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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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..
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.
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?
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.
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.
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).
> 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.
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.
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.
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.
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