Scutoids

What are Scutoids? | Nets for Papercraft Scutoids | Rambling About Scutoids | Mathy Definition

(work in progress)

1. What are Scutoids?

Scutoids are a kind of shape similar to prisms, but where the ends are different polygons. The sides transition from one polygon to the other by having another vertex in there, so there’s a triangle face.

They were first described in 2018 by mathematicians and computational biologists working to understand what kind of shapes cells make in the body.

Scutoids are officially named for the first time in this paper: “Scutoids are a geometrical solution to three-dimensional packing of epithelia”

Scutoids look similar to polyhedra, but not all faces of the scutoid are flat, so they are not polyhedra. Scutoids are a class of shape, so there’s different kinds of scutoids. Some have a pentagon on one side and a hexagon on the other, some have other polygon ends. Some are taller or shorter, more regular or irregular.

Technically you can’t make a perfect scutoid out of paper because of the slightly curved faces, but paper is flexible enough that you can approximate a scutoid fairly closely. It’s certainly accurate enough to enjoy the shape and its properties!

2. Nets for Papercraft Scutoids

This is still technically a draft, but here’s a papercraft scutoid pair available as a pdf or png. I recommend printing the pdf because it will print in a standard size, so that your scutoids will match up with other scutoids.

I originally started writing about Scutoids in Patreon posts, and am re-posting them in order here.

Aug 29 2018:

Hello good humans! Today I would like to talk about Scutoids.

Earlier this month I did a lens clip series looking to see if we could find a scutoid in a pomegranate. The compiled video is embedded above, though it might not make sense without some background, so I thought I’d write an article going through my thinking.

What is a scutoid? It’s the latest viral math shape sensation in the news! As seen in Nature, cells grow in this shape sometimes so that they pack together real snuggly-like.

I’ll admit, it didn’t grab me at the time. As with all news articles based on Nature articles, I like my biological shapes with plenty of salt.

The last viral news shape was, I believe, the gömböc, a shape which has its mass distributed such that it always rights itself no matter how you put it down. This is a good shape too, and I recognize the name from ordering ice cream in Hungary. Hungary is home of many great mathematicians, and where I’ve attended several math events, so I’ve ordered lots of ice cream there.

“Gömböc” is the word used for “scoop” (I don’t know what else it might translate to), so in my head I’ve always thought of the shape as a Scoop, giving it another spiritual connection to the scoot cuteness of scutoid. I imagine that Scoop and Scootoid are probably good friends in the platonic realm.

(If you can think of other viral news math shapes, comment below. It’s an interesting phenomenon.)

I first paid attention to the scutoid because I liked Laura Taalman’s 3d-printable pair, and her work is always worth paying attention to (tweet I saw, shapeways, thingiverse )

But then the scutoid kept popping up. Of course people send me these kinds of things, but I was surprised to see the amount of press. Is the shape really that unexpected or newsworthy? I don’t know, but I’m always happy to see shapes get a moment in the sun. And I do love a snuggly shape that packs.

I ended up in an email thread with some other mathematicians, discussing where and whether you’d expect to find something like the scutoid. I was a bit flummoxed how nature would go about avoiding accidentally making scutoids and similar shapes!

I mean, I know that bees and related insects do a lot of fancy cooperation to get perfect hexagons. I’d just read all about it in Animal Architects by Gould and Gould, which I read for spidermath research purposes.

Luckily I had some hexagony friends I’d just found in the electrical box, so I could really see this cooperation in action! (I believe these are paper wasps, correct me if I’m wrong.)

Packing cells together is an interesting problem with a lot of aspects: do you want the most optimal end result, or the most robust process, or something flexible, or something that can be built on? What if it needs to be repaired? Do we start with all the bits and grow them together, or are the bits spaced out in time? Is there any way to see how the overall pattern is going, or do we choose a process and stick with it no matter what?

Nature finds interesting solutions. I know how plants get their fancy spirally seed arrangements because I’ve studied phyllotaxis (and made the Doodling in Math Class Plants series about it). A spirally packing is the result of plant bits being produced one at a time from a single central source.

But for packing cells in a way that’s not centrally organized and that has to self-repair or grow new stuff around old stuff with all sorts of lumpiness and curvature sometimes, I wouldn’t expect nature to produce perfect hexagonal prisms (or spirals) even in places where it is theoretically possible. All you need is a sheet of cells where there’s five neighbors on one side and six on the other.

There’s a lot of scutoid-like shapes out there. And I’ve seen a lot of packings and 3d voronoi diagrams in my life, so why wouldn’t nature produce similar sorts of shapes when packing cells?

(Note that scutoids are not polyhedra; scutoids have curvy faces that do curvy face snuggles, which is a very good thing about scutoids.)

Basically, as much as I like to see shapes in the news, I was skeptical about the intellectual honesty of the headlines calling it a “new” shape.

I’ve found that skepticism is often used as an excuse to be both lazy and patronizing, basically hiding status-quo apathy behind a mask of intellectual elitism. So if I’m skeptical that scutoids are surprising, my opinion is worth nothing unless I do the work to come to some belief one way or the other.

So when I started thinking about pomegranates and the faceted seed shapes they pack in there, I realized here’s a case where you don’t need a microscope or special equipment to see what the cell shapes actually are, I can just open one up!

When I made the lens series I started with the premise that I expect the pomegranate to yield examples of scutoid-like shapes, and this was published in real time, as a kind of pre-registration of my experiment.

By publicly announcing my hypothesis, I avoid retroactively changing my story to fit the evidence. My skepticism becomes scientific skepticism, rather than safe passive egoistic skepticism, because I have set terms under which I can be wrong. If there’s no scutoids in there, I’ll have to update my understanding of the world!

So the first two clips went live, and then I opened it:

And I’ll admit, I didn’t expect so much of the inside of a pomegranate to be so close to a perfect hexagon packing. Obviously it’s not absolutely perfect, and there were a bunch of scutoidy shapes in there because of exactly the reasons I thought there might be (which I show in the video, not that the resolution is very good). But overall it was more regular than I thought it would be.

And so I did update my understanding of the world! This line of inquiry helped me notice something I hadn’t before. There’s more math in a pomegranate than even I expected.

Now I want to know how pomegranates grow and arrange their seeds! How they pack the different sheets of layers within the sphere! I’d never noticed how they grow in mostly flat sheets butting up against each other at angles, and there’s some sort of interesting geometry going on in there. Do they choose the angle to best facilitate matching the hexagony cells across faces? How does the inner scaffolding form?

It could use further investigation, and maybe a video someday… not that I really need another video project right now! And at least the scutoid angle has been covered by plenty of other people. Matt Parker AKA standupmaths made a great one: https://www.youtube.com/watch?v=2_NZ1ql8B8Y

Anyway, I hope you look at pomegranates a little differently next time you see them! And whether the scutoid is news or not, let us take a moment to appreciate our shape friends and the way they illuminate the world.

Sep 4 2018:

On the recent Scutoid post, Amy Tobol commented asking if there were a papercraft version. I was immediately intrigued.

I knew it wouldn’t be as simple as a polyhedron net, as not all the faces are flat. Which is probably why a papercraft version did not yet exist. But maybe it’s close enough to make it work?

So I tweeted at Laura Taalman (who made some nice 3d printed models) to ask, and she test-fitted some paper to the curved face and thought it was promising, and came up with a draft net design.

Tom Ruen jumped on the thread and tried using Laura’s 3d printable file to do some software and by-hand wrangling and made a few polygonizations.

The first version was difficult to assemble, but did get in a slight amount of curvature:

Then he made a nice connected version that avoids the curvature problem. It’s much, much easier to assemble and still works very nicely. It’s not quite as tight a snug between the two as the one with the extra triangles, but worth it. I recommend trying it out!

(“Scutoid” is officially named after the triangle-shaped “Scutum” of a beetle)

Tom also came out with this variation, which I haven’t tried yet:

I might try some other variations. In the end I’d like there to be a printable version suitable for the classroom (so maybe a little scutoid info and directions in the empty space) because I think a classroom full of scutoids would be just excellent. A scutoid set where all of them fit together (instead of just in pairs) so that a class can collaborate would be even better!

I’ll look into it…

And obviously it would be best with a video that connects the shape to a pile of mathematical concepts (including, now, negative curvature and developable surfaces and computational origami…).

Sep 11 2018:

So with the help of Laura Taalman and Tom Ruen, we’re exploring the possibilities of scutoid papercraft. Here’s some of my tests:

There’s two net sets we’re working with so far (and a third, infinitely-packable one hopefully coming soon, so that a class can put all their scutoids together).

Note: these are drafts, so don’t spread them too far until we get them in shape!

Here’s the classic pair, where the two scutoids are the same and two fit together one way:

And here’s a scutoid quartet, where two copies of two different scutoids fit all together, which we adapted from this model by John Peplinski. This lets people work together in groups of up to four:

As I work towards finalized net designs and instructions, plus probably a video in the future and some teacher-facing instructions, it’s time from some feedback, especially from teachers who have experience with hands-on paper math making activities, and parents with kids who can give me some pointers on what kids at what ages need how much help to create the scutoids.

The challenges are similar to paper polyhedra, but with the extra thing about the paper curving slightly, which may or may not add significant difficulty. Also there may be challenges in different people making pieces meant to fit together.

Sep 18 2018:

This week we have another net pair, two scutoids that can tile infinitely to fill space:

Here’s the PDF version which is recommended if you want to be sure your infiniscoots will be compatible with other people’s infiniscoots.

The original design comes from JohnPep on thingiverse: https://www.thingiverse.com/thing:3042446

As with earlier nets, Laura Taalman and Tom Ruen converted the shape from 3d model to flat polygons, and then I tweaked and arranged the net to hopefully be optimal for a printable paper project.

In an earlier test I tried to keep some of the curvature in the original model by just adding a couple extra seams:

It’s not too hard to make them with the extra curvature, but they don’t fit together nicely so I went with a single-face design again.

Try some with your friends and see if they all scoot together! It’s fun to puzzle them together into different arrangements.

Another note: one of these scutoids has top and bottom faces of 4 and 5, rather than 5 and 6. I think, given that “scutoid” lacks a formal definition, this is within the range of things that can be called scutoids in my book.

4. Mathy Definition

I’m not going to get too specific at the moment because I don’t think an official definition has been published, but here’s some more insight:

Scutoids were created/described using a computational process, so the geometric definition is not as simple as we’re used to with things like polyhedra. Here, “faces” includes the non-flat quadrilaterals that are negatively curved on the sides. The top and bottom are flat polygons, and the graph of the shape is cubic, meaning all vertices are of degree 3.

Scutoids can have any convex polygons as the top and bottom faces. It’s not clear to me whether we’d still call “Scutoid” something where the difference between sides is more than 1, but I think a scutoid with two triangly transitions would still be called a scutoid, as long as it has all the right properties.

Scutoids are the result of a voronoi cell computation, so to be a scutoid a shape should be plausibly voronoi-y. The scutoid itself might not be convex, but any horizontal slice of it should be a convex polygon. (This is a property that the paper versions don’t quite approximate. Because paper is flat instead of negatively curved, it tends to bow inward, making a horizontal slice have slightly curved-in sides.)