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!