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.

Vi