Timeline for Are aperiodic monotiles generalizable to higher dimensions?
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| May 5, 2023 at 14:55 | history | edited | Wolfgang |
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| S Apr 3, 2023 at 1:06 | history | bounty ended | CommunityBot | ||
| S Apr 3, 2023 at 1:06 | history | notice removed | CommunityBot | ||
| Mar 31, 2023 at 17:24 | comment | added | Nick S | Also, I think that one version of the Socolar-Taylor tiling is a topological disk(all you need is to connect the tile to the isolated pieces by the right line segments), but not the closure of its interior. Finally, while the SCD tile is aperiodic by our definition, it does have skew symmetry. | |
| Mar 31, 2023 at 17:19 | comment | added | Nick S | The SCD tile can probably be extended to any dimension $d \geq 3$. The problem lies in what do you mean by aperiodic tile... In the field interested in this question, Aperiodic Order, an aperiodic tile/set of tiles is any set of tiles which can tile $\mathbb R^d$ via translates and rotates, but any tiling of $\mathbb R^d$ has no translational symmetry. By this definition, the new einstein tile is actually not an einstein tile (you need the tile and its reflection) but the SCD tile is aperiodic. The 2-dim Socolar-Taylor tile may or may not be aperiodic depending on what a tile is..... | |
| Mar 31, 2023 at 16:35 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| Mar 29, 2023 at 13:23 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| Mar 28, 2023 at 13:30 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| Mar 26, 2023 at 3:48 | comment | added | Seewoo Lee | @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? | |
| Mar 25, 2023 at 23:47 | comment | added | Will Sawin | @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. | |
| S Mar 25, 2023 at 23:39 | history | bounty started | Nicholas James | ||
| S Mar 25, 2023 at 23:39 | history | notice added | Nicholas James | Draw attention | |
| Mar 24, 2023 at 20:28 | comment | added | Seewoo Lee | 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. | |
| Mar 24, 2023 at 14:37 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| Mar 24, 2023 at 0:36 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| Mar 23, 2023 at 23:57 | comment | added | Nicholas James | @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? | |
| Mar 23, 2023 at 21:51 | comment | added | Terry Tao | 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. | |
| Mar 23, 2023 at 20:52 | history | edited | Nicholas James | CC BY-SA 4.0 |
Fixed wording and listed authors explicitly so proper credit is given.
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| Mar 23, 2023 at 18:58 | history | edited | Nicholas James | CC BY-SA 4.0 |
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| S Mar 23, 2023 at 18:45 | review | First questions | |||
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| S Mar 23, 2023 at 18:45 | history | asked | Nicholas James | CC BY-SA 4.0 |