Z in feedback points to a brand new discovery by David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss, who write:
An aperiodic monotile . . . is a form that tiles the airplane, however by no means periodically. On this paper we current the primary true aperiodic monotile, a form that forces aperiodicity by way of geometry alone, with no extra constraints utilized by way of matching circumstances. We show that this form, a polykite that we name “the hat”, should assemble into tilings primarily based on a substitution system.
All I can say is . . . wow. (That’s, assuming the result’s right. I’ve no cause to suppose it’s not; I simply haven’t tried to verify it myself.)
First off, that is simply wonderful. Much more wonderful is that I had no concept that this was even an open downside. I’d seen the Penrose two-shape tiling sample years in the past and beloved it a lot that I painted a tabletop with it (and ship a photograph of the desk to Penrose himself, who replied with a pleasant little be aware, which sadly I misplaced some years in the past, or I’d reproduce it right here), and it by no means even occurred to me to ask whether or not an aperiodic monotile was doable.
That is the most important information of 2023 thus far (once more, conditional on the end result being right), and I doubt something larger will occur between now and the top of December.
OK, there’s one risk . . .
Penrose did it with 2 distinctive tiles, Smith et al. simply wanted 1, . . . The following frontier in aperiodic tiling is to do it with 0. Whoever will get there would be the actual genius.
P.S. Michael in feedback points out that the Smith et al. sample consists of mirrored tiles. So let’s name it 1.5. Is there a theorem which you can’t do it with simply 1 tile with no mirroring?
Z in feedback points to a brand new discovery by David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss, who write:
An aperiodic monotile . . . is a form that tiles the airplane, however by no means periodically. On this paper we current the primary true aperiodic monotile, a form that forces aperiodicity by way of geometry alone, with no extra constraints utilized by way of matching circumstances. We show that this form, a polykite that we name “the hat”, should assemble into tilings primarily based on a substitution system.
All I can say is . . . wow. (That’s, assuming the result’s right. I’ve no cause to suppose it’s not; I simply haven’t tried to verify it myself.)
First off, that is simply wonderful. Much more wonderful is that I had no concept that this was even an open downside. I’d seen the Penrose two-shape tiling sample years in the past and beloved it a lot that I painted a tabletop with it (and ship a photograph of the desk to Penrose himself, who replied with a pleasant little be aware, which sadly I misplaced some years in the past, or I’d reproduce it right here), and it by no means even occurred to me to ask whether or not an aperiodic monotile was doable.
That is the most important information of 2023 thus far (once more, conditional on the end result being right), and I doubt something larger will occur between now and the top of December.
OK, there’s one risk . . .
Penrose did it with 2 distinctive tiles, Smith et al. simply wanted 1, . . . The following frontier in aperiodic tiling is to do it with 0. Whoever will get there would be the actual genius.
P.S. Michael in feedback points out that the Smith et al. sample consists of mirrored tiles. So let’s name it 1.5. Is there a theorem which you can’t do it with simply 1 tile with no mirroring?