Noble Polyhedra Classification — Full Breakdown & Transcript

[00:00] So we've got some big news from the world of polyhedra, which is that we now have a complete

[00:18] classification of a particular type of polyhedra called noble polyhedra. Right, so whenever I'm talking about polyhedra I will start with looking at a cube. Because a cube captures

[00:30] basically the main ideas of the most symmetric type of polyhedra. All its faces are identical, in fact they're interchangeable, you can get from any face to any other by moving the cube about. And exactly the same thing goes for the

[00:45] edges and for the corners. Mike said the word verticines which means the same thing as corners in this case. So this criteria capture most of what we mean by a regular polyhedron. There's one other point of symmetry we need which is that the

[00:59] faces themselves are highly symmetric. So in this case the faces are squares obviously, but more generally we want the faces to be regular polygons. So let me

[01:11] just quickly sketch a square. I think we all know what it looks like. But the key point about a square is that it too has got those symmetry properties. So the

[01:23] edges are all equal length and the angles are all equal. So in this case they're all right angles and those four edges are all equal. So that's what we mean by a regular polygon. So those kind of things I've described so far describe what we mean by a regular

[01:38] polyhedron and famously the ancient Greeks were able to give a complete list of all possible regular polyhedron known as the Platonic solids. There are five of them, like the cube, the octahedron, the tetrahedron, the dodohedron and the icosahedron.

[01:55] So these are polyhedra that are made of regular polygons and their building blocks are sides. Yes, plus each one only one type of regular polygons and in such a way that all the corners

[02:09] are identical and all the edges are identical and so on. So there's definitely only five, the ancient Greeks proved that, and then in the 17th century Johannes Kepler discovered a 6 and he did that by spotting a possible loophole in

[02:24] what we mean by a regular polygon, okay? So remember I said this square is a regular polygon because all its sides are equal lengths and all its angles are equal and what Kepler said is, okay, what about this shape, a pentagram, right? This has got

[02:39] five edges, they're all equal lengths, all of these angles here are equal, so what about that as a regular polygon? And of course at this point we have a choice. We can say well we didn't want that, we didn't want the edges to crash through each other, we can just

[02:53] outlaw that. And if you do that then, okay, you're left with the original for sonic solids. What's more fun and what we're going to do now is say okay let's run with that, let's allow the edges to crash through each other and see what you get.

[03:06] The shape that Kepler found was this one here. So this is called the small stellated dodecahedron. So an ordinary dodecahedron is built from 12 pentagons.

[03:18] This one here is built from 12 pentagrams. All 12 of them together make up that shape. And he found another one which is similar. This is also built from 12 pentagrams.

[03:30] It's called the great stellated dodecahedron. The difference here is that those pentagrams meet at three per corner rather than five per corner. And then a little bit bit later Louis Poirot came up with two other shapes which if you're going to allow these

[03:44] you have to allow his as well and this is one of them it's called the great dodecahedron and just like the ordinary dodecahedron it's made from 12 pentagons but the difference here is that you can

[03:58] see the pentagons cut through each other so there is one of the pentagons so they just can is configured differently so that the pentagons cut through each other instead of lying on the outside as they do in the ordinary. So it's not just we're allowing sides of the polygon to cut through

[04:13] themselves, we're also allowing the faces of the polyhedra to cut through themselves. Right, and if you're going to allow one it seems reasonable to allow the other, so that's the world we're in, we're allowing that for all of this. Yeah, so that was one of Ponce's and then he discovered one more

[04:27] which is the great icosahedron. So the ordinary icosahedron is made from 20 equilateral triangles, This is also made from 20 equilateral triangles, but those equilateral triangles cut through each other so you can see them there.

[04:42] So it's like a sort of rewiring of the faces of an icosahedron. We're up to nine now, aren't we? Yeah, that's now a complete list of the regular polyhedra, right?

[04:57] So what we're going to now move on to is to discuss the noble polyhedra. So the for the noble polyhedra, what do we need? We're going to hold on to some of those criteria but let others go. In particular what we're holding on to is

[05:13] we want every corner to be identical to every other and we want every face to be identical to every other but we don't require the edges to be identical to each other and we don't require the faces to be regular polygons. That's what

[05:28] makes a noble polyhedron and there are rather mysterious little species of polyhedra that people started looking at in the 19th century so here's an example of one so this is well it's a sort of a tetrahedron a tetrahedron is made from

[05:46] four triangles this is also making four triangles but usually the tetrahedron is made from equilateral triangles this is a sort of wonky tetrahedron so this is not even even made up an isosceles triangle. But nevertheless you can see it satisfies

[06:00] the criteria for a noble polyhedron because every face is identical and indeed interchangeable with every other, every corner is identical to every other, the edges are not identical, that's pretty obvious, and obviously the faces

[06:13] are not regular polygons because this is not equilateral triangles. So we don't care about that. A wonky hedron? A wonky, well a dysphonoid Petro is the term people use but Wank I like better Okay and are there an infinite number of these or is that a very specific triangle that allows you to get away with it No there are infinitely many of these So in fact you can do this with any acute angle triangle It actually an infinite family of noble polyhedra

[06:38] The noble polyhedra were discovered, investigated in the 19th century by Edmund Hess and Max Bruckner in particular, and they found a second infinite class of noble polyhedra, which are,

[06:50] features a little bit more interesting than the Wonka Hedron. So what's on the screen here is a pentagonal prism, but this is not a noble polyhedron because

[07:02] you can see that the faces are different from each other, but there's a process in this area called faceting and what faceting is is where you basically fix all the corners, so you keep them, but then you get rid of the

[07:19] original faces and edges and rewire those corners to make a new shape. And if you do that right, starting with a pentagonal prism, you can get a noble

[07:31] polyhedron out of it and this is what you get. So this is a shape called a stephanoid. So anyone whose name is Stephen has probably at some point looked up what the origins of that name are and it means crown in ancient Greece. So this

[07:47] a crown polyhedron for fairly obvious reasons it looks like a crown. And you can actually see, if you look at it right, you can see that pentagonal prism still lurking in there. So,

[08:00] if you were looking at the top, so there's the pentagon, the squares, so they've all gone, but the corners are still there. So we've got these new edges. Okay, so it's not obviously what the faces are. The faces are shaped like that. So if we're allowing pentagrams then we're

[08:18] also allowing things like bow ties. And there's infinitely many of these stephanoids because you can start with any prism. So there's nothing special about a pentagonal prism here, you could do it with any other kind of prism. And in fact you can also do it with an antiprism. So this is

[08:35] a pentagonal antiprism, meaning that you've got your two pentagons but this time they're joined by a ring of equilateral triangles and in the same way you can get a faceting of those which

[08:49] is another stephanoid, another crown polyhedron and again its faces here are these sort of butterfly shaped or bow ties. So we've got two infinite families now, we've got the disphanoid tetrahedra and we've got the stephanoids. So Hess and Bruckner in the 19th century

[09:05] also discovered a few individual examples. So here's one, discovered by Edmund Hess, and initially it looks like a dodecahedron but you can see the faces are sort of concave. Well actually what you're

[09:17] looking at here is not really the faces. The faces are these triangles which cut through each other. So that's the repeating face and that other look that we saw was almost an artifact of joining the triangles.

[09:31] Exactly right, yeah, yeah, because we're allowing the faces to cut through each other, so when they do cut through each other you get these sort of, yeah, these patterns which are sort of slightly illusory I guess. So if you bring back the, if you colour it all in again, so those kind of little triangles

[09:46] and depressions I'm seeing there, they're not the faces. They're part of the faces. They're part of the faces but that's not the entire face we're seeing there, that's just sort of a, so they don't have to be consistent. No, as it happens in this one they are. So this shape has got more symmetries than noble polyhedra have to.

[10:06] And in particular this is a faceting of the dodecahedron. You can see the dodecahedron, but actually we've also kept the edges of the dodecahedron, which you don't have to do in a faceting. So it's a particularly, I guess, a particularly elegant instance. But that's right.

[10:23] When you look at these noble polygons it's often very very hard to see what the face really is because it's sort of hidden in the inside and in this case it's these triangles here. So that was one discovered by Hess.

[10:37] Here's a nice one. That looks like Superman's spaceship from the original Superman movie. I haven't watched that recently enough but maybe it is that, who knows?

[10:50] Who knows? Maybe the director was a fan of noble polyhedra. So this is an example discovered by Bruckner. And actually, Max Bruckner, he made amazing paper models of a lot of these shapes.

[11:04] Sadly, they no longer exist. I think they've disintegrated in the century that's gone by. You can see photographs of paper models that he made of this. This has its origins actually in the icosahedron. It's what's called a stellation of the icosahedron.

[11:17] There are many stellations of the icosahedron. this is the ultimate installation of the icon, but its faces are these sort of nonograms like that and 20 of them, so it's noble, meaning that all its faces are

[11:32] identical and all its corners are identical. You spoke about this chap making paper models, this was obviously done a long time ago, you know, they wouldn't have used computers. The process they would have used, would it have been

[11:45] like models and visual or drawing or would it have been like logic and numbers and formula that just told them the things existed or what process would they have been using? And there would have been a lot of mathematical

[11:59] reasoning to find out exactly where the corners and edges and the lengths from the areas and all that sort of stuff so there would have been a lot of mathematical calculation going into it but actually the truth is I don't think

[12:11] we really know how he did it. We just see the final results and we feel spectacular. But the full process he went through I think is a mystery. So let's see another a couple more of Max Rooks now. So here's one. So again you look at this and I think it's almost impossible

[12:30] to tell what the face is. We'll probably get some holes in there too. Yes you do get holes in them. There's no rule against having holes in them. So that's what the faces. Each face is one of those. It's an irregular self-intersecting hexagon.

[12:45] This is another book nurse where the face is a regular self pentagon I know this goes against what you told me at the start of the video but this is beginning to feel a little bit like

[13:00] infinite. Like, I can't see where this would stop or why this would stop. Right, well, I mean, we just didn't know. So actually, at the point where Bruton had stopped his work, which was right at the start of the 20th century, it was wide open. Just people didn't know. We knew that there were these two infinite families,

[13:16] and we knew that there are a smashing of increasingly bizarre looking individual examples. And it was left at that. And in fact, it was left at that for over 100 years

[13:28] until Robert Webb, who is someone we've talked about on Numphar before, because he's the guy who created this wonderful software called Stellar, which is just the best thing for looking at all these polyhedra.

[13:41] But as well as designing the software, he also made some important polyhedral discoveries. and one of them was the first noble polyhedron discovered for a century, which he found in 2008. So this is not his shape. This is a classic polyhedron called the snub cube.

[13:56] It's one of the Archimedean solids, so it's been known for a long time. But what Robert Webb did was investigate statetings of the snub cube. So remember what that means is you keep all the corners, but then rejig the faces and the edges.

[14:11] and when he did that he found this shape here which is a noble polyhedron for faceting of the snub cube. The snub cube wasn't noble? The snub cube was not noble

[14:24] because it had some square faces and some triangular faces. For noble we need all the faces to be identical. But from a snub cube he made a noble? Exactly, yeah. And that's how a lot of these noble polyhedra arrive actually by taking other polyhedra which are known and then tinkering with them in some way

[14:40] faceting being one of the main examples. Again, I think very hard to see what these faces are here, but in fact they're these sort of goldfish shapes.

[14:54] Wow. Some irregular self-intercepting pentagons. Wonder how he felt when he found a new one, like what must that feel like? He must feel like an explorer or something. Yeah, this is new terrain. I mean I found a bunch of things which were already known and I didn't

[15:10] necessarily know every one that was already known, so I didn't know which one, but I knew there wasn't one based on the snub cubes. I saw it

[15:22] and I thought, you know, whoa, is this something new? I was kind of surprised no one had found it before, I guess. Yeah, so it's kind of surprising that it was

[15:34] just sitting there, ready to go. I saw some way to call it a noble faceted snub cube. We've got a better name. And so you could call it, we could call it the web polyhedron.

[15:47] And when we talked about the holy monster in a previous video, there was a thing called the web toroid. So we don't want to muddle it up with that. The web noble fastened snub cube. It's not very catchy, is it?

[15:59] No. I don't mean to work on it. Get to the drawing board on that one. It's kind of a bit of an ugly duckling in a way. What, the polyhedra itself? Yeah, it looks a bit kind of random. random and wanty

[16:13] the ugly duckling is also a very good name yeah well we could call it that okay so Rob Webb's discovery here spurred a new flurry of activity and various people

[16:25] kicked in and made some discoveries in the years between 2008 and now and in fact some of them used this very software to do it

[16:37] so this is an amazing looking thing is a discovery made by Ulrich Mikler-Weiss in 2020 and he actually discovered it using the software Stella. So it's not just good for looking at the polyhedra, it's actually pretty powerful

[16:50] and you can discover new polyhedra with it as he did. So we have an increasing list of known examples. I've got to see the face, I've got to see the face. Oh yes, of course, we've got to see all the faces.

[17:03] Yeah, yeah. What do you think it is? I think it's going to be a, like the goldfish, but it's going to have four pieces.

[17:15] What was that? Oh, no, there's six pieces. It's a kind of hexagon, actually. It's got six faces, but it's not what you usually think of as a hexagon. Yeah, well, great.

[17:29] So a young mathematician in America, aged just 17, called Connor Hill, has actually provided a full classification of the noble polyhedra, which means we know the complete list now. And we know that there are those two infinite families,

[17:43] so the disphanoid tetrahedra and all the stephanoids, plus exactly 146 other examples, of which he has now enumerated all of them, and 85 of them are brand new, which are not previously discovered.

[17:56] So he's discovered 85 new noble polyhedra, which he's also shown that completes the list. so we have a complete understanding of Neville Polyhedra which has seemed a long way off for a while because he was very mischievous Polyhedra

[18:08] so it's a fantastic achievement we've got these two infinite families and these ones like this they aren't part of the infinite family are these no these are like these one-offs yep we know there are 146

[18:20] 146 one-offs yep how can you know that how can you how can you know that so it's a clever proof and at the time of writing he hasn't

[18:32] He hasn't published the paper yet, but he's working on it. But I've talked to people in the field and they're very, very confident that this result is accurate. And actually he's been given a big prestigious prize in America for this discovery.

[18:46] Very well deserved. The key idea is where can the vertices be? So he did some quite sophisticated mathematics to determine what possible vertices a noble polyhedron could have and then sort of crunching the possibilities to configure those in different

[19:03] ways. That's a very rough overview. I'm not seeing the details. 17! He was 17 when he did it. Isn't that incredible? He also told me that he's a long-time Numberphile fan. So that good to hear Excellent Oh maybe he let me speak to him and do an interview Definitely without him It feels like you discovering something Like in a way it feels like these are always out there for someone to find

[19:29] And some of them I do think are pretty beautiful. Is he at university? He's not even at university. He's completing his high school. He's going to get in, isn't he? You've got to reckon.

[19:41] So there is a picture of all of them? have they all been like, are they all sitting in some register or index in... We've got them all on Stellar here. So I did ask him of the ones that he discovered which he thinks are particularly interesting or particularly beautiful or

[19:55] notable. He gave me a couple of answers so we could look at those maybe. This is the one he reckons is the most beautiful. So this is a, well I think its technical name is

[20:07] it's the first noble Tituscoidal Hecatomoricosahedron. Of course it is. Of course it is, obviously. So this has got 120 faces. Can you guess what the faces are?

[20:21] No. No, I don't think anyone can. Actually they're relatively simple. They're irregular self-intersecting pentagons.

[20:33] 120 of those. One I thought was quite interesting to look at was this one here. here. What a mess. It looks sort of like a mess of dollars and then you can see all the symmetry. So it's got lots of rotational symmetry but no reflection or symmetry. I think it's quite

[20:49] intriguing and again we better have a look at what the faces are so they're so not non-obvious. So it's again self-intercepting pentagons. You told me 146. 146 individual examples. So that's

[21:03] including the ones previously known, the Webb discovery, the ones by Bruckner's paper models and so on. Is there anything special about the number 146 that it would land on that? I don't think there's any high level explanation as to why you would expect 146.

[21:19] I don't even have a good sense of why you would expect there to be finitely many, why there might not be a third infinite family. But I'm sure Connor will have an answer to that, but I don't know.

[21:32] So another one that Conor picked out as being very surprising is this one. The faces look like that. You like that last one you showed me? Yeah, a little bit.

[21:44] And some of them have got a quite similar structure, but this one's got a very unusual feature. So there's this and one other like it, which are so-called fissary polyhedra. And what fissary means is it's got essentially two corners which exactly coincide so they're

[22:02] one place. So if you look at this corner up here you can maybe see that it's really for two corners it's got this corner for these faces over there and that one up there but actually so it's really

[22:14] two corners but those two corners happen to exactly tell inside in one place so there's only two of the all the noble polyhedra which have got this property. And how many of those special touchy points do they have? Lots of them or? It's got what looks like a lot. In a noble polyhedron

[22:29] all vertices are identical so they're all like that right that's one of the that's one of the defining properties of the noble polygons so all the oh so all the vertices are kissing like that all the exactly all the vertices are kissing like that

[22:41] so everything else with noble polygons when they're self-intersecting like this it's sometimes difficult to tell what the vertices are right because you get these other places where edges cross through each other so those don't count as vertices but yeah all the vertices

[22:53] have these yeah kissing that's a great word for it they're all all pairs of kissing vertices and that's everything I have but I mean I do have a full list if you want to just scan through for any other new fancies

[23:05] looking out there hey Pete you listening Pete let's do something clever at the end with all of them he's cursing you right now I bet you I can hear him let's see them all Pete shall we

[23:29] Thank you.

[23:59] Thank you.

[24:29] Thank you.

[24:59] I think we're sort of...

[25:28] it's like related to the noble polyhedra. I remember there was this one video you did on seredites and like the different possible seredites that can exist in three dimensional space. And that like, that might've been like part

[25:41] of the inspiration for me to just learn about polyhedra in the first place, which would be, this is sort of a full circle moment in that case.