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This question is motivated by a recently released paper written by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. It constructs the first topological disk that tiles the plane aperiodically with no additional constraints or matching conditions. As a note, I am not an expert on this subject in any way, but just a curious observer who would greatly appreciate a thoughtful response!

I'll get right to the question and then elaborate a bit based on what I have found so far:

Question: Assuming the definitions in the paper above, is this generalizable in higher dimensions? That is, is there an aperiodic monotile in some hyperplane of $\mathbb{R}^n$ with dimension greater than $2$?

  • As mentioned in the above paper, Terence Tao and Rachel Greenfeld proved last year that the periodic tiling conjecture is false. That is, they disprove that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations tiles it periodically for sufficiently large $d$. According to Tao and Greenfeld, this implies the same result for $\mathbb{R}^d$ for sufficiently large $d$ as well.

  • According to a paper by the first paper's co-author, Chaim Goodman-Strauss, there is a strongly aperiodic set of tiles in the Hyperbolic Plane $\mathbb{H}^2$.

  • Furthermore, there seems to be an interesting result regarding the so-called "Socolar-Taylor Tile", whose construction in 3-dimensions produces a weakly aperiodic tile (this is due to the fact that its tilings are periodic in one direction).

Besides the above examples, I haven't found anything promising that confirms or negates my question, and due to the non-restrictive terminology used to define a tile and tiling in the paper, I don't immediately see why this wouldn't make sense terminologically. Although this might be an open problem, are there any small insights from the first paper that could give clues on how to solve the "einstein problem" in higher dimensions? Thanks!

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    $\begingroup$ I would imagine that taking the Cartesian product of the new aperiodic monotille in ${\bf R}^2$ with a generic fundamental domain of ${\bf R}^{n-2}/{\bf Z}^{n-2}$ should give aperiodic monotile in ${\bf R}^n$, though proving this claim rigorously may require some annoyingly technical arguments. $\endgroup$
    – Terry Tao
    Commented Mar 23, 2023 at 21:51
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    $\begingroup$ @TerryTao: Thank you for your response! Out of curiosity, what would the “annoyingly technical arguments” that you mentioned involve exactly? Would they be unrelated to the arguments made in the paper in question? $\endgroup$
    – Nicholas James
    Commented Mar 23, 2023 at 23:57
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    $\begingroup$ Adding to Tao’s comment, I think this would give a construction. Consider the “hat prisms” which are just hat x [0,1]^(n-2). The paper says that there are uncountably many different tilings. Now tile each “layers” of R^n with different tilings of hat prisms (just consider a product of tiling in R^2 and $[0,1]^{(n-2)}$. If you randomly shift all the tiles in each layer by random vectors in $\mathbb{R}^2$, I believe this would give aperiodic tiling. Maybe finding an aperiodic monotile in a essentially different way would be more interesting. $\endgroup$
    – Seewoo Lee
    Commented Mar 24, 2023 at 20:28
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    $\begingroup$ @SeewooLee The point is not just to find a tile that admits an aperiodic tiling, but also to ensure it doesn't admit any periodic tilings. It is not at all obvious that the hat prisms don't admit such tilings. $\endgroup$
    – Will Sawin
    Commented Mar 25, 2023 at 23:47
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    $\begingroup$ @WillSawin Thank you for pointing it out. I just realized that we can make aperiodic tiling just with squares in a similar way, but that's not what we want. In case of hat prism, we can fix a single tiling of $\mathbb{R}^2$ by hats and take a product with $[0, 1]^{(n-2)}$ (so just stack it into the other directions without any perturbations) - this gives periodic tiling, right? $\endgroup$
    – Seewoo Lee
    Commented Mar 26, 2023 at 3:48

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