebdomeconta­hexa­exon, OBSA naq, is a uniform polyexon composed of 126 hexecontatetrapeta and 576 heptapeta. Its vertex figure is a 2₂₁ polypeton. It has a circumradius of √32 ≈ 0.866025.

It appears as the vertex figure of the 4₂₁ polyzetton.

Structure

Beginning on a hexecontatetrapetal exon under demihexeractic symmetry, half of its hexateral peta are joined to 32 heptapeta and the other half to 32 other hexecontatetrapeta, both touching the aequator. This leaves gaps at the faces, filled by 160 heptapeta that touch the aequator, at the edges, filled by 60 aequatorial hexecontatetrapeta, and at the vertices, filled by 192 aequatorial heptapeta, the last two spanning across the entire height of the shape. The strucure is then mirrored: 32 hexecontatetrapeta are placed after the first set of hexecontatetrapeta, 160 heptapeta are added at aequatorial tetrahedral junctions after the previous set of 160 heptapeta and 32 heptapeta are placed at aequatorial vertex junctions after the first set of heptapeta, leaving only the opposite hexecontatetrapeton, completing the shape. From this perspective, the shape has a height of 1.

Region	Layer	hop	gee
Near side	1	—	1
	2	32	32
	3	160	—
Aequator		192	60
Far side	1	160	—
	2	32	32
	3	—	1
Grand total		576	126
		702 exa

Beginning on a heptapetal exon, its hexateral peta join with 7 hexecontatetrapeta. This leaves gaps in the cells, filled by 35 heptapeta, in the faces, filled by 35 hexecontatetrapeta which straddle the aequator, in the edges, filled by askew 105 heptapeta which straddle the aequator, and in the vertices, filled by askew 140 heptapeta which straddle the aequator, askew 42 aequatorial hexecontatetrapeta that span the entire height of the shape and 7 heptapeta which straddle the aequator. The structure repeats past this point: 7 hexecontatetrapeta are placed after the previous set of heptapeta and 7 heptapeta are placed after the first set of hexecontatetrapeta. 35 heptapeta are then placed in triangular junctions after the second set of hexecontatetrapeta, and likewise 35 hexecontatetrapeta are placed after the second set of heptapeta. 105 heptapeta and 145 heptapeta are placed in triangular junctions related to their corresponding near set's positions but inverted. This only leaves room for the opposite heptapeton in inverted orientation, completing the shape. From this perspective, the shape has a height of 3√77 ≈ 1.133893.

Region	Layer	hop	gee
Near side	1	1	—
	2	—	7
	3	35	—
	4	—	35
	5	105	—
	6	140	—
	7	—	21
	8	7	—
Far side	1	7	—
	2	—	21
	3	140	—
	4	105	—
	5	—	35
	6	35	—
	7	—	7
	8	1	—
Grand total		576	126
		702 exa

Beginning on a 2₂₁ polypetal vertex, there are 27 hexecontatetrapeta and 72 heptapeta. 72 aequatorial hexecontatetrapeta are then placed after every heptapeton and 216 heptapeta are placed at pentachoral junctions between two hexecontatetrapeta, same for edges opposite from the base vertex. The process is then mirrored, 27 hexecontatetrapeta are placed after every near hexecontatetrapeton and 72 heptapeta are placed after the aequatorial hexecontatetrapeta, completing the opposite vertex. From this perspective, the shape has a height of √3 ≈ 1.732051.

Region	Layer	hop	gee
Near side	1	72	27
	2	216	—
Aequator		—	72
Far side	1	216	—
	2	72	27
Grand total		576	126
		702 exa

As for global vertex structure, the shape is also the convex hull of two mutually inverted rectified octaexa.

Having the same symmetry as the 1₃₂ polyexon and 2₃₁ 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 — naq