# 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.

looks like the future of chairs

These could actually be pretty idiot-proof battery cells styles as long as standards stay consistent

https://youtu.be/u4Y_i5D7WFM

Pomegranates

More and more on the road of endless proof that we were engineered by higher plane being.

Since you cut your hair. You remind me of Mike Phillips. The paintball dude

That’s Sydney, mate!

1:06 "Generic decor" … he said and the thing went out of focus being utterly ashamed of itself.

wait wait wait, that thing Laura is making (7:26, etc), is something else, than Matt's thing, isn't it?

Until now I imagined the Y shape being planar (all 4 vertices and 3 edges of the Y shape being in one plane). Why in Laura's design is the middle vertex sticking out? I don't like it

I want more episodes of MMASOOTHFATPHSWOH !

Is this important to anything or anyone?

I was eating a pomegranate a while ago and I found a bunch of seeds this shape. I tried explaining to the rest of my family why I was so excited, but they didn’t understand me at all.

So could you make other stackable shapes with variations of this concept? For example could you connect a pentagon and octagon with 3 Y edges and the shape still be stackable or is it unique just to this shape?

Is it just me or does non-native english speaking academics have really bad accents? I don't get it, they're smart people, why cant they pronounce clearly. I hate going to lectures and the lecturer has such a thick accent you almost can't understand what they're saying. The funny thing is that in my experience it is worse with academics than many "normal" people.

So we've lost Pluto but we've gained a Scutoid, in modern mainstream academia…

Hi, Matt: I guess if you revealed that the Scutoid is simply a pentagonal prism with one truncated (take your pick) vertice, you wouldn't have had to spend your time, or anyone else's, with all of the "intermediate" shapes, which, in fact, are not intermediate. One simply needs to knock a vertice or corner off a pentagonal prism, et voila.

Squishy science meets hard science…brûlée science.

Did you skip antifrustrom, or is that not a real name?

How is that still a prism(atoid) though? You're adding a vertex, thereby removing the main feature (2 opposing faces)

isnt a frustum just a normal prism but with one side smaller then the other? and if that counts as a new category then a antiprism with one of the sides being smaller should have its own category too

There's no reason that any of the faces have to be curved. Take a pentagonal prism and slightly truncate (chop) just one vertex, forming a new triangular face. Two of the rectangular faces become (irregular but flat) pentagons, and one of the pentagonal faces becomes an (irregular) hexagon, giving a "scutoid". There are plenty of distortions of this shape, of course, but this construction is enough to show that none of the faces has to curved.

Awesome video, thanks for taking time out of your holiday to make this!

6:22 "Stuck in the middle with you" remix. Verrrry clever. Considering the scutoid looks like someone took a straight razor and chopped off a little piece of it.

Epithelia is thin tissue forming the outer layer of a body's surface and lining the alimentary canal and other hollow structures.

The more u know…Bullshite

i have some beef with this shape. it isn’t one. it’s the union of a prismatoid and a prismoid. if it’s just a compound shape, why give it a unique name at all???

Lols at the Little Creatures, chat

are the side faces of the "prismatoid" flat? if not, what's the big deal here?

6:15

wait so are there more undiscovered shapes that work just like this one? that would be really interesting

Actually thought this was bullshit at first, but this turned out to be really interesting! 🙂

Not gonna lie, this just seems so obvious. I feel like I'm missing something. Like this could have been accidentally discovered by a kid in high school or something.

I took wood shop classes in the 7th grade where I learned some basics of cad software, and later in high school I learned 3D modeling. just with this background in 3 dimensional shapes, This shape is so blatantly similar to the prismatoid with the triangle face, that I'm actually kind of shocked they never considered it's existence before.

I wonder what applications this will have in creating more compact brick designs for making walls. Should be able to pack more bricks in: with more structural support.

Arent we gonna talk about how he only says "scotoid" and "s cutoid" but never "scutoid"

So what about scutoids made from other shapes? Do they stack too? Surely pentagon/hexagon isn't the only type investigated right? What about the square/pentagon? Triangle/square? Or what about like a pentagon/octagon? Can you have multiple extra vertices?

Could different types of scutoid even stack with each other?

Can you come to my high school? We have some of the best young mathematicians in the world…

I thought number 3 would be antifrustum

I'm not a native english speaker. I just noticed how "prism" and "prison" sound alike, and I find it funny when he is building cages 😀

come to my school in portugal

Antifrustrum. There! I invented a new shape!

Frustroid. I'm on a roll!

Scrotum. Dammit! I should have quit while I was ahead…

Fine, I will invent a shape too. I call it the anti-frustum. It's the same as the frustum, but with triangle sides. You're welcome.

Generic foreign city (turns camera to opera house)

"Generic foreign city" meanwhile lemme showcase the two most recognizable feats of architecture of the entire country

They are all homeomorphic to spheres, so…boring.

Give me a few genus and twists, give me the double kleinbottle! 😛

shoutout to the girl at 0:46

Been months now and it's still my favorite episode of Matt Makes a Shape Out of Things He Found Around the Place He's Staying While on Holidays.

Scutoid: from the Latin, NOM: Scutum, GEN: Scuti: English translation: shield. My familial name in Latin is Scuta.

And I would like to have a pair of salt and pepper shakers please!

meh

I bet you were so sad when you realized you just HAD to drink 2 sixpacks of pale ale to do this… But you are dedicated so ofc you did it 🙂

6:18 You know what, I just "discovered" another shape, instead of an Y-shape edge, it has a X-shape edge.

Wait, how about those delicious juicy pomegranates….

SCUTOID, sounds like something you would have to go to see a doctor about! That's why pyramids should wear protection.

4:10 So, is a frustum a frustrated pyramid, because it can't get it fully up? O not O/? Maybe the frustum needs the little blue pill..

Very interesting vid, thanks. I suggest a vid on Bayes Theorum. It is an interesting bit of maths with endless applications.

What about an antifrustum

I’ve drawn this in my 5th grade maths class… and everybody clapped.

0:45 That girl says hi

I'm sad that there's no anti-frustum

Aren't frustum's the same as any kind of pyramid ? (if the base is the same ofc) because ∞

This guy got a shape names after him as a meme

Those ain’t pipe cleaners those are bendy strings made for. DIY and stuff

JMU represent!! Love your videos.

OMG Clara has the cutest accent in the WORLD

In my opinion it's impossible to discover shapes; you can discover properties of certain shapes, but discovering a shape is impossible.

Is there an antifrustum ?

6:21 BEST PUN EVER

3d printing these is not easy :')… being better at CAD works a lot

@12:40 – you got me good.. "shapes" and "little creatures".. dammit.. you got me good.

I love you so much. Thank you for making math hilarious.

Liked for the content, commented for the stuck in the middle with you reference 6:25.

So a prism with a different number of edges on each face is a new shape. If that's how loose you want to be with the definition, game developers are sitting on a diamond mine for the world of geometry.

@6:32 Stealer’s Wheel finds a reason to sue Parker

0:45

Anyone else notice the tourists?

Aka, delete a wrong point in blender and fill them in as one face.

What if the frustum is twisted?

HANG ON JUST ONE MOMENT

Clara looks very similar to James Grime. She even has similar mannerisms. AND HER SURNAME IS GRIMA

Is this some ancient lineage of mathematicians spread across the globe?

P H I L A D E L P

2:39

Clowns to the left of me, jokers to the right,

Here I am, stuck in the middle with you

Pentagons to the left of V, Hexagons to the right

Here Y am, stuck in the middle with Scu'….toid.

This is my favorite maths video ever, I think because it isn't about computation but the actual meaning of maths in life. I would encourage you to do more like this, but at the same time I have no doubt it will get the establishment on your back, so only if you're terminal LULZ

Parker square

Parker square

6.40 cracking joke.

That fourth type of prismatoid isn't a polyhedron. At most one pair of edges of the hexagon and pentagon can be coplanar, meaning there is no surface connecting at least three of the sets of four points that the model suggests should be sides.

But then, biologists discovered it? Am I wrong here? It sounds like they found the shape and basically understood what it did, but they needed to model it in a computer so they got some mathematicians and programmers. How did you twist that into mathematicians being the heroes of this story? It's like saying plumbers invented jet engines because there are pipes on them.

Why did you start ordering the shapes with 0? They're not memory offsets!

Well done. This was fascinating, and the story arc is brilliant!

What would it be if you had 4 vertices on the bottom and 8 on the top connected by the anti prism sides? Or any X,X×2, connected with anti-prism sides

how the hell did you cut this precise hexagons without a rule and a compasses….

Shoutout to the couple at 0:46

This is a load of bs. Might as well twist every shape and connect every corner.

I think the mix of a prism and antiprism should be called a hemiprism

0:46 hi back to you too random lady with an infectious smile 🙂

Pomegranates!

@standupmaths The green edges may be straight, but as the 3-D print segment demonstrates, the rectangular faces are not planar. Is the scutoid possible to make with planar rectangular faces? Or is it necessary to divide the rectangular faces into two triangular faces (each) in order to make it happen? Planar faces are required for a 'prismoid', are they not?

Bractagon!

i am so so happy that i found your youtube channel T_T while browsing for exactly this video. i am a fan of you since you appeared on a show at a science channel. OMG so good to see you making videos!

sending love from india!

So it's kind of like you added a pentagonal prism to the top of the generic hexagonal-pentagonal-prismatoid

omg they modeled it in OpenScad….. FLOSS ftw!

Say it with me .. Now that's thinking outside the box!!

hexagon on top | pentagon on bottom

pentagon on bottom | hexagon on top

scutoid!!