7 ms·
First convex polyhedron found that can't pass through itself
https://arxiv.org/abs/2508.18475 https://arxiv.org/abs/2508.18475
Rupert's snub cube and other Math Holes - https://news.ycombinator.com/item?id=45261566 https://news.ycombinator.com/item?id=45261566 - Sept 2025 (10 comments)
Rupert's Property - https://news.ycombinator.com/item?id=45057561 https://news.ycombinator.com/item?id=45057561 - Aug 2025 (23 comments)
- deleted 1y ago[deleted]
- ratelimitsteve 1y agoit intuitively feels impossible because it sounds like the definition of "can pass through itself" is really "has at least one orientation where all of the sides of one instance are at most as long as all of the sides of the other instance" and then however you define an orientation an instance of a shape in orientation X should be able to pass through an instance of the same shape and size in the same orientation
- hyperhello 1y agoYes, and when you think of it that way, it sounds like a partial ordering with a base case. If angle A can pass through angle B, and angle B can pass through angle C…
- strbean 1y agoThe criteria is "pass through itself without cutting in half". Presumably that extends to "without deleting the object entirely", which is what would happen to pass through in the same orientation.
- jibal 1y agoNotably, a sphere is non-Rupert (but a soccer ball is not ... it can pass through a tiny fringe).
- thaumasiotes 1y ago> Notably, a sphere is non-Rupert (but a soccer ball is not ... A soccer ball is a sphere. It has decorative polygons projected onto its spherical surface, but having a color scheme doesn't stop it from being a sphere.
- jibal 1y agoA soccer ball in this context is considered to have planar faces (IRL those faces aren't planar because of the air pressure bowing them).
- jibal 1y agoMy intuition is very different (and happens to fit reality). Note that convex polyhedra can have asymmetries.
- king_geedorah 1y agoRather interesting solution to the problem. You can't test every possibility, so you pick one and get to rule out a bunch of other ones in the same region provided you can determine some other quality of that (non) solution. I watched a pretty neat video[0] on the topic of ruperts / noperts a few weeks ago, which is a rather fun coincidence ahead of this advancement. [0] https://www.youtube.com/watch?v=QH4MviUE0_s https://www.youtube.com/watch?v=QH4MviUE0_s
- anyfoo 1y agoNot that coincidental. tom7 is mentioned in the article itself, and in his video's heartbreaking conclusion, he mentions the work presented in the article at the end. tom7 was working on proving the same thing!
- cubefox 1y agoAnd he tried to disprove the general conjecture, that every convex polyhedron has the Rupert property, by proving that the snub cube [1] doesn't have it. Which is an Archimedean solid and a much more "natural" shape than the Noperthedron, which was specifically constructed for the proof. (It might even be the "simplest" complex polyhedron without the property?) So if he proves that the snub cube doesn't have the Rupert property, he could still be the first to prove that not all Archimedean solids have it. 1: https://en.wikipedia.org/wiki/Snub_cube https://en.wikipedia.org/wiki/Snub_cube
- brabel 1y agoWouldn’t this problem be related to the problem of finding whether two shapes collide in 3d space? That would probably be one of the most studied problems in geometry as simulations and games must compute that as fast as possible for many shapes.
- MyOutfitIsVague 1y agoA test for this one is a bit simpler, I think, because you just have to find a 2D projection of the shape from multiple orientations so one fits inside the other. You don't technically have to do any 3D comparisons beyond the projections. It's pretty easy to brute force most shapes to prove the property true. The challenge is proving that a shape does not have the Rupert property, or that it does when it's a very specific and tight fit. You can't test an infinite number of possibilities.
- moralestapia 1y ago* * *
- greenchair 1y agoI aspire to be a gentleman scientist!
- mrguyorama 1y agoFans of "Tom7" should be very recently familiar with this! He released a video about the Ruperts problems and his attempt to find a Nopert on just Sept 16th! https://www.youtube.com/watch?v=QH4MviUE0_s https://www.youtube.com/watch?v=QH4MviUE0_s With this and the Knotting conjecture being disproven, there are have some really interesting math developments just recently! Tom regularly releases wonderful videos to go with SIGBOVIK papers about fun and interesting topics, or even just interesting narratives of personal projects. He has that weird kind of computer comedy that you also get from like Foone, the kind where making computers do weird things that don't make sense is fun, the kind where a waterproof RJ45 to HDMI adapter (passive) tickles that odd part of your brain.
- chaps 1y agoHis videos are some of the best out there. Super funny, depth that's rarely seen elsewhere, and a refreshingly scrappy academic approach. His video on kerning being an incomputable problem is filled with rigor and worth a watch. Highly recommend all of his videos!
- biot 1y agoPresumably a simple sphere would trivially qualify as being unable to pass through itself.
- smokel 1y agoThe puzzle applies only to convex polyhedra.
- biot 1y agoThe article says: > The full menagerie of shapes is too diverse to get a handle on, so mathematicians tend to focus on convex polyhedra The phrase "tend to focus on" suggests it's not an exclusive thing. However, you're right -- it appears that the Rupert property only applies to convex polyhedra, so the article title and text is at the very least incomplete given that a sphere is a shape.
- LostMyLogin 1y agoA sphere is not a convex polyhedron
- guelo 1y agoAt the limit of faces they are.
- jibal 1y agoA sphere has no faces so it's not a convex poloyhedron.
- burnt-resistor 1y agoCorrection: a sphere has infinite faces so it's not an "convex poloyhedron [sic]." A convex polyhedron must have finite faces, so apeirotopes aren't allowed.
- jibal 1y ago
- jmkd 1y agoLayperson question: aren't the nopert candidates just increasingly close to being spheres, which cannot have Rupert tunnels?
- tmiku 1y agoYes, they get visually more sphere-like as more faces are added. But spheres are obviously/trivially non-Rupert, while the question of whether a convex polyhedron can be non-Rupert is more interesting.
- gitaarik 1y agoWould be interesting to see how much sides you can keep adding before the shape can't pass through itself. Or maybe you can indefinely keep passing them through, occasionally encountering noperts. Or maybe the noperts gradually increase, eventually making the no-nopperts harder to find. Who knows, let's find out.
- KernalSanders 1y agoYou'd probably end up with tighter and tighter tolerances such as they mention with the triakis tetrahedron. The challenge is that it gets computationally intensive the more sides that you add if you don't have shortcuts like ruling out entire blocks of orientations in their parameter space (they figured out that if one shadow, projection, protrudes significantly, then you'd need a large rotation to get that protrusion into the other shadow, thus removing all of those rotational angles and reducing the number of orientations needed to check). More sides and more symmetry make it much harder to test a candidate, but you have an interesting idea.
- maplant 1y agoBut importantly, they’re NOT!
- dnw 1y ago> Noperthedron (after “Nopert,” a coinage by Murphy that combines “Rupert” and “nope”). A good sense of humor to go with the math.
- 867-5309 1y agothis logical falsehood annoyed me since nopert is no+Rupert, whereas nope+Rupert would in fact be nopepert
- strbean 1y agoThat's not how portmanteaus work.
- gary_0 1y agohttps://xkcd.com/739/ https://xkcd.com/739/
- stephenlf 1y agoTom7 also has a couple of videos about portmanteaus
- foobarbecue 1y agoVery true. Portmanteaus work by holding your luggage for you.
- thaumasiotes 1y agoThis is actually a really interesting point. English portmanteaus usually work by combining all of one word with "half" (broadly construed) of the second word. Nopert fits the pattern precisely, including all of nope and half of Rupert. The reason I find this so interesting is that Mandarin Chinese portmanteaus take a different standard form: instead of combining all of one word with half of the other word, they combine half of one word with half of the other word. Think about how much you'd need to know about the structure of an arbitrary language before you'd feel confident predicting how it creates portmanteaus.
- tempestn 1y agoI really like the level of detail in this article. It was enough that I felt like I could get an actual understanding of the work done, but not into such mathematical detail that it was difficult to follow.
- teo_zero 1y agoMisleading title. Other shapes have been well known for years, like a sphere. The novelty here is the first polyhedron that can't pass through itself.
- jibal 1y agoconvex polyhedron (but your point about the title is valid)
- cluckindan 1y agoA sphere can be approximated by a polyhedron. Somewhat obviously, all such polyhedra would seem to have the Rupert property. This new Nopert seems to differ in one key detail: some of the vertices near the flat top/bottom are at a shallower angle to the vertical axis than the vertices below/above them. Can you pass the T-shaped tetromino through itself?
- mkl 1y agoThe T-shaped tetromino is not convex, so not part of the conjecture. There are many nonconvex shapes that don't have the Rupert property.
- boothby 1y agoNevertheless, the t-shaped tetronimo (assuming four glued cubes) has a shadow shaped like a bar of length two. I believe that such a shadow will pass through a bar of length three, with a tilt similar to the cube's.
- stephenlf 1y agoHe did it!!
- cyode 1y agoI'd love to have an in-print magazine with articles of this subject matter and level of detail. Especially for older kids...accessible and interesting content without all the internet's distractions. Googling says Quanta is online only. Anyone know of similar publications that print?
- TheOtherHobbes 1y agoPrince Rupert was an incredibly interesting character. This problem was a minor footnote in an impressively rich life.
- somat 1y agoDoes it have to be straight through? I can imagine a scenario where the moving shape has to be rotated as it passes through. sort of analogous to some of those block puzzles or getting a sofa around a corner. The article does say straight through and most analyses has been done with variation of the shadow technique, which has to be straight through. But the original bet. The thing that started this whole line of thought just said you had to get one through its copy, I think rotating is is an acceptable technique in this problem.
- jstanley 1y agoThis is specifically about convex polyhedra, I don't see how rotating could help.
- somat 1y agoI don't really know, I am currently farting around with blender trying to see, but that is far from rigorous. and going poorly. but let me explain my thought process. Note the egg shape in the article. specifically the widest band around the equator. now imagine one passing straight down through the other. one edge ring would pass through the shadow if it has a slight rotation offset but it is blocked by the next edge ring up, which could also fit but requires a different offset, so if you could change that rotational offset while it is passing through would it fit?
- boothby 1y agoI have the same question -- the problem of moving a couch around a corner is a nonconvex problem, but I suspect that pivoting, or perhaps a helical "rifling" motion, may avoid a vertex:face contact.
- JoeAltmaier 1y agoImagine a fat corkscrew, for instance.
- celeritascelery 1y agoThat is not convex
- zem 1y agoI'd only heard of Prince Rupert because of his eponymous "prince Rupert's drops", but apparently he had not just one but several dazzling careers https://en.wikipedia.org/wiki/Prince_Rupert_of_the_Rhine https://en.wikipedia.org/wiki/Prince_Rupert_of_the_Rhine
- KernalSanders 1y agoThat military career is quite a rollercoaster. Quick-thinking but also youthfully impatient, clearly disciplined enough to rise in the ranks but kicked all around based on how history went. It's pretty amazing that his achievements spanned quite different areas beyond just the military.
- andy99 1y agoThis was discussed on HN previously https://news.ycombinator.com/item?id=45057561 https://news.ycombinator.com/item?id=45057561 And I thought that the paper http://arxiv.org/abs/2508.18475 http://arxiv.org/abs/2508.18475 had also been discussed but can’t find it so could be wrong
- beefman 1y agoAlso discussed here: https://news.ycombinator.com/item?id=45261566 https://news.ycombinator.com/item?id=45261566 Paper was posted twice: https://news.ycombinator.com/item?id=45075566 https://news.ycombinator.com/item?id=45075566 https://news.ycombinator.com/item?id=45041978 https://news.ycombinator.com/item?id=45041978
- dang 1y agoThanks to you both! Macroexpanded: Rupert's snub cube and other Math Holes - https://news.ycombinator.com/item?id=45261566 https://news.ycombinator.com/item?id=45261566 - Sept 2025 (10 comments) Rupert's Property - https://news.ycombinator.com/item?id=45057561 https://news.ycombinator.com/item?id=45057561 - Aug 2025 (23 comments)
- ohyoutravel 1y agoSo disappointing to not have the 3D printer STL file for this shape. Wish they would have uploaded it to thingiverse or something.
- dwrensha 1y agoMoritz Firsching made an STL file: https://github.com/mo271/models/commit/85495b9329be3455a5e3c1b3316c757bdb0f2c42 https://github.com/mo271/models/commit/85495b9329be3455a5e3c...
- dyauspitr 1y agoWhat does this mean? Does it mean that an object can pass through the largest 2D projection of itself?
- Havoc 1y ago> a researcher at A&R Tech, an Austrian transportation systems company Austrian transport companies research this stuff?!? I’m both impressed and confused
- megablast 1y agoYou should see what their patent office researchers get up to.
- cool_dude85 1y agoIt seems like both the authors on this paper were hobbyists (though, to be fair, trained mathematicians/statisticians, as one has a masters and the other a PhD).
- tug2024 1y ago[dead]
- willmadden 1y agoThe sphere and anything cylindrical...
- MichaelDickens 1y agoThe title says "first shape found" but the article clarifies that it's really the first convex polyhedron. A sphere isn't a convex polyhedron, so it doesn't quality for the (now-disproven) conjecture.
- fuglede_ 1y agoThe triakis tetrahedron fit really is crazy close: https://youtu.be/jDTPBdxmxKw https://youtu.be/jDTPBdxmxKw
- n4r9 1y agoGiven how hard it was to find one example, the next result is bound to be something like "almost all convex polyhedra cannot pass through themselves".
- halapro 1y agoSorry for the silly question, but why spend time on this? Is it just for fun or is all mathematical exploration eventually useful? This feels closer to art than engineering.
- staplers 1y agoThings like this sometimes lead to practical inventions like velcro or self-locking mechanisms that could be useful. All it takes is someone to connect the dots or find a use case for it and change the world in a small way.
- kvdveer 1y agoThe problem itself might not be very applicable, but the techniques used to solve it might be. That said, researching something solely for the sake of curiosity can be a valid endeavour. Many profound scientific discoveries have been made by researching topics with no obvious application.
- mikepurvis 1y agoMathematicians spent decades agonizing about matrix transformations and surface normals, all entirely in the abstract, and then in the 80s that math turned out to be suddenly extremely practical and relevant to the field of computer graphics.
- halapro 1y agoWho pays for this abstract exploration? I get that "in the future" it could be useful, but today these researchers need money. Is it like open source where people just do it because they want to?
- ndsipa_pomu 1y agoSome maths is done just for fun/curiosity and other maths is more directed towards a specific goal. Both types of maths can end up being useful or useless (outside of maths, that is) and it's pretty much impossible to predict which areas are going to be incredibly useful or mere curiosities. Sometimes, the uses for obscure maths aren't even discovered for more than a century afterwards.
- psychoslave 1y agoWhat, I can't believe no one came with a term like "anisotransient" for such a property.
- juris 1y agoBah, but with those two flat sides I cannot use it for D&D! I’m really rooting for you, rhombicosidodecahedron!
- sans_souse 1y agoActually, perhaps you're onto something there - didn't Rupert's original conjecture specify polyhedron dice? Perhaps symmetry is one of the requirements for the law..
- dotancohen 1y agoWhat about the sphere? Surely a hole bored through a sphere, no matter its size, could not pass a sphere of equal size?
- mcv 1y agoIndeed. A sphere is obviously a nopert shape. The question is whether there are polyhedral shapes with the property.
- ashvardanian 1y agoIn case someone is searching for the computational part of the proof, its on GitHub, implemented using SageMath: https://github.com/Jakob256/Rupert https://github.com/Jakob256/Rupert
- n1b0m 1y agoAre there other mathematical discoveries that came from the result of a wager?
- mcv 1y agoI bet there are.
- n1b0m 1y agoYou’re on :)
- someguyorother 1y agoIt's only mathematics-adjacent, but Stephen Hawking was known for making quite a few bets. https://www.science.org/doi/10.1126/science.359.6382.1317 https://www.science.org/doi/10.1126/science.359.6382.1317
- PunchyHamster 1y agoShowing animation for every other shape but one that found, why
- teekert 1y agoI feel like the next Voyager type vessel should include this shape made from gold or titanium or something. Also add an Einstein-tile. What more should we include? Perhaps that knot that has a none additive “unknot” from a recent Stand-up Maths episode as well… All Easter-eggs from our universe we found so far.
- bArray 1y agoImagine them being discovered by a less-advanced alien race, where they find a box full of "impossible" facts. I wonder what a box would look like for us.
- ultratalk 1y agoIf it was that less-advanced of a race, then they probably wouldn't be able to figure out that they were "impossible" in the first place. For a race at or around the level of humans, these would just be counter-examples to their conjectures.
- bpodgursky 1y agoIt would be challenging to be a less advanced race that's still able to intercept a Voyager probe (not like it will survive atmospheric re-entry).
- hshdhdhehd 1y agoWell it is a nice looking shape. Im gonna print the STL linked in the article. Needed an excuse to fire up the Bambu after months.
- diffuse_l 1y agoFor some reason, it really bothers me that under one of the images there is a caption that says "the pink cube", but the cube is in a shade of blue...
- vismit2000 1y agoRelevant Dimensions chapter: https://dimensions-math.org/Dim_CH2_E.htm https://dimensions-math.org/Dim_CH2_E.htm Video: https://www.youtube.com/watch?v=AhM9JH5GNiI&list=PL3C690048E1531DC7&index=2 https://www.youtube.com/watch?v=AhM9JH5GNiI&list=PL3C690048E...
- daxfohl 1y agoSomething still feels off if the formal proof can't be understood. I don't dispute its correctness, but there's a big jump from 4 color theorem, where at least mathematicians understood the program, to this, where GPT did the whole thing. Like if GPT ceased to exist, nobody would have a clue how to recreate the formalization. Or maybe there's a step in there that's a breakthrough to some other problem, but since it was generated, we'll never notice it.
- dwrensha 1y agoWhere do you see any mention of GPT? The computer-assisted component of the Noperthedron proof is a reasonably small sagemath program that was (as far as I know) written by humans: https://github.com/Jakob256/Rupert https://github.com/Jakob256/Rupert Perhaps you have confused this article with a recent unrelated announcement about a vibe-coded proof of an Erdos conjecture? https://borisalexeev.com/pdf/erdos707.pdf https://borisalexeev.com/pdf/erdos707.pdf
- daxfohl 1y agoOops you're right! I read these both yesterday and they blended together in my memory by the time I made this comment this morning. I knew something felt "off". Tangentially I'll have to reconsider my position on long but lossy context LLMs.