This is a question about the einstein problem. As has been well known, in 2023, work of
David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss (https://arxiv.org/abs/2303.10798) showed the First example of an aperiodic monotile (see picture below). As you can see, this example is not convex. Is there any theorem in tiling that implies no convex aperiodic monotile exists?
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The set of convex shapes which tile the plane has recently been determined by work of Rao (Exhaustive search of convex pentagons which tile the plane), and all of them admit periodic tilings. Rao even lists the non-existence of a convex aperiodic monotile as a corollary to his result, so there should not be any easy reason why such tiles do not exist besides the classification.
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$\begingroup$ Does "convex shape" have a technical meaning here? A convex subset of $\mathbb{R}^2$ of finite, positive measure? (EDIT: I guess en.wikipedia.org/wiki/Convex_body is an answer.) Certainly I agree that the work of Rao resolves the case of convex polygons, but maybe there are some other very pathological convex "shapes" which could be said to tile $\mathbb{R}^2$. $\endgroup$2026-01-22 20:44:08 +00:00Commented Jan 22 at 20:44
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7$\begingroup$ @SamHopkins: the portion of the boundaries that separate two tiles must be straight (convexity on both sides) so you cannot have convex tilings with non-polygons. $\endgroup$Willie Wong– Willie Wong2026-01-22 20:58:07 +00:00Commented Jan 22 at 20:58
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1$\begingroup$ @WillieWong A half-plane is convex and tiles the plane but is not usually called a "polygon." $\endgroup$Timothy Chow– Timothy Chow2026-01-23 03:21:19 +00:00Commented Jan 23 at 3:21
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2$\begingroup$ The definition at wikipedia.org/wiki/Convex_body was my intent, sorry for being imprecise. Compactness is certainly necessary, the plane is covered without translational symmetry by four quarter planes for example (think the standard four quadrants). Once you have compactness Willie's argument will work- any given tile is separated from all the others by finitely many lines, and so is a compact intersection of half planes. I am less certain that the positive measure assumption is needed but one would need to say exactly what it means to tile the plane by intervals for me to be certain. $\endgroup$Matthew Bolan– Matthew Bolan2026-01-23 03:40:50 +00:00Commented Jan 23 at 3:40