This is the main page for the hyperbolic tesselations, with links to all of the many dimensions they can exist in. It also contains the definition of a hyperbolic tesselation and what is listed in the categories.
A hyperbolic tesselation is a filling of the entirety of an n-dimensional hyperbolic space with polytopes, including other tesselations. A tesselation is uniform if all vertices are the same under the symmetry group, all elements are uniform, and all edge lengths are the same.
For a tesselation to be counted in the listing, all half-spaces must contain the center of at least one element of the tesselation. An example forbidden by this is the hyperbolic honeycomb {7,3,3} and its dual {3,3,7}, due to there being half-spaces that contain the centers of no elements. Also, no pseudogons. ABSOLUTELY NO PSEUDOGONS.
The other conditions are that the tesselation must be a faithful realization of an abstract polytope, and the tesselation must not be dense. Also, to allow a listing at all without Cantor getting mad at me, there must be a finite number of element orbits.
These go up to really high dimensions, but I doubt I will ever reach even 5D tesselations.
1D Hyperbolic Tesselations - This contains the pseudogon. We do not like the pseudogon.
2D Hyperbolic Tesselations - This is where the infinitude of regular tilings appear. There are many symmetries here, including symmetries without triangles as a fundamental domain, and symmetries that aren't even generated by reflections.
3D Hyperbolic Tesselations - This dimension includes only a finite amount of regulars (that fit the requirements to be listed), but still a large amount of symmetries, some non-simplicial.
4D Hyperbolic Tesselations - The 5D version of the 120-cell and 600-cell are here. They're hyperbolic now. This is also home to a prominent non-simplicial symmetry, which some tesselations have, like the tesselation contit. The last simplicial compact honeycombs appear here.
5D Hyperbolic Tesselations - The last regular hyperbolics appear here. There are none after this. Somehow. Meanwhile, there are hyperbolic knockoffs of the E6 symmetry, containing some squares rather than triangles.
6D Hyperbolic Tesselations - This dimension seems kind of boring, with the symplicial symmetries being strange variants of E7.
7D Hyperbolic Tesselations - The symmetries here are strange variants of E8, just like the dimension above being strange variants of E7. There is also a strange non-simplicial symmetry here as well, with pentagons even.
8D Hyperbolic Tesselations - E9 knockoffs, based off of symmetry mutations of the E8 lattice (E9).
9D Hyperbolic Tesselations - The horrifying E10 is found here. This contains the last of the simplicial symmetries, but there exist non-simplicial symmetries higher up.
10D Hyperbolic Tesselations
11D Hyperbolic Tesselations
12D Hyperbolic Tesselations
13D Hyperbolic Tesselations
14D Hyperbolic Tesselations
15D Hyperbolic Tesselations
16D Hyperbolic Tesselations - A weird symmetry with a Coxeter-Dynkin diagram that seems like a joke is here. Blame Tumarkin.
17D Hyperbolic Tesselations - The mythical HONSE exists here. It has a prominent tesselation with a vertex figure being the E8 lattice * E8 lattice. Its symbol is oo8oooooxooooo8oo. Blame Vinberg.
18D Hyperbolic Tesselations
19D Hyperbolic Tesselations
20D Hyperbolic Tesselations
21D Hyperbolic Tesselations
22D Hyperbolic Tesselations
23D Hyperbolic Tesselations
24D Hyperbolic Tesselations
25D Hyperbolic Tesselations - For some reason, you can turn the 24D Leech Lattice into a Coxeter-Dynkin diagram. This is basically the final boss of hyperbolic tesselations. Blame Conway.