[0:00] Breaking Maths News!
[0:01] Some of you will have seen the headline, "New Shape Found That Cannot Pass Through Itself."
[0:06] And I know some of you have seen it, because you've all e-mailed me.
[0:09] I mean what a year to have declared, "The year of breaking maths news."
[0:13] When this Noperthedron was discovered, everyone started e-mailing me;
[0:19] two people walked up to me in real life to tell me about it;
[0:22] one of them had printed out the paper to show me,
[0:25] so thank you everyone for alerting me to the breaking maths news.
[0:29] But, I've got some terrible news: this, this is not my Nopert.
[0:34] This is my Nopert.
[0:37] [Shopping channel music plays]
[0:49] I say my Nopert, not as in I discovered it, Tom 7 discovered this.
[0:53] My Nopert, just as in, this is the one I vote to be Nopert.
[0:58] I believe in this Nopert.
[0:59] Look at that, it's got a pentagon top and bottom.
[1:02] I think, I think it's really neat, check it out.
[1:04] I, I, um, added the colours myself just to make the faces more obvious.
[1:08] Twenty vertices, twenty-seven faces, this shape cannot go through itself, probably...
[1:16] To expand on the word "probably" and "cannot go through itself" we need to go back to the cube.
[1:25] Which I also 3D printed, as, like, as the greatest misuse of 3D print-time ever.
[1:32] But, you know, sometimes you just, you just need a really precise cube.
[1:37] First-up, we're going to talk about what it means to be a Nopert.
[1:40] Indeed, we're gonna recap what it means to be Rupert.
[1:43] And some of you will already know this.
[1:45] I wrote about this in a book back in 2014; the, the idea of being Rupert has been around for a while.
[1:50] And the question was: Prince Rupert, centuries ago wondered, could you fit a cube through itself?
[1:58] And so here, I've also 3D printed another copy of the cube, slightly more complicated this one.
[2:03] And it's exactly the same size, so if I line them up: look at that, boom!
[2:06] Exactly the same in every dimension, these are identical cubes,
[2:10] other than this one has a removable piece.
[2:12] Because we can fit this cube through a cube exactly the same size.
[2:17] If I remove that section, there, there's a cube shaped hole and so, while it seems counter-intuitive,
[2:24] it is possible to fit a shape through itself.
[2:27] Or, at least, that used to be counter-intuitive.
[2:31] Turns out there's loads of room to fit a cube through a cube.
[2:34] So if I, very carefully, align this so you can look straight down the, like, the diagonal axis there.
[2:39] You're looking at the projection of the cube that is a hexagon, that's kind of fun.
[2:43] And it turns out that hexagon is bigger than that square.
[2:46] In fact, you could fit a square 3.5% bigger by length through that hexagon.
[2:53] Oh, if you went off-axis slightly, there's actually another projection that's ever-so-slightly bigger.
[2:58] And you can fit a cube that's 6% bigger by length through a smaller cube.
[3:04] So, um-um, the Rupert property is just:
[3:07] "are there two projections of a polyhedron, convex,
[3:11] such that one of them can fit entirely inside the other one?"
[3:15] And it turns out - yeah, if something's not a sphere, if you move it around its projection changes.
[3:19] Some of the projections are big, some are small, turns out small things can often fit inside big things.
[3:25] So, I mean for the longest time I was like, "oh, would you believe it: a cube fits through a cube!"
[3:29] And after a while everyone's like, "oh yeah, actually it's quite obvious that a cube fits through a cube."
[3:32] A couple of years ago, 2021 I think, I did a video about this where,
[3:36] for Halloween, I put a pumpkin through a pumpkin.
[3:39] So I bought the two most identical pumpkins I could find in the shops
[3:43] and then I cut a hole through one pumpkin that would fit the other pumpkin.
[3:46] So pumpkins are Rupert, although not convex.
[3:49] Anyway, the point is the story went from being,
[3:52] "ooh isn't it amazing that you can fit a cube through itself, that's stunning."
[3:57] And eventually kind of became, "oh turns out it's quite easy to fit shapes through themselves."
[4:02] And now it's gone full cycle and everyone's like,
[4:04] "oh my goodness! Can you believe there are shapes that don't fit through themselves?"
[4:09] So that's the world we live in now.
[4:11] I mean a decade after I wrote "Things to Make and Do in the Forth Dimension", the fact has inverted.
[4:17] So after centuries of showing that shapes are Rupert,
[4:20] it was only this year that humans managed to prove that there is a convex,
[4:25] polyhedron that is "not Rupert" or "Nopert".
[4:29] And I actually saw this one coming because I was in New York to do
[4:35] "An Evening of Unnecessary Detail" last July.
[4:38] And I caught up with Tom 7 and he'd already sent me the paper he was working on.
[4:42] And we talked about how he was trying to research shapes that were Nopert,
[4:47] and he'd found this one at that point.
[4:49] And he was working on a way to prove that, definitely.
[4:53] Now, I'm not gonna lie: this video, you don't need to watch this,
[4:57] you can go and watch Tom 7's excellent video.
[5:00] [Tom] So this one is Nopert No. 214, or potential Nopert No. 214.
[5:06] [Matt] The only issue is, it's an hour and eighteen minutes long.
[5:10] And I love, I love Tom's videos, they're incredible, you should go and watch that video.
[5:15] But I know a lot of you aren't gonna watch that video.
[5:17] And so I thought I would make a Rupert video, 'cos my video is just gonna go through Tom 7's video.
[5:25] Um, so I'll recap some of it and then you should really go watch the rest.
[5:29] Er, but to explain the Nopert journey,
[5:32] Tom 7 first got a bit distracted when he saw an interesting sentence on Wikipedia.
[5:38] [Tom] I was reading about the dodecahedron, which is my favourite, er, Platonic solid.
[5:44] Maybe even my favourite solid.
[5:46] Er, and it came, the Wikipedia page, at least at the time, contained a puzzle.
[5:52] [Matt] On the Platonic solid page,
[5:53] Tom had come across the statement that all five Platonic solids have the Rupert property.
[5:58] And on the page for Archimedean solids, which arguably contains the Platonic solids,
[6:02] It has the mysterious, puzzling sentence that,
[6:05] "at least ten of the Archimedean solids have the Rupert property."
[6:10] Why some? Which got Tom wondering, are there shapes that don't have the Rupert property?
[6:16] [Tom] And I found a claim, a conjecture, that every convex solid has the Rupert property.
[6:24] Which would be amazing.
[6:25] We can do this for eleven of the thirteen Platonic/Archimedean, er, Platonic and Archimedean solids.
[6:33] But there's like two that we, somehow, didn't check.
[6:35] [Matt] It is weird that we have shapes of unknown Rupertness that are Archimedean solids or Catalan solids.
[6:42] After Platonic solids, with their regular faces and vertices, Archimedean solids, they're up there as well;
[6:51] they've got identical vertices and every face is a regular polygon, but you can mix and match the polygons.
[6:59] Then you've got the Catalan solids, the duals to the Archimedean solids; they all have identical faces.
[7:05] These are blockbuster shapes.
[7:08] And yet there are three Archimedean solids and two Catalan solids for which we just don't know.
[7:15] And that seems strange.
[7:18] So Tom 7 put together a numerical solver to take a polyhedron and work out a Rupert solution for it.
[7:25] And for all the solutions that we already have, he found them in milliseconds, just was churning them out.
[7:31] And the ones we couldn't find, he couldn't find with billions of attempts, trillions of attempts.
[7:39] Which means, either the solutions are really obscure or just there aren't any.
[7:46] [Tom] Sometimes this happens, there's a phase change between, like, two cl-two classes of problem.
[7:52] And so, mathematically you know this, it's not enough to have that empirical evidence and say,
[7:57] "well, it can't be solved."
[7:59] But it is suggestive and it kind of taunts you because y-you kinda know.
[8:04] So, err, yes that was part of the suspicion, is that it's very easy to solve them.
[8:11] [Matt] This is a shape that came out of Tom 7's code.
[8:15] 'Cos once he had a solver he could then pair that up with a generator that's producing random polyhedra
[8:23] and then see if he could produce or evolve a Nopert.
[8:28] And he found loads of candidate Noperts via this method,
[8:31] the best of which is candidate number 2 1 4, the 214th Nopert.
[8:37] It's the best in that its only got twenty vertices and 27 faces
[8:43] and that's the fewest of all, candidate Noperts out there.
[8:47] And I like it because it's got this cool, kind-of pentagonal, rotational symmetry.
[8:51] I think it's a really cool shape.
[8:53] And Tom 7 tried literally trillions of different possible alignments of this shape with itself
[8:58] and none of them went through, his solver could not find a solution to this.
[9:02] And the solver's very good.
[9:04] Now that doesn't prove that is is Nopert, but it makes it exceedingly likely that it is.
[9:11] But in maths, we want a proof.
[9:13] One avenue Tom 7 explored to find a provably Nopert polyhedron
[9:19] was to look at the family of prisms that go from "manhole cover" to "churros".
[9:28] [Tom] This really simple case.
[9:30] So take an n-gon and extrude it.
[9:33] And if you extrude it a very shallow amount, then you get a "manhole cover", as I call it.
[9:38] And we know that, you know, "manhole covers" that aren't round can fall through themselves;
[9:43] it's just, it's proof by idiom.
[9:45] And then there, er, if you extrude it a lot, then you get a, a "churro", as I call it, which is a long one;
[9:51] and you can stick that through its own side.
[9:53] And, possibly, there's a s-, a simple point at which they s-, they don't have solutions.
[9:58] And if that's the case, then this could be a really good route to
[10:01] proving the existence of simple Noperts that no one ever thought to check.
[10:06] [Matt] Here are Tom 7's solutions for the 5-churro.
[10:08] And on the horizontal axis, we have how long the churro is, and each of these lines is an integer.
[10:15] So this is a length of 1, or a 'd' of 1, where 'd' is the length of the sides of, in this case, the pentagon.
[10:22] And, as you can see, as the length goes up to 2, to 3, to 4,
[10:25] what you have here is the quality of the solutions.
[10:28] So, sometimes the numerical solver is finding a very high-quality solution.
[10:33] That means that the residue, what's left after the shape's gone through itself,
[10:36] has reasonably, ahh, for some definition of "thick" connectors.
[10:40] Down here, they're not as good, but still connected.
[10:43] And, because it's a numerical solution, sometimes it finds one of these,
[10:46] sometimes it finds one of these, depends how it randomly orientates the two shapes
[10:49] and occasionally they're a little bit better or worse.
[10:51] Overall you can see two, very clear trends:
[10:53] it often find the "drain-hole cover" solution here;
[10:56] Particularly when the churro is very thin, the drain-cover arrangement.
[11:00] And it sometimes still finds solutions like that when it's much-much longer,
[11:05] but once it does get long, it's way more likely to come in through the side.
[11:09] You can see here, these are the churro solutions and they take over.
[11:12] And Tom 7's theory was maybe there's a gap in between.
[11:15] But no, they cross here and, in this case exactly (well, pretty much) on one,
[11:20] and there's a seamless handover.
[11:22] So there's always a solution.
[11:23] And you may have noticed this extra spike, here.
[11:25] That's a different class of solution; that, I don't think it's been investigated yet.
[11:30] So if you want to go down another maths-hole, you can start to look at this.
[11:33] And, we can look at the other churros.
[11:35] If we go up one, to a 6-churro: ahh, hey, look at this!
[11:38] We get all these extra spikes, we still got the "drain-hole cover" solution here,
[11:43] we still got the "churro solution" there.
[11:45] But we now got, I mean in this case, three new bonus ones.
[11:48] And that's way more pronounced on an even-'n' "churro".
[11:51] So when we go up to seven, this is much closer to the five,
[11:54] there's still a spike there but now not as clear.
[11:57] And we got the classic handover.
[11:58] Ooh, look, there are other, small, sub-spikes down here, and we went back to five,
[12:02] you can see they're down there as well; isn't that interesting?
[12:04] But all the even ones have a whole bunch of spikes.
[12:08] So eight, oh, it's always half as many, so eight has these four extra spikes here.
[12:12] The nine looks like the classic odd-case.
[12:15] And then ten, five spikes appearing here.
[12:18] So I'm imagining these are when you've got the "drain-hole solution",
[12:23] these are some orientation effects that we're seeing, but again, I'm purely speculating.
[12:29] Oh, what's going on down here?
[12:30] And as we go up to more and more complicated churros, approximating a cylinder,
[12:36] you can see these patterns continue but start to get, you know, whoa - so messy!
[12:40] As well as prism churros, Tom 7 also looked at anti-churros.
[12:45] These are like a prism, but instead of rectangles on the side, you use triangles.
[12:49] And check out the 5-anti-churro, look at all of this!
[12:53] There's so many more complex solutions happening down there.
[12:56] Is this still the "drain-cover solution"?
[13:01] I mean, we don't know, what's going on here?
[13:02] There's still, you know, a thing's crossing at about 1, in this case.
[13:06] The even ones, now, look at this!
[13:09] Now the fact that this is so covered means, because it's numerical we're not getting exact coverage.
[13:16] I would say, in theory, this should be block covered;
[13:18] which means there's a continuous series of solutions for any given 'd'
[13:23] up to some threshold, which is the top of this.
[13:25] And just the number of numerical solutions that Tom 7 ran means that we've got this kind of spotty coverage,
[13:29] but it should be a lot, erm, more thorough.
[13:31] But you can see there's another one here, we get these extra spikes appearing there.
[13:34] Look at these: seven, eight, nine.
[13:37] Ooh, there's a lot we don't know about how you intersect anti-churros.
[13:43] So, if you want to look into this, go for it!
[13:45] But the point is, for all our original churros, there's always something going on,
[13:49] there's no point where it's gonna hit zero.
[13:50] So, while there's other interesting stuff going on,
[13:53] we're gonna need a different method to try and find a Nopert.
[13:57] [Matt] So when you, ooh-err!
[13:59] [Business Matt] Mind if I borrow this?
[14:00] [Matt] Yeah, fine...
[14:01] [Business Matt] Excellent.
[14:02] I meant the desk.
[14:03] [Matt] Oh, unbelievable.
[14:04] [Business Matt] Because this video is brought to you by Bambu Lab,
[14:07] the 3D printer we used to print all these fantastic objects to explain the Nopert property.
[14:14] Yes, I use my Bambu Lab X1C printer, which is so quick and easy to use, to print the cube you saw before.
[14:24] Of course, the Rupertable cube where the middle bit comes out.
[14:29] And, because it can take multiple filament colours at once,
[14:33] I could print the Nopert with different colours for each of the faces.
[14:39] And if you're looking for inspiration of what to print, you can head over to Maker World;
[14:43] look at all of these fantastic models and, of course, Tom 7 has put up their full set of .pac models.
[14:51] You can open them up in Bambu Studio, oh look at this!
[14:54] There are Platonic, and Archimedean and Catalan solids and you can print them all out.
[15:01] I mean, what home is complete without a full set of these wonderful shapes?
[15:07] Fun fact, as well as these, I did try to print out the churro example, with a very long pentagonal prism.
[15:14] And I added in these support structures, because I knew the walls would be so thin.
[15:20] But I, I totally printed it too small and so there're little tiny gaps there.
[15:25] But you can, you kind of get the idea, like that will like up there, er, and slide through.
[15:30] And I had a lot of fun in Bambu Studio
[15:33] arranging the negative one of these to get the correct hole and add that in.
[15:38] And it were, it was a fun learning experience, n-not quite good enough to be in the main video, though.
[15:43] So do check out Bambu Lab printers!
[15:46] Hey, I am gonna balance these, it will really annoy Regular Matt.
[15:48] Erm, they are a trouble-free, 3D printing experience and if I can use it to print this, you can too.
[15:56] Link in the description.
[15:58] [Matt] Ok, back to the, what happened here?
[16:00] Oh my goodness, those are fragile.
[16:02] Back to the video.
[16:04] Given there are so many ways to arrange a shape in 3D space;
[16:09] I mean, uncountably, infinitely many ways.
[16:11] How can we search them all, how would we prove that something is Nopert?
[16:17] Well, if you look at all the orientations of a shape,
[16:21] all we really care about are the 2D projection and the convex hull.
[16:25] Like, the collection of points and edges that are around the outside.
[16:28] And as I move this around, if I was to move it like a very-very small amount;
[16:34] the same edges from your point of view, would always be on the outside.
[16:38] And if I do a big enough movement, one of the edges might pop behind and so you can no longer see it.
[16:43] And so you could classify every possible projection
[16:48] by unique sets of which vertices are on the convex hull in a specific order.
[16:54] [Tom] I was tying to visualise what is the solution-space like here.
[16:58] And, so what that, erm, what the snub-cube, the textured snub-cube that's in the thumbnail of the video.
[17:05] What that is, is I take all the different ways of looking at the snub-cube.
[17:10] And a different way is basically, like, what is the shape of its shadow, erm, topologically?
[17:16] So which points from the snub-cube are on the shadow and in what order, that convex hull.
[17:21] And there's, oh I forgot the number,
[17:24] 32 or so different ways of looking at the snub-cube that are fundamentally different.
[17:28] [Matt] And he coloured in a snub-cube to show that.
[17:31] There are 36 different colours on this snub-cube and each one,
[17:35] if you're looking straight at it towards the very centre of the snub-cube,
[17:39] that's the particular collection of vertices that are around the outside of its 2D projection.
[17:43] Now I just think it's an amazing visualisation.
[17:46] And that's a way to kind of tame and contain the collections of possible projections.
[17:53] So what do we do with these 36 different classifications
[17:56] of types of arrangements or projections of the snub-cube?
[18:01] Well Tom 7 realised, if you've got your kind of target "outer" snub-cube and then you've got the moving,
[18:08] hopefully "inner" snub-cube, you've now got 36^2 pairings.
[18:15] So that's 1296 ways you can arrange the pair of them together.
[18:21] And then within that, I mean you can fix one and the other one relative.
[18:25] You've still got all its orientations and you can translate it.
[18:30] So, actually there's still seven different variables within each of those 1296 combinations
[18:38] and you've gotta check all of them, even though they're continuous.
[18:43] If you've got seven different continuous variables and you want to search that entire space;
[18:48] you can think of it as a 7D cube.
[18:51] I mean, thinking about it that way is not helpful.
[18:54] But if you want to divide it up, it is kind of helpful.
[18:56] What Tom 7 did was realise if he could take different chunks of 7D volume
[19:04] and within them cover the entirety of the 7D cube.
[19:07] He can show that the entire search place does not contain a solution
[19:11] and each of individual chunks he just had to be able to show,
[19:15] instead of doing individual points within them, that the entire sub-volume wasn't possible.
[19:20] It was a very clever way to go about it and if you want it explained much better than I just did,
[19:26] check out Tom 7's video, link below.
[19:29] [Tom] The main thing to realise is that the extreme values of the output,
[19:32] which is what we are trying to compute, will happen at extreme values of the inputs.
[19:36] But not necessarily in the same order.
[19:38] If I have the interval 2 to 3 and I multiply it by an interval that's just -1, then I get -3 to -2,
[19:45] so they get swapped around.
[19:46] So I basically compute all the end points and do a min and a max over those.
[19:49] [Matt] Spoiler for Tom 7's video, he was beaten.
[19:52] When I spoke to Tom in New York, he was still working on it, wasn't aware anyone else was doing it.
[19:57] Him and his buddies were crunching away to get the whole proof together.
[20:02] And then I got an e-mail from him in September this year saying, "someone else had found a Nopert."
[20:09] And they were doing exactly the same style of proof,
[20:13] by pairing up the different possible orientations and going through it all.
[20:16] However, whereas Tom 7 was trying to prove that a snub-cube, an already existing famous shape was Nopert.
[20:26] These other mathematicians, this is perfectly valid,
[20:28] came up with their own, custom shape that made the proof easier to show that it was Nopert.
[20:35] And that's the Noperthedron that you saw reported in the media.
[20:40] And if you read any of the articles, at least the ones I saw, they didn't go into a lot of depth.
[20:46] Now, you can look up their paper on The ArXiv if you want to go through in excruciating detail what they did.
[20:52] But, because their technique was so close to what Tom 7 was doing,
[20:56] if you watch Tom 7's video it's the best explanation I've seen
[21:00] to how you'd go about proving something like this.
[21:02] Oh and the reason that their Noperthedron was better,
[21:06] was they got the space down to a 5D hypercube
[21:10] and they tackled the issue of the diagonal where you have very similar orientations.
[21:16] So the shapes coming at each other almost exactly match and that's just very hard to deal with.
[21:22] Thanks for watching my video that does just go through Tom 7's video.
[21:27] So I guess in the analogy, this is my video here, and it's gone through Tom 7's video.
[21:33] But now we've got what's called "the residue", the bit that wasn't part of the whole.
[21:38] These are all the bits in Tom 7's video that I didn't talk about,
[21:41] there's some really good stuff in there, I'm gonna link to it.
[21:44] I know it's long, you should watch it and enjoy it.
[21:46] Like a lot of maths videos on YouTube, I mean due the nature of maths, can be a bit of lacking in personality.
[21:53] Let's say, Tom 7's videos have the opposite problem.
[21:58] If you want to watch a different one, I mean his video about harder drives,
[22:03] that makes the most sense in terms of audience overlap with people watching this one.
[22:06] But I like his, his ridiculous chess algorithm video
[22:11] is the one that got me into his channel in the first place, so I highly recommend that one.
[22:16] Erm, I'll link to it below.
[22:17] And please, if someone can prove that this is definitely Nopert, I really want.
[22:23] For me, I mean I know I said this is "my Nopert",
[22:26] but I really want this to be the smallest provably-Nopert, convex polyhedron.
[22:34] So, if we could prove that, that would be incredible.
[22:37] Er, I will link to everything I've mentioned in this video, er, below.
[22:41] Er, please do check it out and go watch Tom 7's videos.
[22:44] Thanks for watching my summary of one of his.
[22:49] That in there.
[22:55] It's just a little bit rough on that one edge, there.
[22:58] Look at that!
[23:00] Push.
[23:03] Slides right through.
[23:05] Woo.
[23:08] And I'll do a bit of this.