4₂₁ polyzetton

The 4₂₁ polyzetton, also known as the 2160-17280-zetton, long Greek dischilia­hecaton­hexeconta­myria­heptachilia­diacos­ogdoëconta­zetton, OBSA fy, is a uniform polyzetton composed of 2160 hecatonicosoctaexa and 576 octaexa. Its vertex figure is a 3₂₁ polyexon. It has a circumradius of 1.

Structure

Beginning from a hecatonicosoctaexon under demihepteratic symmetry, half of its heptapetal exa join to 64 other hecatonicosoctaexa and 64 octaexa. This leaves gaps in the cells, filled by 560 octaexa, in the faces, filled by 280 hecatonicosoctaexa that touch the aequator, in the edges, filled askew by 1344 octaexa that touch the aequator, and in the vertices, filled askew by 2240 octaexa that touch the aequator, askew by 448 octaexa that touch the aequator, askew by 448 hecatonicosoctaexa that straddle the aequator and 14 aequatorial hecatonicosoctaexa. This can be understood from the 3₂₁ polyexal vertex figure. 64 octaexa that straddle the aequator are then placed after the first set of hecatonicosoctaexa. 2240 octaexa are then placed in tetrahedral junctions corresponding to the vertices of a runcic hepteract (oriented closest to the first set of hecatonicosoctaexa) in relation to the base, 560 aequatorial hecatonicosoctaexa are placed in tetrahedral junctions corresponding to the base's tetrahedra and 3360 aequatorial octaexa are placed in edge junctions corresponding to the vertices of a stericated hecatonicosoctaexon. All straddle the aequator. The structure repeats past this point.

448 octaexa are placed in (pre-aequatorial) hexateral junctions, inverted with respect to the last set of hecatonicosoctaexa, in relation to the base, 64 octaexa are placed in vertex junctions after the first set of octaexa, 2240 octaexa are placed in aequatorial triangular junctions corresponding to the vertices of a runcinated hecatonicosoctaexon in relation to the base, 280 hecatonicosoctaexa are placed in aequatorial edge junctions after the first set of 280 hecatonicosoctaexa, 448 hecatonicosoctaexa are placed in aequatorial vertex junctions corresponding to the vertices of a hexic hepteract (oriented closest to the first set of octaexa) in relation to the base, 1344 octaexa are placed in aequatorial vertex junctions corresponding to the vertices of a steric hepteract in relation to the base, 560 octaexa are placed in tetrahedral junctions corresponding to the base's tetrahedra, 2240 octaexa are placed in vertex junctions corresponding to the vertices of a runcic hepteract (oriented closest to the first set of octaexa) in relation to the base and 64 octaexa are placed in vertex junctions corresponding to the first set of hecatonicosoctaexa in relation to the base. Finally, 64 hecatonicosoctaexa are placed after the second last set of 64 octaexa, only leaving room for the opposite hecatonicosoctaexon, completing the shape. From this perspective, the shape has a height of √2 ≈ 1.414214.

Region	Layer	oca	zee
Near side	1	—	1
	2	64	64
	3	560	—
	4	—	280
	5	1344	—
	6	2240	—
	7	448	448
	8	64	—
	9	2240	—
Aequator		3360	14+560
Far side	1	2240	—
	2	64	—
	3	448	448
	4	2240	—
	5	1344	—
	6	—	280
	7	560	—
	8	64	64
	9	—	1
Grand total		17280	2160
		19440 zetta

Due to missing sources, its octaexon-first structure is unavailable, although it can be inferred it has a height of 32 = 1.5.

Beginning from a 3₂₁ polyexal vertex, there are 126 hecatonicosoctaexa that touch the aequator and 576 octaexa. 576 hecatonicosoctaexa that touch the aequator are then placed after the last set of octaexa. 4032 octaexa are then placed in triangular junctions corresponding to the base's pentachora and likewise 2016 octaexa are placed in hexateral junctions corresponding to the edges of a 2₃₁ polyexon in relation to the base, both touching the aequator. 756 aequatorial hecatonicosoctaexa are then placed in edge junctions corresponding to the base's edges and 4032 aequatorial octaexa are placed askew in vertex junctions corresponding to the vertices of a demified 3₂₁ polyexon in relation to the base.

The structure mirrors past this point: 576 hecatonicosoctaexa are placed after the second set of hecatonicosoctaexa, 2016 octaexa are placed in aequatorial edge junctions after the set of 2016 octaexa and 4032 octaexa are placed in aequatorial pentachoral junctions after the first set of 4032 octaexa. Finally, 126 hecatonicosoctaexa are placed after in vertex junctions corresponding to the vertices of a 2₃₁ polyexon in relation to the base and 576 octaexa are placed after the previous set of 576 hecatonicosoctaexa, completing the opposite vertex. From this perspective, the shape has a height of 2.

Region	Layer	oca	zee
Near side	1	576	126
	2	2016+4032	576
Aequator		4032	756
Far side	1	2016+4032	576
	2	576	126
Grand total		17280	2160
		19440 zetta

As for global vertex structure, the shape is the convex hull of a demiocteract and a rectified diacosipentecontahexazetton as well as the convex hull of two mutually inverted birectified enneazetta and a small exioctadecazetton.

Having the same symmetry as the 1₄₂ polyzetton and 2₄₁ polyzetton, both with E₈ symmetry, their components have neat correspondences:

Component	142	241	421
E7	lin	laq	pt [naq]
D7	hesa	pt [inv. hesa]	zee
A7	pt [broc]	oca	inv. oca

Source: Incidence matrices — fy