I'm not sure where to post this, but I ran into an interesting programming problem recently, that is somehow related to higher-dimensional geometry.
Basically, I'm trying to design an efficient storage scheme for a dense set of points (vectors with integer coordinates) in an n-dimensional state space of some problem that the program is trying to solve. (Oh btw, I'm using "dense" in the programming sense of having many adjacent points, not in the mathematical sense of having a point between every pair of points.) Initially, I considered the usual partitioning of n-space into n-cubes, with each n-cube serving as a "bucket" to contain some set of points. The problem with this approach is that when the set is dense, the average number of points per bucket is kn, which quickly becomes unmanageable as n grows large. Furthermore, upon closer investigation, in each iteration of my algorithm, it actually mostly only cares about points that differ only in a small number of coordinates, so a bucket shaped like an n-cube will actually contain mostly irrelevant points, because in high dimensions, most of the n-cube's bulk is concentrated near its vertices! Instead, a more ideal shape for my buckets is closer to the n-cross, perhaps some manner of truncated n-cross -- basically, the set of points around some given reference point P that differ only in a small number of coordinates from P. Since an n-cross only has 2n vertices, and truncated n-crosses only have k*n vertices for some small constant k, this is far more manageable from a storage efficiency POV -- only O(n) space complexity, as opposed to O(k^n) complexity!
This led me to consider how to partition n-space into buckets of n-cross-like facets. In the ideal case, the partitioning should be facet-transitive (makes the code more manageable if we only have to deal with one shape of buckets). So we're basically dealing with some kind of tiling of n-space with n-crosses (or some kind of n-cross truncates). But I'm guessing probably there's no such partitioning that's facet-transitive, so I'm willing to settle for a uniform tiling instead, or some kind of tiling that requires only 2 or 3 different facets (preferably only 2), and preferably all the facets will have relatively small n-bulk (i.e., not exponential like the n-cube).
I tried searching online but I didn't find any tilings involving the n-cross or truncated n-crosses. What are the known tilings involving n-crosses or low-order n-cross truncates? Are there any good candidates for my program?