[HN Gopher] Two Twisty Shapes Resolve a Centuries-Old Topology P...
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       Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle
        
       Author : tzury
       Score  : 49 points
       Date   : 2026-01-26 20:12 UTC (1 days ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | abetusk wrote:
       | 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
        
         | matheist wrote:
         | 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).
        
           | OgsyedIE wrote:
           | 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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       (page generated 2026-01-28 07:01 UTC)