1

Does there exist a set of m tiles that can tile a large finite area without gaps or overlaps (say, it can cover a disk of radius N * m times the longest dimension of any tile where N>10), but cannot tile the whole plane? If so, how large can N be, where N is the ratio of the disk radius to m * max tile diameter? Any tile is OK but it would be nice if the tiles are polygonal so it can be made into a physical jigsaw puzzle that has a built-in size limit.

May be related to aperiodic tilings. It may look similar to, but is not the same as Heesch's problem because constructions with long thin tiles or many different tiles (for the many-tiles version) can increase the number of surrounding layers but may not increase the ratio N.

(I originally wrote N instead of N * m, but that would allow loopholes using arbitrarily many different tiles and is against the spirit of the question. This is not the same as Tiling arbitrarily large portions of the plane implies tiling the plane? which asks about tiling of arbitrarily large areas where this question is about one finite area. This might also be related to https://math.stackexchange.com/q/3965182)

alices_and_bobs
  • 119
  • Your question is hard to put into a mathematicaly rigourous shape. An attempt : do you mean finding, for a certain "bounded" (a less ambiguous term than "finite") gathering of tiles, a proof of the fact that this tiling with diameter $D$ cannot be extended into a tiling with diameter $D+ \Delta D$. – Jean Marie Sep 23 '24 at 21:05
  • 1
    @JeanMarie I just want to find a tile or a set of different shapes of tiles that can tile an area much larger than individual tiles in every direction, without relying on many "non-reusable" different shapes of tiles. Finding a (gap-less non-overlapping) tiling of large diameter D (relative to the size and number of different tiles) can give a lower bound of the ratio N, but proving it can't be extended by ΔD might not be necessary except to find an upper bound; proving the set of tiles cannot tile the whole plane is needed though. "Bounded" area is OK but for a disk it's the same as finite. – alices_and_bobs Sep 23 '24 at 21:45
  • The only thing that is clear to me is that $m$ is the number of different tiles. What is $N$ ? Take $m$ different pie shapes to tile a disk with some radius. You can make that radius as large as you want but never be able to tile the whole plane. – Kurt G. Sep 24 '24 at 08:21
  • 1
    With only one tile allowed, this is very much an open question... https://en.wikipedia.org/wiki/Heesch%27s_problem – caduk Sep 24 '24 at 08:45
  • @KurtG. Thanks, I will edit to add a definition of N. It's the ratio of the disk radius to (number of different tiles * max tile diameter). The question is about finding a set of tiles with as large N as possible (or prove N can be unbounded). Pies wouldn't work because the radius of the disk is proportional to the diameter of each tile. – alices_and_bobs Sep 24 '24 at 13:06

1 Answers1

3

Edit: The problem is in fact not equivalent but only related.

When restricted to one tile, this is very similar to an open problem, known as the Heesch's problem. Instead of trying to cover a large disk (with a radius much larger than the diameter of the tile), we try to surround the shape with as many possible layers of tiles as possible. As of today, we cannot find tiles that can tile more than $6$ layers, but that do not tile the plane. The difference is that the tile for a given number of layers could get more and more elongated.

A generalization of the Heesch problem to a fixed number of tiles exists. The problem seems still open for two tiles, but for $k\geq 3$, we can get an arbitrary number of layers: Bojan Bašić, The Heesch number for multiple prototiles is unbounded, Comptes Rendus Mathematique, Vol. 353, No. 8, pp. 665-669 (2015).. However, a key difference is that we must surround around a given tile.

caduk
  • 5,621
  • 1
  • 5
  • 23
  • Thanks! Cool to know that a variant of my question has been answered, and in a way that feels like programming an automata (maybe because tiling is related to decidability after all). Though the example construction in the Heesch number paper relies on increasing the max tile diameter or number of different tiles, so does not give an unbounded ratio N for my question. I was motivated by jigsaw puzzles construction and did initially consider asking about the # of surrounding layers, but decided that disk size and size and number of tiles matter more to me. Guess I dodged a solved problem :) – alices_and_bobs Sep 24 '24 at 12:57
  • Also, the Heesch problem is not equivalent to this question for 1 tile, but I noticed the Heesch's problem page has a nice example (Robert Ammann's 3 layer construction) that gives a ratio N of about 3 for my question, probably the highest I've seen so far. Are there more examples like that? – alices_and_bobs Sep 24 '24 at 13:20
  • @alices_and_bobs Indeed, this variant of Heesch problem is not equivalent: one of the tile grows, but also, the tile to be surrounded is forced (in the paper, we could just tile with the square). This paper seems to be the top of the art of the subject, I guess it's safe to consider the problem open. – caduk Sep 24 '24 at 14:59