First convex polyhedron found that can't pass through itself
quantamagazine.orghttps://arxiv.org/abs/2508.18475
Rupert's snub cube and other Math Holes - https://news.ycombinator.com/item?id=45261566 - Sept 2025 (10 comments)
Rupert's Property - https://news.ycombinator.com/item?id=45057561 - Aug 2025 (23 comments)
Rather 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
Not 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!
And 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
Wow this is such a well made video. So many great insights, just the right level of simplification and also funny as hell!
Wouldn’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.
A 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.
Truly a special gem of a channel.
I 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.
I'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
That 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.
Something 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.
Where 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
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
This was discussed on HN previously https://news.ycombinator.com/item?id=45057561
And I thought that the paper http://arxiv.org/abs/2508.18475 had also been discussed but can’t find it so could be wrong
Also discussed here: https://news.ycombinator.com/item?id=45261566
Paper was posted twice:
https://news.ycombinator.com/item?id=45075566
https://news.ycombinator.com/item?id=45041978
Thanks to you both! Macroexpanded:
Rupert's snub cube and other Math Holes - https://news.ycombinator.com/item?id=45261566 - Sept 2025 (10 comments)
Rupert's Property - https://news.ycombinator.com/item?id=45057561 - Aug 2025 (23 comments)
In case someone is searching for the computational part of the proof, its on GitHub, implemented using SageMath: https://github.com/Jakob256/Rupert
Does 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.
This is specifically about convex polyhedra, I don't see how rotating could help.
If you passed a cube vertex-first through a regular triangle, you'd want to rotate it, right?
I 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?
I 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.
Imagine a fat corkscrew, for instance.
That is not convex
if a convex shape is rifled while it is moving the resulting cut volume could become concave(example 1). does this new concave cut volume free up enough space to meet the rupurt cut challenge where a non-rotating sweep would not?
There are also other rotation profiles, hard to say if they would help or not.
example 1. a cube rifling faces parallel would not generate a concave cut volume but moving diagonally point to far point it would. really whenever a point sticks out. Unfortunately a cube already fits and any help a convex cut volume provides is not needed.
Layperson question: aren't the nopert candidates just increasingly close to being spheres, which cannot have Rupert tunnels?
Yes, 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.
Would 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.
You'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.
But importantly, they’re NOT!
I'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?
Scientific American
Bah, but with those two flat sides I cannot use it for D&D! I’m really rooting for you, rhombicosidodecahedron!
Actually, 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..
Sorry 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.
Mathematicians 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.
The 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.
I remember someone (G.H. Hardy?) saying something along the lines of "the only bad thing about math is that it can be useful."
George Boole might have some opinions on this.
Things 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.
Some 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.
Given 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".
The triakis tetrahedron fit really is crazy close: https://youtu.be/jDTPBdxmxKw
I don't understand why this is "hard". Doesn't a donut have this coveted property? I can't think of a way to drill a hole in a donut that would allow a donut through.
A donut is nonconvex. The title leaves that very crucial word out.
I 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.
Imagine 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.
If 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.
It 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).
> a researcher at A&R Tech, an Austrian transportation systems company
Austrian transport companies research this stuff?!?
I’m both impressed and confused
It 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).
You should see what their patent office researchers get up to.
Relevant Dimensions chapter: https://dimensions-math.org/Dim_CH2_E.htm Video: https://www.youtube.com/watch?v=AhM9JH5GNiI&list=PL3C690048E...
Showing animation for every other shape but one that found, why
For 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...
Are there other mathematical discoveries that came from the result of a wager?
It'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
I bet there are.
You’re on :)
Presumably a simple sphere would trivially qualify as being unable to pass through itself.
The puzzle applies only to convex polyhedra.
The 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.
A sphere is not a convex polyhedron
At the limit of faces they are.
Sure, and pi is the limit of a sequence of rational numbers, but lots of properties that hold for rational numbers don't hold for pi.
As you approach sphere you lose Rupertness.
A sphere has no faces so it's not a convex poloyhedron.
Correction: 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.
Limiting behaviour can be counterintuitive. As you add vertices to a polyhedron, some properties approach those of a sphere (volume, surface area), but others just get further and further away (number of surface discontinuities). It's not at all obvious which way "Rupertness" will go, or even whether it's monotone with respect to vertex addition.
Convex polyhedra are required to be finite polytopes.
What, I can't believe no one came with a term like "anisotransient" for such a property.
Well 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.
> Noperthedron (after “Nopert,” a coinage by Murphy that combines “Rupert” and “nope”).
A good sense of humor to go with the math.
Tom7 is one of my favorite people, he is hilarious, has an amazing technical depth, and so much whimsy to go along with it. I'll proselytize for him all day!
relevant video: https://www.youtube.com/watch?v=QH4MviUE0_s
less relevant, but I think my favorite: https://www.youtube.com/watch?v=ar9WRwCiSr0
this logical falsehood annoyed me since nopert is no+Rupert, whereas nope+Rupert would in fact be nopepert
That's not how portmanteaus work.
Very true. Portmanteaus work by holding your luggage for you.
Tom7 also has a couple of videos about portmanteaus
https://xkcd.com/739/
This 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.
Perhaps you should review what "logical falsehood" means, because that's not one.
The coiner gets to pick the combination that sounds the best, there is no correct choice. We could have gotten breakfunch and mototel, but some person or collection of people decided that brunch and motel work better.
Portmanton't.
Prince Rupert was an incredibly interesting character. This problem was a minor footnote in an impressively rich life.
What about the sphere? Surely a hole bored through a sphere, no matter its size, could not pass a sphere of equal size?
Indeed. A sphere is obviously a nopert shape. The question is whether there are polyhedral shapes with the property.
Fans 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
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.
His 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!
The sphere and anything cylindrical...
The 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.
What does this mean? Does it mean that an object can pass through the largest 2D projection of itself?
He did it!!
So disappointing to not have the 3D printer STL file for this shape. Wish they would have uploaded it to thingiverse or something.
Moritz Firsching made an STL file: https://github.com/mo271/models/commit/85495b9329be3455a5e3c...
it 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
The 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.
Notably, a sphere is non-Rupert (but a soccer ball is not ... it can pass through a tiny fringe).
> 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.
My intuition is very different (and happens to fit reality). Note that convex polyhedra can have asymmetries.
Yes, 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…
Misleading 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.
convex polyhedron
(but your point about the title is valid)
A 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?
The T-shaped tetromino is not convex, so not part of the conjecture. There are many nonconvex shapes that don't have the Rupert property.
That’s true. I digress, but you could give tetrominos convex hulls, and the result would still be somewhat Tetris-compatible.
Nevertheless, 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.
If the remaining edge has exactly zero thickness , it means it doesn’t fit. I think that would be the case in that example?
It fits if it is tilted. If the 3-cube bar of the T is tilted 45° there will be a 3×√2 rectangle part of the shadow, which the 1×2 shadow fits through.
Somewhat unrelated question, a lot of the folks replying to the parent comment read to me like they're really good at visualizing things in their minds eye, when you talk like this is it because you can think about math really well? Can you visualize what you're saying? Sorry if this question doesn't make sense!
I do not have a mind's eye*, outside of the dimmest, briefest flash of people's faces if I've known them for several years. I do have a peculiarly strong sense of imaginary touch, which gets used when contemplating problems like these. Also, a significant component of my job is arranging things in 3-space. I'd be one of those people who say "I'm good at tetris" while packing a moving truck, but I am not actually good at tetris per se.
* https://en.wikipedia.org/wiki/Aphantasia
Thank you for your reply. Extremely interesting to me. My thinking and memory is pure motion picture and sound, it makes me think that I could be good at geometry as I can do spatial thinking well, however the downside of my thinking style is I've never been able to find a framework that allows me to hold fine detail symbols in space and work with them usefully, I suppose hence dyslexia + dyscalculia diagnosis. Maybe people like me who are in math use whiteboards and notes a lot or something? Maybe people like me don't go into maths so much. Imagination of touch is also very interesting, I've read about kinetic leaners before, I like to touch things when I'm learning as it helps with the recall later, but absolutely zero sense of touch is present in the recall. Sorry for the massive tangent, I'm just very curious about this stuff!
I don't think that's the case, how can you turn a bar of length two but still have it fit within the width of the bar of length three?
The long side is three cubes long, the short side two. You can easily move the short side through the long side's shadow if you tilt the latter so it becomes wider than a cube's side.
For laymen's sake I think the title should say "First shape (without curves) found that [...]"
And not "pass through itself" but "pass through its copy"
The article isn't really for the layperson. It's confusing why several people are nitpicking at the title.
Because not-laypoeople ale precisely the kind of people who would nitpick the technically incorrect title.
I'm a lay person w.r.t. this topic, and I assumed the great exposition was meant exactly for persons like me.
This is quanta magazine. It is for lay people. The reason people are "nitpicking" the title is that "shape" is not a technical term. The technical term for what was found is "convex polyhedron". I read so much of the article before I was sure that it was talking about convex polyhedra specifically because the title is so ambiguous.
Oh come on. Quanta Magazine basically writes for HN. They have very little online footprint elsewhere, but they feature here multiple times a week and I'm sure they know it. The titles are almost always in this mold, implying some profound yet vague discovery. Some real, recent examples:
I don't necessarily mind it, even if I don't find the articles very informative. But it's certainly fair game to nitpick this borderline-clickbait style.Quanta Magazine is very much designed for non-technical lay people.
From their About page: Quanta Magazine is an editorially independent online publication launched by the Simons Foundation in 2012 to enhance public understanding of science.
Does a sphere not pass through itself (with zero margin?)
There needs to be nonzero margin, else the question is pretty trivial.
Ah yes correct you just use same orientation and you can get anything through itself.
You can do it regardless of orientation given you can apply enough pressure
How thick would the remaining walls be after you’ve made the hole required?
Infinitesimal I guess! You can fit an arbitrarily slightly smaller sphere for vanishing values of smaller.
Why wouldn't a sphere pass through itself? The projected shadow has the same size as its diameter
A polyhedron has the Rupert property if a polyhedron of the same or larger size and the same shape as can pass through a hole in the original polyhedron.
A sphere is a surface of constant width, which the polyhedron approximation is not.
> The projected shadow has the same size as its diameter
Thus this is exactly why the sphere doesn't have the Rupert property.
Ok, so by that definition a geodesic sphere has the Rupert property, as the sphere is an approximation made up of equilateral triangles. What if we perform isotropic subdivision on the equilateral triangles, such that each inserted point lies on the sphere, centred on each base triangle. We then subdivide each base triangle by constructing 3 new triangles around the inserted point. Thus at each iteration, geodesic sphere of N triangles is subdivided into 3*N triangles. If we continue with the subdivision, each iteration is a refinement of the geodesic sphere, and the geometric approximation gets closer to the shape of a true sphere. As N approaches infinity, the Rupert property holds true (according to the definition). What happens at infinity?
At infinity, the shape becomes a sphere and all orientations of it are identical. It is no longer a convex polyhedron and, thus, not subject to consideration.
I would guess the margin goes toward 0.
Why do you say that the Rupert property applies for all finite N?
A sphere is not an infinity-sided polyhedron. It's a sphere. It's also the limit of a sequence of polyhedra, each of which does not have infinity sides. Just like aleph-null is the limit of the sequence of natural numbers, but is not a natural number.
Wouldn't you need a little material "left over" to claim that it can pass through itself? Two spheres of equal size wouldn't work because they would occupy exactly the same space.
Yes!
The "pass through itself" criteria is the same as "has one shadow that fits entirely inside another shadow". If you allow "one shadow equals another shadow" then it's trivially true for every shape because a shadow equals itself.
Note that this "shadow" language assumes a point light source at infinity, i.e. all the rays are parallel.
That's trivially true for every shape, so it's probably not interesting in the context of this puzzle.
I think Sphere is a outlier for this context.
Yeah I’m confused
The shadow has to be bigger for the other shape to pass through. There's no way to orient a sphere so its shadow becomes bigger. For a cube there is.
Make a 2” inner diameter cylindrical hole in a 2” diameter sphere.
That would depend on the light source and its size and distance.
[dead]
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I conspire to be a colonial governor!
I’d be happy just winning a bet!
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