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].
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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