Scutoid Papercraft and Video Instructions (Patreon)

Printable PDFs for three different kinds of scutoid papercraft are attached!

Video instructions are above.

Scutoid Pair: 

Two identical scutoids on one printable sheet.

These are based on a 3d model by Laura Taalman, which you can find here: https://www.thingiverse.com/thing:3024272

“Scutoid” is a general term, but the Taalman Scutoids have been seen representing Scutoids all over the internet.

The papercraft version was made with her help, as well as additional help from Tom Ruen.

Scutoid Quartet:

Four scutoids that all fit together. Print out the same sheet twice, to get four scutoids total.

These are based on 3d models by John Peplinski, which you can find here: https://www.thingiverse.com/thing:3033602

Like the most popular scutoid, these each have pentagons on one side and hexagons on the other. But they aren’t regular.

This arrangement illustrates some of the properties talked about in the original scutoid paper.

Infiniscoots:

Two different scutoids on one sheet of paper. Print out infinite copies and make infinite scutoids to fill up the universe.

These are also based on 3d models by John Peplinski, which you can find here: https://www.thingiverse.com/thing:3042446

This infinite tiling may not imitate nature so exactly, but it is very mathematically cool. One side is a Cairo Pentagonal Tiling.

As with the others, the papercraft version was made with help from Laura Taalman and Tom Ruen. Thanks!

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 together 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.

Paper is flexible enough that you can approximate a scutoid fairly closely, even if it’s missing some subtleties of the curvature. It’s certainly accurate enough to enjoy the shape and its properties!

When this shape first went viral in 2018, a lot of people were asking about it and I posted some thoughts. A commenter Amy Tobol asked about a papercraft version, and I thought it should be possible. I started looking into it, got some help, and 6 years later here we finally are :)

What’s the actual technical definition though?

Scutoids were created/described using a computational process, and I don’t believe a geometric definition has been published (unless my information is out of date). But as mathematicians have taken the shape and run with it, maybe we’re due to define the mathy version of it soon.

When the Scutoid Paper first dropped, I was briefly in contact with one of the mathematician authors, Clara Grima, who helped me understand a little bit more about the process. So thank you Clara! But I still might not have everything right.

For the computational version of the definition, “faces” includes the non-flat quadrilaterals that are negatively curved on the sides. The top and bottom are convex polygons, and the graph of the shape is cubic, meaning all vertices are of degree 3.

Scutoids can have a variety of different polygons as the top and bottom faces. I don’t believe the process described in the paper cares whether the difference between sides is more than 1. I believe if there were two transitory triangles, such as if the top and bottom were a quadrilateral and hexagon, that would still be a scutoid. I guess we could all decide to name it a double-scutoid, or whatever we wanted. But no one has, yet, unless it’s happened recently and I’m out of date.

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 papercraft 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.)

The mathefied non-computational-biology version of scutoid might be more fun if we’re less strict on the voronoi part of the definition, I don’t know. 

As with all things that are named, the names are a result of community and culture and use. No superhuman authority can tell us what scutoids are. We can probably agree that first claim to authority on scutoid definitions goes to the humans who wrote the original paper and coined the word. But if mathematicians take the shape further and have extra fun with it, and the original authors don't object, I think there's plenty of room in the world for more scutoid fun.

And, as with all things that are named, if no one makes up a technical definition that we all decide to honor, the technical definition won’t exist.

Maybe you’ll be the one to define it?