THE SCUTOID: did scientists discover a new shape?

If you’ve been following the nerdy end of the news you will have seen headlines this week along the lines of “Scientists discover new shape!” Now th—w—hang on who were these scientists discovering a new shape? What d’you mean by new shape? Are any mathematicians involved and obviously I wasn’t the only person wondering this because people start to email me and send me messages saying “Matt, can you explain what this new shape the scu-toid or SQ-toid is?” and I was like You know what? I want to find out. The problem is I’m currently travelling through I don’t know, generic foreign cities somewhere in the world because I’m doing shows for high school students. Details below. So, to be able to explore this shape and see what it is and how it behaves and if it is indeed new, it’s time for another installment in everyone’s favorite stand-up math segment Matt makes a shape out of things he found around the place he’s staying while on holidays. Let’s do it Welcome to the rather echoey dining room of my Airbnb. It’s furnished with all the most generic decor that IKEA has to offer and a lot of their walls. So anyway, I’ve got some cardboard I found around the house. I’ve got all the Australian staples There’s BBQ Shapes and Little Creatures Pale Ale. Thankfully, I’ve got tape with me and I found a pair of scissors and I was able to locate some pipe cleaners. Now the scutoid is a member of the wider family of prism or prismatoid shapes. So, before we get into the scutoid itself, we’re gonna start with shape number 0: the prism. I’ve now got a collection of pentagons that I’ve cut out of shapes and I’ve got hexagons from Little Creatures. Now, a prism is just when you’ve got two identical parallel faces and you join up all of the vertices. So I’m going to use the pipe cleaners to do that. Okay, and there we go. There’s my pentagonal prism and my hexagonal prism. Now, of course, these faces should be filled in rectangles. I’ve just done the edges and left these empty, but they should be filled in the same way that the ends are filled in and these your standard-issue prisms. Nothing that exciting yet. However, that’s not the only option for making a prism if we actually take our pentagonal prism and give it a twist, I can sneak a few more edges in there. So our original prism here had hexagonal faces and then all these faces joining them are rectangles. Because of the twists, I’ve now got in twice as many edges going around and two edges meet at each vertex. I mean, plus the, you know, edges on the top. And so what I’ve got here is an antiprism. If your two parallel identical faces line up and you’re using rectangles as a prism, hexagonal prism. If things are skewed and you use triangles, antiprism. So that’s now shape number one in the prism family: the antiprism. Slight detour off the prismatoid path to a pyramid. It’s a, you know, hexagon based pyramid. This is not a prismatoid because it hasn’t got a pair of parallel faces anywhere. But you could imagine, like if you sliced off the top and put a new face in there, it would be parallel to the bottom one. That’s exactly what I’ve made here. So this is a frustum. That’s shape number two: the frustum. This is what happens if you get a pyramid-like shape and you lop the top off. It works with cones as well. You get a frustum shape It is a prismatoid. It’s yet another variation on the thing. So now we have the prism, the antiprism, and the frustum. One more to go. Okay, so what I’ve done now is I’ve put a Pentagon on the top and a hexagon on the bottom. But to join them together, well there’s more vertices at the bottom than the top so two of the edges, which I’ve done in yellow here, have to meet. And so this bit here, to patch the six vertices at the bottom to the five on the top I’ve used an antiprism triangle here whereas the rest of it is prism rectangles. Now in maths, there’s not a special name for this. This is just back to the generic name prismatoid, right? It’s got the parallel faces all the ones in between have four or three edges around them. It’s a prismatoid. But it’s some combination of, it’s a bit prism-y me in some bits, it’s a bit antiprism-y in others. It’s a mix of the two. Doesn’t get a special name. So I guess our final category is just prismatoid, which is kind of everything else. But now, there’s one more. So this is it! This is our scu-toid or es-cu-toid. It is the same with our generic prismatoid before, but instead of the antiprism triangle face all the way out, you’ve got like a Y. So you’ve got a little triangle face and then over here you’ve just got your standard single edge coming into the top. So you could say a scutoid is a prismatoid at one end and then it’s a prism at the other. prism and a single triangle prismatoid Though, there you are. I mean, that’s– that’s it. I guess the most important bit is this bit here where it merges. A little like Y or V shapes. So in short, there’s pentagons to the left of V, hexagons to the right. Here, Y am stuck in the middle with scu- toid. Hey, that’s pretty good. That’s so good, I’m taking five minutes off. ahh Actually, while I am taking a break, earlier on today I had a chat with my friend Laura Taalman who was working out how to 3D print one of these. Which is not easy, because these faces no longer all flat you’ve got this concave bit here. This weird kind of 5 edged face is no longer well behaved and so 3D printing that is not trivial. And we’ll get to in a moment, these things are meant to kind of stack together. So I asked, basically, I asked Laura how she went about designing a 3D model for these. So when I first tried to make the scutoid, I did sort of the most boneheaded thing you can do which is just to put vertices in the shape of a hexagon, and then below it in the shape of a pentagon, and then have an extra vertex, and take the convex hull of all those vertices. That didn’t work. Some of the sides have to be curvy. They’re not actually planar faces. The scutoid isn’t a polyhedron. The next thing I did after that is I divided the object vertically into slices and kind of lofted a slice at a time so each slice was not curved but then it approximated this nice curved surface going up. And then, I did get an object that was nice and curvy and looked like a scutoid that looked pretty much just like the scutoid that’s in the picture in the article. However, it didn’t pack with itself. It turns out that the locations of the vertices and, in particular, the location of the extra vertex are really key for determining whether a scutoid will pack with itself. But there’s ways that you can constrain it to make that happen and the final model does have this property that it packs with itself. I was doing all this in OpenSCAD. You can delete that if that’s not interesting, but I was. I did this an OpenSCAD. So what does a scutoid have to do with packing and biology? Well, there are cells in biology called epithelium cells and they form an epithelium layer and biologists just always assumed that they would be prisms that have f– well, obviously distorted prisms. The cells would grow and fill the space and just, you know have a face at each end of be joined up normally in the middle and whenever there was some kind of curvature or the surface was distorted then it’s a frustum. It’s a truncated pyramid arrangement. But then, biologists spotted patterns and curvature in these layers of cells which couldn’t be explained by truncated pyramid shaped cells. And so they got some mathematicians involved. Yeah, thank goodness There were some mathematicians involved once the biologists saw these strange shapes, some mathematicians came in and they thought well how would we model what shape these cells would form if they were packed into these various arrangements? And they used, well, a variation on a thing called the Voronoi Technique which is where you put a bunch of points in space and then you expand them until they fill all the space and each region around each point are all the other points which are closest to that original starting seed point. And then they looked at, in the 3D arrangement, how these cells would form if they were trying to like minimize the energy, but find a stable arrangement where they pack together as efficiently as possible? And that is where they came up with the scutoid. The mathematicians predicted that these epithelium cells would have a different number of edges on one face compared to the other face and the edges would merge somewhere in the middle giving this prismatoid arrangement. I was actually lucky enough to talk to one of the mathematicians involved in the project, because I knew they were in there. So, I had a chat with Clara Grima and she explained a bit more about how they found this shape based on what the biologist told them. Hey Matt, this has been the first time for me that we work together with biologists, physicists, and computer scientists and I think that the role of everyone was quite important. In short, we the mathematicians had to formalise how to define the new shape that biologists had observed in the microscope. The group leaded by Luis M. Escudero, biologists had discovered that the usual representation of cells or the epithelial tissues prisms or truncated pyramids could not explain some properties that they observed in the microscope. So they ask us to try to find the correct model. With our first model we made some predictions, but they will not still accurate. So, we have to change our model and finally with the scutoid everything worked fine. Finally, we have the seal of approval of physicists and he proved that the scutoids are stable when they pack together. The mathematics didn’t predict that all epithelium cells would be scutoids just enough to give you the right curvature. So for the case of a cylinder, which happens in animals apparently, or some kind of spheroid, you would have a certain percentage of these and it’s quite nice that it’s built out of pipe cleaners because it’s not some fixed rigid shape, right? But it’s not, it’s not like a cube or something like that but nor is it quite topology because there is some geometry. It’s somewhere between topology and geometry. They worked out the structure that would be required in some cells to give these sorts of shapes and then they found it. They looked in fruit flies. They took what the mathematicians told them and the biologists were actually able to find these shapes inside little creatures. It’s ap– a phenomenal collaboration between biology and mathematics. There may be applications as well. I was chatting to Clara and she said that one day if they look at tissue actually in, let’s say, humans and they know what structure to expect, they can tell if the cells are growing normally or not. Or if we ever make artificial cells, we need to know what kind of structure they form. So at the moment, it’s all very abstract. It’s all very exciting. You can read the full nature paper if you want. I’ll put a link to it below. It’s publicly available. One last bombshell that Clara told me when we were talking is that this shape here which gives it the name scutoid in the media. Officially, it’s because of a beetle, right? It’s named after a very similar shape you will see in beetles. Biologists! But when I was chatting to her, it turns out that was not the original reason for the name scutoid. This is funny and cute for me Officially, it’s because part of the body of a beetle is very similar to our shape the scutoid. okay, but there is a hidden reason and I will tell you secret. The biologist leader of the project is called Escudero Translated to Latin is “escudo”. So from “escudo” as a joke, we start to call it escu-toids Later, we cannot think another name for our shape, but scutoid. Bye Matt, May the scutoid be with you. Thanks so much for watching the video. There are more videos of me building things that are stuff I find while traveling. You can subscribe to my channel for all your mathematical needs and thank you so much to all my patreon supporters who make these videos possible. If you come across more maths in the news or something interesting, drop me an email. I’ll see if I can make a video about it and finally the shows for high school students I was doing in Sydney are over over but I do them all the time right across the UK and we’ll be in New York for the first time in October. Details below.