I suspect they've spent much longer looking at 2d shapes than 3d ones (because they're easier to imagine and reason about, and there are fewer variations in how "aperiodic monotile" can be defined in 2d).
Also, the tricky bit for this kind of shape (similarly to Penrose's tiles) isn't finding the shape, but finding the aperiodic tiling and adding gadgets that rule out any periodic tilings (and proving this). The actual aperiodic monotile shown looks like it has 201 faces (although that could trivially be reduced by quite a bit).
I got confused ... IIUC it's not a perfect 7-cubes-shapes, but it has a few bump and dents here and there to ensure it only can be assembled in some position(???).
Yeah, the seven cubes shape tiles R^3 by itself following the aperiodic tiling, but it also tiles R^3 in other ways, some of which are periodic. The bumps prevent any of the periodic tilings. See [1] for a better explanation with a picture of the shape as a rep-tile, and [2] for insight into how a rep-tile becomes an aperiodic tiling.
> However, the paper itself and the formalization that it claims for support are not
in a form that is readily usable to anyone who wants to understand or check it. This is
not just because of the needless length or complexity of the argument (which is indeed
overwhelming as presented), or because the formal repository is not in normal form, or
because much of the text is misleading or poorly organized, but mainly because there
is so much irrelevant puff to push through.
I don’t get it. Mathematicians that have spent decades studying 2d and 3d tiles overlooked this super simple 7 cube chair shape?
I suspect they've spent much longer looking at 2d shapes than 3d ones (because they're easier to imagine and reason about, and there are fewer variations in how "aperiodic monotile" can be defined in 2d).
Also, the tricky bit for this kind of shape (similarly to Penrose's tiles) isn't finding the shape, but finding the aperiodic tiling and adding gadgets that rule out any periodic tilings (and proving this). The actual aperiodic monotile shown looks like it has 201 faces (although that could trivially be reduced by quite a bit).
I got confused ... IIUC it's not a perfect 7-cubes-shapes, but it has a few bump and dents here and there to ensure it only can be assembled in some position(???).
Yeah, the seven cubes shape tiles R^3 by itself following the aperiodic tiling, but it also tiles R^3 in other ways, some of which are periodic. The bumps prevent any of the periodic tilings. See [1] for a better explanation with a picture of the shape as a rep-tile, and [2] for insight into how a rep-tile becomes an aperiodic tiling.
[1] https://arxiv.org/abs/2609.24779
[2] https://en.wikipedia.org/wiki/Rep-tile#Examples
See also: https://www.scientificamerican.com/article/a-unique-shape-a-...
See even more: https://arxiv.org/abs/2609.24779
> However, the paper itself and the formalization that it claims for support are not in a form that is readily usable to anyone who wants to understand or check it. This is not just because of the needless length or complexity of the argument (which is indeed overwhelming as presented), or because the formal repository is not in normal form, or because much of the text is misleading or poorly organized, but mainly because there is so much irrelevant puff to push through.
Zing