Timeline for Does the concept of connective constant make sense for any tiling of the plane?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2014 at 8:01 | comment | added | Ritwik | @Carl: My question was primarily for periodic tilings. Otherwise it is quite conceivable that $\mu$ doesn't exist. | |
| Feb 24, 2014 at 7:39 | comment | added | user25199 | I think the answer is probably yes for periodic and quasiperiodic in the sense of cut-and-project (eg Penrose). But for others, the limit may not exist. Consider rectangular tiles $1\times 1$ and $1\times 10$. Any arrangement of vertical strips of width 1 is a tiling, but you could probably find a pathological choice of tiles for which the quantity oscillates and doesn't converge. | |
| Feb 23, 2014 at 15:06 | comment | added | Wlodek Kuperberg | Yes, your question is meaningful and natural. But it should not be too surprising if it has not been studied before - this happens quite often. A new discovery often leads to many new questions. I guess there are more such questions than all mathematicians in the world can handle or even think about. By the way, if you're interested in tilings, I recommend "Tilings and Patterns" (1987) - a great book by Grünbaum and Shephard. | |
| Feb 23, 2014 at 5:28 | comment | added | Ritwik | @Kuperberg: So calculating $\mu$ for other tilings is at least a meaningful question. I was just wondering why this has not been studied by others (because apriori it seems like a "natural" question to me and also something others are likely to have thought about). | |
| Feb 23, 2014 at 5:24 | history | edited | Ritwik | CC BY-SA 3.0 |
Asking why the concept has not been studied for other lattices although it is a meaningful question
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| Feb 23, 2014 at 4:00 | comment | added | Wlodek Kuperberg | If rotations are allowed (all tiles convex and congruent), then there is a great variety of possible tilings. The problem of classifying all of them is still open. However, it seems that for your purpose the tiles need not even be congruent, but then the assumption of uniformity seems reasonable: there should be an upper bound on their diameters and a positive lower bound on their areas. Without this assumption $\mu$ could be undefined or infinite. | |
| Feb 22, 2014 at 19:37 | comment | added | Ritwik | @Kuperberg: What are the possible tilings I can have if I do not require the second condition (i.e. rotations are allowed)? | |
| Feb 22, 2014 at 19:28 | comment | added | Ritwik | What you are saying seems to make perfect sense. The reason I asked this question is, it seems all the conjectures or numerical simulations done so far are for Lattices. Is there any particular reason why only Lattices have been studied? | |
| Feb 22, 2014 at 19:10 | comment | added | Anthony Quas | I'm not sure what you have in mind, but the question probably makes perfectly good sense for non-congruent tilings of the plane - whether periodic or not. I'm sure you could make a definition of $\mu$ that would work for any periodic tiling of the plane. I would be very confident that it would also work for aperiodic tilings such as the Penrose tiling also (without having any details in mind). | |
| Feb 22, 2014 at 18:56 | comment | added | Wlodek Kuperberg | Then the only tilings you can have are by polygons with at most six sides, and if you want the tiles to be parallel translates of each other (rotations not allowed), then triangles and pentagons are out. | |
| Feb 22, 2014 at 17:44 | comment | added | Ritwik | I think I was mistaken; I can't really think of a tiling of R^2 even for an 8-gon. I do want the tiling to be congruent. | |
| Feb 22, 2014 at 17:42 | history | edited | Ritwik | CC BY-SA 3.0 |
Removed the comment about genus g surface
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| Feb 22, 2014 at 15:36 | comment | added | Wlodek Kuperberg | Could you describe explicitly an example of a tiling of $\mathbb{R}^2$ by $4g$-gon with $g>1$? (I assume the tiles should be convex and congruent, but I could be wrong.) | |
| Feb 22, 2014 at 8:48 | history | asked | Ritwik | CC BY-SA 3.0 |