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Clarified the notion of "possible" animals.

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edited Nov 24 2010 at 2:34

Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile types as $t_1,\ldots,t_k$. Say that an {\em animal} using tiles $t_1,\ldots,t_k$ is a connected subset of the plane that can be obtained by gluing a finite number of tiles together along their edges; identify congruent subsets. If there is only one type, this is often called a polyomino; here are some pictures of polyominoes in the square lattice (which has only one type of tile and is not in fact an interesting lattice from the point of view of this question).

Say that a tiling of the plane using (distinct) tiles $t_1,\ldots,t_k$ is universal if it contains every possible lattice animal using tiles $t_1,\ldots,t_k$. To explain what I mean by "possible", if one of the tiles suppose that $k=2$, that $t_1$ is the "thin diamond" from the Penrose tiling , then by and that $t_2$ is the "thick diamond. By gluing together four thin diamonds copies of $t_1$ one can obtain the following "animal"animal".

This animal can't be contained within any tiling using the two (Penrose tiles below. or otherwise) using $t_1$ and $t_2$. So it makes sense to restrict to animals which, for example, are contained within some tiling of the plane with the given tiles.

My question is: are there $k \geq 2$ for which (aperiodic -- adjective added in edit) universal tilings exist?

Edit: when I first posted the question I omitted the adjective aperiodic above. As pointed out in comments, in this case the answer is obviously yes, which is good to have had pointed out.

We can also restrict the allowed animals. For example, we could restrict to animals which exhibit some form of symmetry.

One could then ask: do aperiodic tilings exist which are universal for animals in a (non-trivial) restricted class? Are there any interesting results along these lines? Is the Penrose tiling itself known to be universal for some interesting class of animals?

added important adjective "aperiodic"

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edited Nov 24 2010 at 0:33

Louigi Addario-Berry
3,392●1●11●24

Are Penrose tilings universal? Do aperiodic universal tilings with more than one tile exist?

Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile types as $t_1,\ldots,t_k$. Say that an {\em animal} using tiles $t_1,\ldots,t_k$ is a connected subset of the plane that can be obtained by gluing a finite number of tiles together along their edges; identify congruent subsets. If there is only one type, this is often called a polyomino; here are some pictures of polyominos polyominoes in the square lattice (which has only one type of tile)tile and is not in fact an interesting lattice from the point of view of this question).

Say that a tiling of the plane using (distinct) tiles $t_1,\ldots,t_k$ is universal if it contains every possible lattice animal using tiles $t_1,\ldots,t_k$. To explain what I mean by "possible", if one of the tiles is the thin diamond from the Penrose tiling, then by gluing together four thin diamonds one can obtain the following "animal"

This animal can't be contained within any tiling using the two Penrose tiles below. So it makes sense to restrict to animals which, for example, are contained within some tiling of the plane with the given tiles.

My question is: are there $k \geq 2$ for which (aperiodic -- adjective added in edit) universal tilings exist?

Edit: when I first posted the question I omitted the adjective aperiodic above. As pointed out in comments, in this case the answer is obviously yes, which is good to have had pointed out.

We can also restrict the allowed animals. For example, we could restrict to animals which exhibit some form of symmetry.

One could then ask: do aperiodic tilings exist which are universal for animals in a (non-trivial) restricted class? Are there any interesting results along these lines? Is the Penrose tiling itself known to be universal for some interesting class of animals?