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.
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.
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?
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.
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.
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.
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.
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..
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.
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.
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?
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.
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.
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.
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.
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?
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).
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!
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.
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.
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.
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.
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.
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 ...
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.
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…
>Prince Rupert of the Rhine, a 17th-century army officer, naval commander, colonial governor and gentleman scientist, won a bet about whether it’s possible to pass a cube through another.
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
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.
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.
For laymen's sake I think the title should say "First shape (without curves) found that [...]"
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.
Why do you say that the Rupert property applies for all finite N?
I would guess the margin goes toward 0.
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.
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
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.
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..
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
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".
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.
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.
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?
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.
But importantly, they’re NOT!
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
What, I can't believe no one came with a term like "anisotransient" for such a property.
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.
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.
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'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
> 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.
> 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.
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.
Perhaps you should review what "logical falsehood" means, because that's not one.
Portmanton't.
Presumably a simple sphere would trivially qualify as being unable to pass through itself.
The puzzle applies only to convex polyhedra.
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.
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.
Prince Rupert was an incredibly interesting character. This problem was a minor footnote in an impressively rich life.
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.
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!
What does this mean? Does it mean that an object can pass through the largest 2D projection of itself?
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...
He did it!!
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…
>Prince Rupert of the Rhine, a 17th-century army officer, naval commander, colonial governor and gentleman scientist, won a bet about whether it’s possible to pass a cube through another.
Based.
I aspire to be a gentleman scientist!
I conspire to be a colonial governor!
I’d be happy just winning a bet!
Good news: you can start today.