User:Double sharp/List of uniform tilings by Schwarz triangle
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Yes, I'm fully aware that this is a misnomer: Schwarz triangles are spherical. I should really be referring to Coxeter polytopes (since we're referring to tilings of E2/H2, I could say Coxeter polygons.)
This is the Euclidean and hyperbolic version of List of uniform polyhedra by Schwarz triangle.
What exactly sabotages {5,10/3}, {5/2,10}, {7,14/5}, {7/2,14/3}, {7/3,14} etc. and makes them have infinite density though they'd fit in the plane? – ah, I see, from Regular Polytopes (3rd ed.), p.108: they would have rotational symmetries that are not 2-, 3-, 4-, or 6-fold, which we know to be impossible.
Basically what we need is:
Add and subtract interior angles of {3} (60°), {4} (90°), {6} (120°), {8} (135°), {8/3} (45°), {12} (150°), {12/5} (30°), {∞} (180°) to yield some multiple of 360°. This will determine a candidate vertex figure. We may reject everything containing consecutive positive and negative terms of the same angle as degenerate. Some candidates may also be excluded by symmetric concerns, such as {8/3, 8} above. (Though we need to prove that only the above tiles are possible.) However such a bald listing ends up creating lots of junk as well...