A recent Quanta article [1] discusses a new result of Greenfeld & Tao [2]:
There is some dimension $d$ with a $d$-dimensional tile which aperiodically fills $\mathbb{R}^d$ but cannot do so periodically, allowing translations, rotations, and reflections.
Bhattacharya [3] had earlier proved that this is not possible (for $d=2$) with translations alone:
Any tile which aperiodically fills $\mathbb{R}^2$ can also fill it periodically, allowing only translations.
Is anything known about the remaining cases? In particular, I am curious about allowing translations and rotations but not reflections, but any other collections of allowed isometries would be of interest.
Edit: Dan Rust brings up disconnected tiles in the comments. I'm happy to allow them, but it might be interesting to consider both cases.
[1] Jordana Cepelewicz, ‘Nasty’ Geometry Breaks Decades-Old Tiling Conjecture, Quanta Magazine (Dec 2022)
[2] Rachel Greenfeld and Terence Tao, A counterexample to the periodic tiling conjecture, arXiv preprint (2022). arXiv:2211.15847 [math.CO]
[3] Siddhartha Bhattacharya, Periodicity and decidability of tilings of $\mathbb{Z}^2$, American Journal of Mathematics, Volume 142, Number 1 (Feb 2020), pp. 255-266. arXiv:1602.05738 [math.CO]