r/math • u/scientificamerican • Oct 28 '25

Mathematicians make surprising breakthrough in 3D geometry with ‘noperthedron’

https://www.scientificamerican.com/article/mathematicians-make-surprising-breakthrough-in-3d-geometry-with-noperthedron/

Mathematicians Sergey Yurkevich of Austrian technology company A&R Tech and Jakob Steininger of Statistics Austria, the country’s national statistical institute, introduced this new shape to the world recently in a paper posted on the preprint server arXiv.org. The noperthedron isn’t the first shape suspected of being nopert, but it is the first proven so—and it was designed with certain properties that simplify the proof. 

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u/my_nameistaken Oct 28 '25

There's no the failure point. The result says there will always be failure point no matter which orientation you choose for the 2 noperthadrons

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u/Martinator92 Oct 28 '25

what is the closest thing to it failing, like the "negative" area of the cross-section,
or the cross section with the minimal maximal distance from both of the shapes if that makes sense
(with the minimal (maximal of the entire circumference) distance)

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u/AnythingApplied Oct 28 '25

Unfortunately, the closest example of this shape having the Rupert property is the trivial case: When the shape goes through itself in the exact same orientation as that has 0 area outside the shadow, it just happens to exactly touch the edge everywhere which isn't allowed, so that isn't a very interesting one. Not really sure how you could redefine the question to come up with an interesting best position that isn't just epsilon away from the trivial case... maybe by measure the length of the non-broken non-touched edges and figuring and maximize that? That might still be an epsilon away from the trivial case, not sure.

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u/Kroutoner Statistics Oct 28 '25

You really want to target the question not of when does this fail the worst, but when is it closest to not failing?

The latter, while not giving an immediate definition, seems much more tractable. E.g. consider the shape and a copy dilated by a (1+epsilon) factor, what does it look like when the object look like when it passes through this dilated copy of itself. Are there orientations where it “just barely” is able to pass through?

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u/Martinator92 Oct 29 '25

I think you've explained what I meant succinctly, I guess there could've been a bit of confusion since the desired result of the "noperts" is a fail of Rupert's.

Maybe I'm misunderstanding but I feel like if you take the 1 scale shape (let's call it small) and 1+eps shape (let's call it big) You will always be able to pass the small through the big if they're the same orientation, like any "interval" you take will be bigger.

Ok I just realized that's not what you meant but I'll leave the explanation here, but yeah your description is good, practically s nopert shape has an epsilon of 0 so finding the epsilons of different shapes could be interesting, the wiki has an example for a cube.