Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle

56 points by tzury 3 days ago|4 comments

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abetusk 2 days ago

I'm no expert but, from what I understand, the idea is that they found two 3D shapes (maybe 2D skins in 3D space?) that have the same mean curvature and metric but are topologically different (and aren't mirror images of each other). This is the first (non-trivial) pair of finite (compact) shapes that have been found.

In other words, if you're an ant on one of these surfaces and are using mean curvature and the metric to determine what the shape is, you won't be able to differentiate between them.

The paper has some more pictures of the surfaces [0]. Wikipedia's been updated even though the result is from Oct 2025 [1].

[0] https://link.springer.com/article/10.1007/s10240-025-00159-z

[1] https://en.wikipedia.org/wiki/Bonnet_theorem

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matheist 2 days ago

To be precise, the mean curvature and metric are the same but the immersions are different (they're not related by an isometry of the ambient space).

Topologically they're the same (the example found was different immersions of a torus).

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OgsyedIE 2 days ago

Is it the case that 'they' are simply two ways of immersing the same two tori in R^3 such that the complements in R^3 of the two identical tori are topologically different?

If so, isn't this just a new flavor of higher-dimensional knot theory?

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matheist 17 hours ago

They don't appear to care about the images of the immersions or their complements, aside from them not being related by an isometry of R^3. They're not doing any topology with the image.

In other works, they have two immersions from the torus to R^3, whose induced metric and mean curvature are the same, and whose images are not related by an isometry of R^3. I didn't see anything about the topology of the images per se, that doesn't seem to be the point here.