# Curvahedra is a construction system for arty mathsy structures

Edmund Harriss is a very good friend of the Aperiodical, and a mathematical artist of quite some renown. His latest project is CURVAHEDRA, a system of bendable boomerang-like pieces which join together to make all sorts of geometrical structures.

A few years ago I commissioned some art from Edmund for Newcastle University’s maths department. He sent me some rather mesmerising layered laser-cut patterns.

To get the first production run going, Edmund’s set up a Kickstarter campaign. Here’s his pitch video:

https://www.kickstarter.com/projects/2100742888/curvahedra-a-new-way-to-make-beautiful-geometry

The CURVAHEDRA system is a set of flippy-floppy shapes which can each work as half-edges coming out of a vertex in a polyhedron. The ends fit together with a very simple interlocking buckle which reminds me of old-fashioned Scalextric track.

The Kickstarter campaign is looking for \$15,000, and at time of writing with a week to go they’re already past the \$10,000 mark. The smallest kit begins at \$10, heading up to a few hundred dollars for much bigger packs with lots of different shapes. I had a chat with Edmund to learn more about the process of inventing CURVAHEDRA and how it works. I’ve got a very practical question to start with: is there a reason they’re not die-cut? Do you just like laser-cutting so much? Well, yeah, that’s part of the reason my life is so crazy right now. So, before we started I even got a quote from a printer to get them die cut which would’ve been about 1.5¢ per piece, which would’ve been quite reasonable. That printer went out of business, possibly because they were giving out such low quotes and so we hunted around and we eventually found one die-maker who was willing to give us the sort of tolerances that could be used. We tried a version of it that worked in those tolerances but it was a compromise on the die side because it worked in those tolerances and a compromise on the design side because the pieces didn’t always fit together. You don’t want to do something that’s sort of not really working either way in order to force it to work. We did eventually get one quote back but it ended up being more expensive after you made the die than laser-cutting. Part of the factor is I had my own laser cutter and I know how to use a laser cutter and so the cost for us for laser cutting is much lower than for someone else. That’s probably a lot longer answer than is really interesting but I am talking to people who know manufacturing and there are some cutting technologies but you’re talking about a$50,000 cost per die, which would be great if we were selling millions. There are some possibilities down the road now we’ve done the basic product development.

The good thing about laser cutting is you’re not tied to manufacturing certain things.

You can change it?

You can change it and make bespoke parts – make pieces that create spheres with arbitrary polyhedra. Make pieces that allow you to do non-orientable surfaces. Right now you can’t use it to make a Möbius strip for example because when you turn the pieces over, they don’t fit together any more.

At some point I’d like to try to make a Klein bottle. At least for me. Other options include doing things like Boy’s surface, and other projective planes.

So that’s a good lead on to this question: what can you actually make with the current set?

You get 3-,4-, 5- and 6-edge pieces in the current kit. We’ll have a stretch goal coming out soon — you can have an exclusive on that — 7- and 8-edge connectors are possible and we might have those in a stretch goal.

The first thing I can think of for 7 is there’s a good hyperbolic tiling.

A skew apeirohedron. Image by Tomruen [CC BY-SA 4.0], via Wikimedia Commons

And there are also some toruses you can do. And with 7- and 8-edge connectors you can make any mesh where each vertex has up to 8 edges coming into it. Once you get to 9, the edges start to overlap, so it’s not quite so workable. The other thing you can do is make various sorts of minimal surfaces and branching surfaces. Have you heard of skew apeirohedra?

Now I like that!

Especially on The Aperiodical! It’s a collection of space filling things that are either regular or uniform and you can make these infinite surfaces.

So it’s like a unit that tiles 3d space.

Well, it’s a surface, so at every vertex you have the same pieces together and it links up with itself in a higher way. The way it works out is that the way you link up faces forms various shapes, like there’s one where the cells are icosahedra. For that you need 7-connectors because 7 triangles meet at every vertex. The covering space will be the hyperbolic plane because you need 7 triangles coming into every vertex.

OK. We’re getting to the end of my geometrical knowledge. I sort of know what a covering space is.

If you want to know, this is actually a good example! It’s sort of what happens when you remove all the branching, so every branch becomes its own part of the space. The classic example is a torus – if you start unfolding the torus, so every time you go round the torus you go into a new bit of space, the covering space becomes the whole of the euclidean plane.

Right… So I can see that these surfaces with lots of holes in, that would make lots of space, which makes it hyperbolic. Is that a good handwavey description?

Yep.

I’ve seen the model on your kickstarter page where you’ve got two models interwoven with each other, made by Andrea Hawksley. Was that something you knew would be possible?

Nope. I gave it to her and she came back and said “Hey, you can do this with it!” And I said “Wow, that’s cool”.

Woven curvahedra discovered by Andrea Hawksley

I guess when you give her something and she starts weaving with it, it’s not a surprise.

Yeah. Whenever you create any sort of unit you give it to Andrea and she finds some exciting new way of using it.

The reason it happens is that the spiral breaks the reflection symmetry and so the two balls you have are one and its reflection.

You can get into some group theory, but it’s too late on a Friday for me.

Yeah, I was looking at it and thinking there’s some kind of group action but I couldn’t put it into words

Yeah, let’s just say it’s a ball woven into its mirror image.

Have you thought about if it’s possible to put something inside its dual, like an icosahedron inside a dodecahedron?

That won’t work, because the edge lengths need to be different. But on the campaign page there’s a model which is an icosahedron and a dodecahedron woven together on two layers in a weird way. I know that doesn’t make much sense but if you just look down on the 5-connectors, you can see that there’s a dodecahedron, and the 3- and 5-connector pieces are woven together. To keep the covering theme, you can think of this as a double-covering of the sphere.

A dodecahedron and icosahedron woven together.

I’ve heard those words before, so I’ll just let that slide. So, is it like a dodecahedron and an icosahedron inscribed into the same sphere?

Yeah, that’s right.

I see nobody’s signed up for the museum pack yet (a $500 package containing 100 sets of various designs). I suppose there are only so many museums that would potentially be up for that. Yeah. And also you might notice that nobody’s gone to bring me out to give an actual talk. On the other hand, surprisingly a few people have backed it at the$250 level.

That’s sort of the level where if you work in a university maths department you can go to your head of school and ask for that much money for an outreach project.

It makes a really fun activity for a group of people.

You showed that in your video – a group of people just sat around a table putting it together.

Yeah, we had an event at a sort of meet-and-greet thing, and people said it was the best icebreaker they’d ever had, because you ask questions because you need the answer rather than just for the sake of asking a question. So conversations just sprang out after you’ve broken the ice.

Questions like “can I make a shape just using 3-connectors and 4-connectors”?

Yeah, like that.

So what’s in the kit? It comes as some laser-cut sheets of paper and you press the pieces out?

Yeah, for 12 pieces you’ll get 4 sheets, each with three of them on it. We’re currently working out packaging and getting plastic bags the right size and envelopes the right size and so on.

Did you think very hard about the size of the pieces?

I have, but it’s one of those things where the very first size I made is the piece that’s now available. And I’ve made it in lots of different sizes and I always don’t like them. It just feels like I hit the right size the first time. This is a thing that seems to happen in art. You get an idea and you do it, and it’s just right the first time. You do a lot more experimentation and try different things, but you just end up with the first version you had, but now validated by all the extra work you did.

About that photo of you with your party hat on – what happens if you keep building it?

It’s just a bigger and bigger cone.

It’s just a cone, it’s not going to flare out?

A pseudosphere. Image by Leonid 2 [CC BY-SA 3.0], via Wikimedia Commons

Well, it depends on what you mean by ‘keep building it’. 6-connectors make a cone, but if you add in the occasional square, then you’ll get a pseudosphere.

A pseudosphere is…

It’s a model of hyperbolic geometry. Its relationship to hyperbolic geometry is the same as the sphere’s to Euclidean geometry, except it can’t do the infinite sheet.

OK. I don’t understand that. Moving on…

One of my first questions was a sort of skeptical one – once you’ve got this kit and you’ve made the usual polyhedra, what else can you do? And that hat is a good example of one way you can go.

As I said, you can make any mesh.

Are you expecting someone who buys the museum pack to build a model of, like, a computer game character?

Of course you need to make a rabbit at some point. Although Blender seems to be replacing the rabbit with a monkey’s head, so that’s also possible.

Or that dancing baby!

Maybe that claim is a little overblown because making models that way is going to make models which are a bit too large. The other thing is that there are non-obvious polyhedra that can happen. This system can do bigons.

Let bigons be bigons?

That joke has to be made, yeah. So it’s a nice thing – when you’re with a group of bright kids, someone’s going to say “I know what this is, you can do all the regular polyhedra”, so you say “OK, yeah, what will you do next?” “OK, I’ll do the uniform polyhedra”. And if they’re a real smart-Alec they’ll know that, but then you can say “no, that’s not everything” because they won’t be thinking about the fact you can do an infinite number of things because you can replace every edge with a bigon.

What I like about this is that it looks like a much cheaper and more robust way of making polyhedra than other construction systems, like Zome.

The downside with this is that it doesn’t come apart and go back together too many times – the card starts to delaminate a bit. You can take it apart three or four times, but not infinitely many – or arbitrarily many, I should say – which is what Zome will support. However, the Zome pieces will only give you straight lines, and I think a lot of the attractiveness of these models is that they have these interesting curves, and the paper naturally responds to the geometry that you’re putting it into, by coming up with these forms.

Was that how you started, were you trying to go for something that would give those curves?

Well, I was trying to go for something that was flat and would become three dimensional, so the bending naturally comes out of that.

The first models I made were trying to make a torus which, as you know, though it doesn’t have total curvature it does have local curvature, in order to not be a cylinder. And so, the motivating idea was trying to take a sheet of paper which was cut and folded in certain ways, that you could put together to make a torus. That led to thinking about how the junctions could work and then from there actually going ‘well, a torus is nice, but there are simpler things like spheres.’

These bits are made of paper, so you could draw on them, to add some more structure, or art. Did anybody start doing that in your test sessions?

You definitely could. We haven’t actually done it, but there’s definitely a possibility.

What are the kinds of things you’d want them to do, or hope they’d do?

Well you just let them do what they want! A lot of the motivation is thinking about getting a system where people can start playing and discovering stuff without being told what to do.

People could start sketching things or doing scribbles or they could start to explore the structure of what happens when you draw certain patterns on each face and how that’s going to connect up.

The CURVAHEDRA Kickstarter campaign will end on Tuesday the 15th of November, and it looks like it’s going to comfortably reach its goal.