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Yes, as discussed in the article, the shapes are not piecewise smooth curves (i.e., not the sort of shapes you can construct by making a finite number of straig
by jessriedel 5y ago
Yes, as discussed in the article, the shapes are not piecewise smooth curves (i.e., not the sort of shapes you can construct by making a finite number of straight and smoothed cuts). Furthermore, as also mentioned, the areas of the shapes are non-measurable, so the perimeter is probably non-measurable too.
- Sniffnoy 5y agoThe areas aren't non-measurable, that's part of the point. Doing it with non-measurable pieces was already known.
- jessriedel 5y agoThe Quanta article says that doing it non-constructively with non-measurable pieces was already known, and that later results did it partially and then fully constructively (without leaving a measure-zero set). But it does not say in the Quanta article whether or not the pieces in these later results were individually measurable. (The union of non-measurable sets can be measurable.) Are you saying that you know they are?
- Sniffnoy 5y agoWell, when I wrote that I was just inferring it from the article, which while it doesn't explicitly say that they are, is written in a way so as to clearly suggest that they are. But you're right that it doesn't eplicitly say that, now that I look again. But if you click on the link to the actual paper, it says right in the abstract that they're Borel, so yes, they are indeed measurable.
- jessriedel 5y agoThanks!