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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", suppose that $k=2$, that $t_1$ is the "thin diamond" from the Penrose tiling and that $t_2$ is the "thick diamond. By gluing together four copies of $t_1$ one can obtain the following "animal".

This animal can't be contained within any tiling (Penrose 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?

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 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).

(source)

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", suppose that $k=2$, that $t_1$ is the "thin diamond" from the Penrose tiling and that $t_2$ is the "thick diamond. By gluing together four copies of $t_1$ one can obtain the following "animal".

This animal can't be contained within any tiling (Penrose 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?

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", suppose that $k=2$, that $t_1$ is the "thin diamond" from the Penrose tiling and that $t_2$ is the "thick diamond. By gluing together four copies of $t_1$ one can obtain the following "animal".

This animal can't be contained within any tiling (Penrose 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?

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", suppose that $k=2$, that $t_1$ is the "thin diamond" from the Penrose tiling and that $t_2$ is the "thick diamond. By gluing together four copies of $t_1$ one can obtain the following "animal".

This animal can't be contained within any tiling (Penrose 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?