Rectified 1₂₂ polypeton

The rectified 1₂₂ polypeton, also known by its OBSA ram, or as I like to call it, the 54-72-peton, long Greek penteconta­tetra­hebdomeconta­di­peton, is a uniform polypeton that is the rectification of the 1₂₂ polypeton, composed of 54 penteractitriacontaditera and 72 dodecatera. Its vertex figure is a triangular duoprismatic prism. It has a circumradius of √3 ≈ 1.732051.

Structure

Beginning from a penteractitriacontaditeron, its icositetrachoral tera join to 10 other penteractitriacontaditera that touch the aequator, while half of its rectified pentachoral tera join to 16 other penteractitriacontaditera that straddle the aequator and the other half joins to 16 dodecatera. 40 aequatorial dodecatera are then placed in tetrahedral junctions corresponding to the edges of a triacontaditeron in relation to the base. The structure repeats past this point: 16 penteractitriacontaditera are placed after the first set of penteractitriacontaditera, 16 dodecatera are placed after the second set of penteractitriacontaditera and likewise 16 penteractitriacontaditera are placed after the first set of dodecatera. This then only leaves the opposite penteractitriacontaditeron, completing the shape. From this perspective, the shape has a height of √6 ≈ 2.449490.

Region	Layer	dot	nit
Near side	1	—	1
	2	16	10+16
Aequator		40	—
Far side	1	16	10+16
	2	—	1
Grand total		72	54
		126 peta

Beginning from a dodecateron, all its rectified pentachoral tera join to 12 penteractitriacontaditera, leaving gaps at the vertices, filled by 20 other dodecatera, both touching the aequator. This leaves gaps at octahedral junctions corresponding to the edges and cells of a hexateron in relation to the base, filled by 30 aequatorial penteractitriacontaditera. 30 aequatorial dodecatera are then placed in octahedral junctions corresponding to the vertices of a small cellidodecateron. The structure repeats past this point: 12 penteractitriacontaditera are placed after the first set, while 20 dodecatera are placed at vertex junctions after the first set of dodecatera. This then only leaves the opposite dodecateron, completing the shape. From this perspective, the shape has a height of 3.

Region	Layer	dot	nit
Near side	1	1	—
	2	—	12
	3	20	—
Aequator		30	30
Far side	1	20	—
	2	—	12
	3	1	—
Grand total		72	54
		126 peta

Due to missing sources, its vertex-first structure is unavailable, although it can be inferred it has a height of 2√3 ≈ 3.464102.

Having the same symmetry as the 1₂₂ polypeton and the 2₂₁ polypeton, both with E₆ symmetry, their components have neat correspondences:

Component	122	221	221	0221
D5	hin	tac	pt [hin]	nit
D5	hin	pt [inv. hin]	tac	nit
A5	pt [dot]	hix	inv. hix	dot
A4×A1	pen	inv. pen	line [rap]	rap
A4×A1	pen	line [rap]	inv. pen	rap
A2×A2×A1	line [triddip]	trig [trip]	trig [inv. trip]	pt [tratrip]

Source: Incidence matrices — ram