This is the landing page for my list of uniform euclidean tesselations. It contains a list of links to all of the dimensions, with a rough description of what sorts of things each one contains, as well as the definitions for what counts as a valid tesselation and what makes one uniform. As always, credit to Eric Binnendyk's old euclidean pages for being a base for my work, and credit to Jonathan Bowers' uniform polytope pages as inspiration.
In simple terms, a tesselation is an infinite filling of euclidean space by polytopes (or other tesselations), such that all polytopes meet exactly one other polytope at all of their facets. The facets of a tesselation are these polytopes, and they may self intersect, intersect eachother, be tesselations themselves, etc. A tesselation is uniform if all of its vertices are the same, meaning they can all be moved to one another without changing the appearance of the tesselation, all of its elements are also uniform, and all of its edges are the same length.
Formally, a tesselation is a faithful realization of an abstract polytope of finite rank but infinite element count onto a euclidean space. It must be strongly connected, so it cannot be a compound or have compound elements (though compounds may be listed as asides in categories), and it must be dyadic, so exactly two facets meet at every ridge. Furthermore, the requirement of a faithful realization disallows things like digons and overlapping vertices (cases that are faithful on facets and ridges but nonfaithful on other elements are called fissary and may be listed as asides in categories). Furthermore, we also disallow two unique elements to share the exact same set of ridges, which stops things like the ike+gad combocell, and we disallow the set of vertices being dense, meaning there must be some non-zero distance D such that a ball of radius D centered on any vertex contains no other vertices. We also require the realization to be planar, meaning all d-elements reside in a d-dimensional subspace (or a (d-1)-dimensional subspace if the element is a tesselation).
A tesselation is considered uniform if all of its elements are uniform (either uniform polytopes by their definition, or uniform tesselations by this definition. The apeirogon will be defined to be uniform as a base case.), and the vertices are transitive under the symmetry group of the tesselation's realization. Notably, this requires all elements to be inscribed either in a hypersphere or hyperplane, allowing us to make a final exclusion: we disallow any elements sharing an inscribed hypersphere or hyperplane from sharing any of their ridges. This prevents Exotic Combo Elements, like a compound of two cuboctahedra that share a central hexagon that has 4 faces on all its edges, and also prevents cases called subdivisions, in which an element that is a tesselation is split into an infinite stack of tesselations, like a squat into an infinite tower of azips.
The highest dimension I can conceive ever making a page for is 6-D, but honestly I won't go past 4-D anytime soon. Nonetheless, I wanted to write down some discussion of the happenings in these higher dimensions, so they're listed here anyway.
-1 Dimensional Uniform Tesselations - No.
0 Dimensional Uniform Tesselations - No. I mean technically the point does fill its space, but I don't think it counts.
1 Dimensional Uniform Tesselations - 1-D tesselations are also rarely known as "sequences". This dimension contains aze, the apeirogon, in all of its glory. NOT A CIRCLE.
2 Dimensional Uniform Tesselations - 2-D tesselations are usually called tilings. This is where things first get interesting. This dimension contains squat, the square tiling, our first real experience with a hypercubic tesselation, as well as containing the exceptional regulars of trat, the triangular tiling, and hexat, the hexagonal tiling. Even here we already receive complex blended and nonwythoffian tesselations.
3 Dimensional Uniform Tesselations - 3-D tesselations are usually called honeycombs. This dimension is surprisingly unexceptional. Our only regular is chon, the cubic honeycomb. This is the first dimension where the demicubic and cyclosimplex symmetries start making distinct uniforms. Despite the lack of exceptional symmetries, there are still a lot of interesting objects, and many known blends.
4 Dimensional Uniform Tesselations - 4-D tesselations are sometimes called "tetracombs". This dimension gives us our last exceptional regulars, hext and icot, the hexadecachoric and icositetrachoric tetracombs. This leads to an extraordinary amount of exceptional objects, along with more hypercubics and cyclosimplex items than ever. This is also the first dimension where quarter-cubic symmetry becomes distinct, but it has too much supersymmetry to make any convex cases (like how there aren't convex demitessic uniform polychora). Here also are cyclotriangular gyrotrigonisms. Some blends are known, but nonwythoffians have not been extensively investigated from here on out. Already in this dimension we begin to see combinatorial explosion, some wythoffian regiments here have millions of members!
5 Dimensional Uniform Tesselations - 5-D tesselations are sometimes called "pentacombs". This dimension is relatively boring, like 3-D. No exceptional symmetries, our only regular is the penteractic pentacomb, and this trend of only one regular will continue to all further dimensions. However, this is the first dimension where quarter-cubic honeycomb symmetry finally gets its own distinct convex members, which continue to exist in higher dimensions. This is also the first dimension where there are convex cyclosimplex symmetrics with no supersymmetry, which will continue to exist as well.
6 Dimensional Uniform Tesselations - 6-D tesselations are sometimes called "hexacombs", you can see the pattern. This dimension contains the very nice exceptional E6 hexacomb symmetry, whose triangular supersymmetry makes for very interesting properties. I think there are also cyclotetrahedral gyrotetrahedronisms here.
7 Dimensional Uniform Tesselations - A.K.A heptacombs. Here we get our next exceptional symmetry, E7, which has only mild amounts of supersymmetry. E6 alterprisms live here.
8 Dimensional Uniform Tesselations - A.K.A octacombs. Here we get our very last exceptional symmetry (for wythoffians), E8, which has no supersymmetry and is famously a very nice lattice. This dimension has E7 alterprisms as well as the very large set of E6 gyrotrigonisms. I think there are also cyclopentachoric gyropentachoronisms here. This will surely be one of the most interesting dimensions, if only we could make any real progress at enumerating it.
Higher - Tesselations continue to be given goofy dimension-specific names based on greek number prefixes, like "dodecacombs". The last exceptional wythoffian symmetry was E8, but there will continue to be exceptional atypical symmetries, like those of the leech lattice in 24-D (although we do not know of any uniforms with that symmetry). The last exceptional wythoffian alterprism symmetries were in 8-D as well, but there will continue to be hypercubic, demihypercubic, and quarterhypercubic alterprisms, and cyclosimplical alterprisms and gyrosimplexisms in their respective dimensions. Good luck studying any of these, even hexacombs are hard enough.
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