Timeline for Tradeoffs in translation-invariant tilings of $\mathbb{R}^3$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 23, 2019 at 19:39 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
typo in the title
|
| Nov 23, 2019 at 19:05 | comment | added | squiggles | This is certainly true, thanks! I guess I'm interested in more fine-grained bounds, but this is definitely the right way to phrase the problem. I've posted a more general follow-up question here: mathoverflow.net/questions/346768/…. | |
| Nov 23, 2019 at 14:57 | comment | added | Dan Rust | Well once everything is in the torus, then you can start using Euler characteristic arguments and homology to relate these quantities . | |
| Nov 23, 2019 at 13:19 | comment | added | squiggles | That's correct. I'm equivalently interested in tools for relating these values in any regular CW-decomposition of the $3$-torus. | |
| Nov 23, 2019 at 12:25 | comment | added | Dan Rust | By translation invariant, presumably you mean that there exists a full-rank sublattice of the set of translations on $\mathbb{R}^3$ such that the tiling is fixed under translations in the sublattice. We would call such a tiling periodic. I believe if you mod out $\mathbb{R}^3$ by this lattice (or some sublattice of that so that you don't identify too many $n$-cells together) and consider the induced tiling of the torus, then all the quantities you consider will be reflected there. | |
| Nov 23, 2019 at 1:04 | comment | added | squiggles | Sorry that the question isn't clear. When I say tiling, I mean one in the most general sense: a regular CW-filtration of $\mathbb{R}^3$. The tiling gives us a discrete group of isometries that preserves the filtration. This group action partitions the flags into equivalence-classes, and a regular tiling is, to me, one that only has a single equivalence class. For this question, any partition of space (into non-convex bodies or otherwise) is allowed, so long as its translation invariant. | |
| Nov 23, 2019 at 0:19 | comment | added | Joseph O'Rourke | May I ask: What is a "regular tiling"? Tiling by regular polyhedra? And could you expand upon the phrase "symmetry-inequivalent flags."? | |
| Nov 22, 2019 at 23:22 | history | edited | squiggles | CC BY-SA 4.0 |
edited body
|
| Nov 22, 2019 at 23:05 | history | asked | squiggles | CC BY-SA 4.0 |