"Apeiratization" is an operation that I found when playing around with uniform Euclidean tilings. It can be applied to any finite uniform polytope P of positive dimension, and the result is a uniform Euclidean tiling of the next rank up with two P's and a single apeiratization of each of P's facets meeting at each vertex. The base case is that the apeiratization of the line segment is the apeirogon.
The vertex figure of the apeiratization of P is the bowtie prism of the vertex figure of P, where the edges through the center are vertex figures of apeirogons.
Sometimes the apeiratization of a polytope is an infinitely dense tiling (these are excluded under my definition of a proper tiling), but this rarely happens unless the polytope's symmetry is not part of any Euclidean symmetry group.
| Polytope | Apeiratization | Head of regiment |
| Triangle (x3o) | Tha (o3x~x3/2*a) | That (o3x6o) |
| Square (x4o) | Sha (o4x~x4/3*a) | Squat (x4o4o) |
| Hexagon (x6o) | Hoha (o6x~x6/5*a)/2 | That (o3x6o) |
| Tet (x3o3o) | Datha (o3o3x~x3/2*b) | Batatoh (o3o3x3x3*a) |
| Cube (x4o3o) | Dacha (o3o4x~x4/3*b) | Chon (x4o3o4o) |
| Oct (o4o3x) | Doha (o4o3x~x3/2*b) | Rich (o4x3o4o) |
| Co (o4x3o) | Cuhada (x3o3/2x4/3o4*a~*c)/2 | Rich (o4x3o4o) |
| Trip (x3o x) | (unnamed, non-Wythoffian) | Thiph (o6x3o □ x~o) |
| Pen (x3o3o3o) | o3o3o3x~o3/2*c | Cytopit (x3x3o3o3o3*a) |
| Rap (o3x3o3o) | o3o3x3o3/2x3/2*b *c~*e | Scyropot (x3o3x3o3o3*a) |
| Spid (x3o3o3x) | (unnamed, non-Wythoffian) | Scyropot (x3o3x3o3o3*a) |
| Tes (x4o3o3o) | o3o3o4x~x4/3*c | Test (x4o3o3o4o) |
| Hex (o4o3o3x) | o4o3o3x~x3/2*c | Rittit (o4x3o3o4o) |
| Ico (x3o4o3o) | (o3o4o3x~x3/2*c)/2 | Icot (x3o4o3o3o) |
| Hix (x3o3o3o3o) | o3o3o3o3x~x3/2*d | x3x3o3o3o3o3*a |
| Rix (o3x3o3o3o) | o3o3o3x3o3/2x3/2*c *d~*f | x3o3x3o3o3o3*a |
| Dot (o3o3x3o3o) | o3o3x3o3/2x3/2*b *c~*e *d3o | x3o3o3x3o3o3*a |
| Scad (x3o3o3o3x) | (unnamed, non-Wythoffian) | x3o3x3o3o3o3*a |
| Pent (x4o3o3o3o) | o3o3o3o4x~x4/3*d | x4o3o3o3o4o |
| Tac (o4o3o3o3x) | o4o3o3o3x~x3/2*d | o4x3o3o3o4o |
| Rat (o4o3o3x3o) | (o4o3o3x3o3/2x3/2*c *d~*f)/2 | o4o3x3o3o4o |
| Hin (x3o3o3o *b3o) | o3o3x~x3/2*b3o3o | x3o3o3x *b3o *c3o |
| Hop (x3o3o3o3o3o) | o3o3o3o3o3x~x3/2*e | x3x3o3o3o3o3o3*a |
| Ril (o3x3o3o3o3o) | o3o3o3o3x3o3/2x3/2*d *e~*g | x3o3x3o3o3o3o3*a |
| Bril (o3o3x3o3o3o) | o3o3o3x3o3/2x3/2*c *d~*f *e3o | x3o3o3x3o3o3o3*a |
| Staf (x3o3o3o3o3x) | (unnamed, non-Wythoffian) | x3o3x3o3o3o3o3*a |
| Ax (x4o3o3o3o3o) | o3o3o3o3o4x~x4/3*e | x4o3o3o3o3o4o |
| Gee (o4o3o3o3o3x) | o4o3o3o3o3x~x3/2*e | o4x3o3o3o3o4o |
| Rag (o4o3o3o3x3o) | (o4o3o3o3x3o3/2x3/2*d *e~*g)/2 | o4o3x3o3o3o4o |
| Hax (x3o3o3o3o *b3o) | o3o3x~x3/2*b3o3o3o | x3o3o3o3x *b3o *d3o |
| Jak (x3o3o3o3o *c3o) | o3o3o3o3x~x3/2*d *c3o | x3o3o3o3x *c3o3o |
| Mo (o3o3o3o3o *c3x) | (o3o3o3x~x3/2*c3o3o)/2 | o3x3o3o3o *c3o3o |
| Oca (x3o3o3o3o3o3o) | o3o3o3o3o3o3x~x3/2*f | x3x3o3o3o3o3o3*a |
| Roc (o3x3o3o3o3o3o) | o3o3o3o3o3x3o3/2x3/2*e *f~*h | x3o3x3o3o3o3o3o3*a |
| Broc (o3o3x3o3o3o3o) | o3o3o3o3x3o3/2x3/2*d *e~*g *f3o | x3o3o3x3o3o3o3o3*a |
| He (o3o3o3x3o3o3o) | o3o3o3x3o3/2x3/2*c *d~*f *e3o3o | x3o3o3o3x3o3o3o3*a |
| Suph (x3o3o3o3o3o3x) | (unnamed, non-Wythoffian) | x3o3x3o3o3o3o3o3*a |
| Hept (x4o3o3o3o3o3o) | o3o3o3o3o3o4x~x4/3*e | x4o3o3o3o3o3o4o |
| Zee (o4o3o3o3o3o3x) | o4o3o3o3o3o3x~x3/2*e | o4x3o3o3o3o3o4o |
| Rez (o4o3o3o3o3x3o) | (o4o3o3o3o3x3o3/2x3/2*e *f~*h)/2 | o4o3x3o3o3o3o4o |
| Hesa (x3o3o3o3o3o *b3o) | o3o3x~x3/2*b3o3o3o3o | x3o3o3o3o3x *b3o *e3o |
| Naq (o3o3o3o3o3x *c3o) | o3o3o3o3o3x~x3/2*e *c3o | x3o3o3o3o3o3x *d3o |
| Laq (x3o3o3o3o3o *c3o) | (o3o3o3o3o3x~x3/2*e *d3o)/2 | o3x3o3o3o3o3o *d3o |
| more to come |
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