Curvahedra

My friend Edmund Harriss, who is a mathematician and artist, has a kickstarter. Usually I don’t blog about commercial things. And Edmund didn’t even ask me to do this.[1] I emailed him letting him know I’d be happy to write up a post on his kickstarter. Why?

It’s the same reason I think he’s an awesome person. He is invested in getting people to see the joy that is inherent in doing mathematical work, and the creativity associated with it. To the point that years ago, when he happened to be in NYC, he came and talked to my Algebra II classes about aperiodic tilings (and brought a ton of these tiles for my kids to work with).He co-authored/illustrated a fabulous adult coloring book, Patterns of the Universe, which inspired one of my students last year. (And led me and a colleague down a crazy fun rabbit hole this summer.)

And more recently, this year, when I started curating a math-art show at my school, he immediately agreed to be a featured artist and sent not only artwork, but some extra tiles that I requested just because I knew our lower school kids would enjoy playing with them. (A walkthrough video of the gallery is here.) Here are some of the tiles he sent… Yup… fractal tiles!

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One of the pieces he sent for the show were these curvahedra. I had fun making and playing with them (and the sheets that the pieces popped out of).

 

So feel free to check out his kickstarter if you are interested…

As an added bonus, I emailed Edmund gads of questions about his work, his process, and his curvahedra. And below is our Q&A!

What are the feelings you hope kids/adults/students/teachers have while using Curvahedra

I hope they feel curious, but relaxed, looking to see what can be discovered and enjoying the pleasure of discovery. Many people report that it can also be quite a mindful experience, which I had not expected.


What are the different types of things that I can make with Curvahedra?

There are a whole collection of different balls and eggs that are the most obvious things to make. Donuts are the next option. I have made some exotic pieces which branch and rejoin. I discovered when making pieces for the kickstarter that you can make some nice cones. In theory though there are few limitations, anything that can be made as a mesh can be made from Curvahedra. As meshes make many of the creatures and objects in CGI the limits are hard to find!


What do you love about mathematics? How does that tie-in with your creation of Curvahedra?

I liked mathematics initially because I could do it, but that never gave passion. What fired my love of the subject was the way you could create little worlds and then explore them. Once the rules were fixed you had constrained the behaviour, yet that did not mean that you knew what could happen. I often feel that the most valuable thing I learnt from my PhD was flexibility of thinking. In mathematics you have to be flexible to think about what you can control and how that works with other rules. In this vein Curvahedra is a system that can do many things. The fun is in discovering the most beautiful and making mathematical rules to help find that.


What is the intellectual provenance of Curvahedra? In other words: how did you come up with the idea?

I wanted to make 3d objects on a machine that cuts essentially in 2d, a laser cutter. I knew that paper could bend, so that was my goto material. I tried a lot of options, not always successfully. Then with the right connector and the right idea suddenly I had Curvahedra. It was a revelation when I made the first ball.


When you were playing with Curvahedra, what unexpected discoveries did you make?

It has really helped my understanding of space and surfaces. In particular getting an idea of the curvature of a surface, and when that is rigid or not. That understanding helped when I was studying 2D crystals with a Physicist. I have also used Curvahedra to investigate minimal surfaces, these are surfaces that minimise strain and cannot be  improved by local change. This is exactly what the paper seeks out, you impose some geometry and the paper pieces find the most efficient form.


What do you do with your creations when you’re done making it?

They litter my house, many hang from string in my windows. Luckily my wife likes them too, she has been really supportive helping put together the kickstarter and will be playing a key role in running it. I could not have done it without her.


Where is the mathematics in Curvahedra?

The notions of curvature and minimal surfaces that I discussed above, as well as other concepts from multivariate calculus are really shown off well by the system. In fact many of these advanced topics can be easily introduced to elementary students with Curvahedra. There is also some excellent topology to discover. On the other hand that mathematics is maybe a little too advanced for a general audience. There is a lot of geometry to be done. The regular polyhedra turn up naturally and general ideas about space will come from playing with the system.


How can a math teacher use Curvahedra in their classrooms? What about an art teacher?

I would hope they would both use it in the same way. As a low direction activity exploring, where people initially make a basic ball and then are just left to it. This is how I usually run sessions with the system. As people discover things that can be done it is natural to try to find out if you can discover everything that could be done. That gives natural mathematics questions. I always feel that when people have a question and mathematics is the answer they are far more motivated to look at the mathematics.


What are your favorite mathematical ideas? Who are your favorite mathematicians?

I love all sorts of visual mathematics and geometry more generally. The discovery of non-Euclidean geometry was such a revelation and is incredibly beautiful to me. It is hard to list favourite ideas though, as it is all so cool. As an undergrad I swore off differential equations, for example. I did not look at any mathematics involving them for years. Then I finally started to see how they linked to geometry, made art with them and did research and now I love the ideas especially in Differential Geometry. The best answer is often that my favourite maths is what I have just been studying! For mathematicians, I have been fortunate enough to spend time with John Conway and both he and his work are phenomenal. The humour and insight in Archimedes’ writing is also a great favourite. For geometric insight Alicia Boole-Stott is incredible, she was able to work out all sorts of properties of four dimensional figures that I could not approach without a computer. Similarly Felix Klein’s visual insight is amazing. Finally you cannot be a geometer, interested in the more elementary side of the subject, without loving the work of Donald Coxeter.


Who are your favorite mathematical artists?

Max Bill is my favourite artist of any sort and I also love the work of Kenneth Martin. They both worked in concrete art which  attempts to create form out of nothing, rather than refining essentials from something real as in abstraction. That is a notion that works very well with mathematics. Moving in the direction of mathematics, George Hart has many inspiring pieces, his use of symmetry and shape to create intricate structures definitely played a role in the development of Curvahedra. Finally I am very lucky to be able to work regularly with Chaim Goodman-Strauss who brought me to Arkansas. His work which he describes as the illustration of mathematics was probably the single biggest inspiration for me. For a time I felt like all my best ideas were stolen from him.


Besides earning zillions upon zillions of dollars, why did you want to do this Kickstarter?

I want more people to get their hands on Curvahedra. Every place I have taken it is seems to inspire and delight and I wanted to spread that further. Sadly I doubt that Math Art will make anyone rich, but I do hope it can get into more hands and inspire more love and enjoyment of the subject.


What do you think — in an ideal world — would every math teacher have in their classrooms?

All sorts of geometry and math toys and games, and time to explore them. Things waiting to be explored by the students on their own terms rather than to satisfy a rigid curriculum. Though the problem with ideal worlds is that they also have ideal students! This might not be practical in all sorts of ways.

 

[1] To be very, very clear: I asked Edmund if I could promote his kickstarter. He did not approach me. And I am also getting nothing from him for posting this. In fact, I’m about to pledge money to his kickstarter!

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