Category Archives: Math Art

Hypernom – a 4D VR Game

Hypernom is a 4D VR game that I’ve been working on with Vi Hart, Henry Segerman, and Marc ten Bosch. We’ve already talked about it on the eleVR blog. You can see the code on Github, read about the math behind the game in our paper, and see our presentation about it at Bridges 2015 in flat and spherical video … amongst other things. So I thought I would try something a bit different in my blog post…hypernom

In which I explain our new game using only the ten hundred most used words.

I work in a group that makes things for the pretend world that you wear on your head. We just made a game where you eat four-world things, one three-world thing at a time. My favorite thing to eat is made up of my favorite three-world thing.

My favorite thing in the three-world is made from ten and two faces  with five sides each that are put together.

My favorite three-world thing is like a different three-world thing with twenty faces with three sides each that are put together. The points of my thing are like the faces of the other thing and the faces of my thing are like the points of the other thing. Both things have ten times six edges and share a group with 10 times six times two parts. Notice that 10 times six times two is also five times (ten and two) times two and three times twenty times two.

If you get a hundred and twenty of my favorite three-world things put together then you have another of my favorite things. It does not fit in the three-world and instead lives in the four world. In the fun game that we wrote, you can eat all hundred and twenty of my favorite three world things by moving your head around.

Go try it out now!


Overshadowed by the Packaging


If companies wanted to be serious about getting even slightly more environmentally friendly (and save themselves money as well), they really ought to consider using less packaging. The amount of useless plastic being made just to securely package things in obnoxiously difficult to open clamshell packaging is rather mind-boggling.

Fortunately, sometimes that packaging doesn’t need to just get thrown out.

2012-07-02 18.55.05A couple years ago, shortly before people stopped selling them because of potential health risks to children, I bought a lot of those neat little magnetic ball toys. The packaging for these things was rather absurd. A large plastic box many times larger than the size of the balls contained within it. A few more plastic bits to keep everything held “just so” in the packaging. A small paper box with writing on it. And, of course, a small sturdy plastic box to store the magnetic balls in. It wasn’t just over-packaging. It was, over-over-packaging. And it made it rather hard to get at the part you wanted to play with too.

My friend Aviv Ovadya was over as I unpackaged them. As we fumbled with opening the packages, he proposed that we re-use some of the excessive packaging in math art (my friends are awesome like that).

I know Aviv through origami circles originally, and I think we both share a bit of the origami “aesthetic”. In particular, neither of us wanted to cut attachment holes or just glue the boxes together and be done with it. Fortunately, I happened to have a large package of rubber bands handy, so we experimented with different non-damaging ways of connecting the boxes using rubber bands, settling on creating a nice icosahedral structure with 30 boxes and 40 rubber bands.

If I’d made it more recently, I might have made a blog post about it then, but back then I rarely touched my blog, so that was the end of this particular diversion. The project might have been left undocumented forever, if I hadn’t realized a few weeks ago that the rubber bands holding it together had mostly disintegrated, and the structure needed to be trashed (boo!) or completely rebuilt. For the rebuild, I used small hair bands, which should last longer than regular rubber bands. The hair ties were also much more secure, so I only needed 20 of them.


Probably the best thing about the transparent packaging material is how amazing the shadows through it look. Regular geometric objects often have cool shadow projections, but I think the ones here are particularly spectacular.IMG_0715

Do you have any packaging trash that could be transformed into something that might (like this) literally or figuratively overshadow the original packaging contents?

Sand Hill-bert Curve


Although I’ve never managed to make a particularly castle-y sand castle, I’ve always loved making sand castles at the beach. The last couple of times that I’ve been to a sandy beach, I’ve been particularly inspired by the idea of creating fractal sand castles. Something about a fractal sand castle on a fractal coastline just feels right.

Long time readers of my blog might remember the Sand-pinski Triangle that I created on a previous trip to the beach.

On my most recent beach trip, I hilled up some more sand to create a Sand Hill-bert curve. The Hilbert Curve is a fractal space filling curve discovered by David Hilbert. It’s shown up in line-enveloped form in my blog before. The Sand Hill-bert Curve is a third order Hilbert Curve, although I’ve taken the liberty of adding some ellipsis so that you know that it could keep going and filling all of space.


Although you can get a pretty good sense of the 3-dimensionality of the sand hill from the shadows in the picture, I love the effect created by this awesome 3D rendered animated GIF that Emily Eifler made for me. Also, there is something delightfully weird about the idea that the sand castle might be hollow.



Bridges Math Art 2014

It’s conference season, and I just concluded a week in Korea at the Bridges Math Art conference. This was my third time there, although my previous blogging of it has been minimal. Bridges is *the* conference for people interested in mathematical art, and the diversity of work that is shown and presented there is incredibly impressive.


This year I presented a workshop on Binary Dance, which, by participant request, was spontaneously renamed the “Binary Dance Party” to make it  more “fun”. I certainly had fun moving around, trading partners, stomping, clapping, and performing assorted binary operations. So much fun, in fact, that I forgot to take pictures. Oops! You can find the full paper on the Bridges archive.

I also showed my origami star polyhedra in the juried art exhibit, helped jury the short movie festival (go watch some of these now!), and acted in the play. By the way, the play, “Scrambled Legs and Bacon”, was a world premiere written by Steve Kennedy and directed by Steve Abbott, both regular Bridges attendees. I played a female graduate student who seemed to spend most of her time solving logic puzzles.

As you can already tell just from the things that I personally had a hand in, Bridges really puts an effort into being about all kinds of math, all kinds of art, and all sorts of connections between them. In addition to the events that I’ve mentioned, there was a mathematical poetry night, a math dance performance by Karl Schaffer, Erik Stern, and Saki, and a few large construction “barn raisings”.

Given my fondness for the 120-cell, it’s no surprise that I was a fan of this giant Zometool omni-truncated 120-cell that was constructed on site, mostly by some highly motivated high school students.


Another highlight of the conference was seeing Noam Elkies and listening to him give his talk on Canons. If you ever get a chance to see Noam Elkies talk about music you absolutely need to go. People were buzzing about his talk for days.

DSC_9801Unlike 6OSME, where the exhibition is supplementary to the accepted papers and presentations, the Bridges exhibition mostly showcases artwork for which there is no presentation (and vice versa). Indeed, at Bridges, the artwork and the papers are juried/refereed separately. However, a few of the talks did match up with pieces in the art exhibition.

For example, Carlo Séquin talked about his lego-compatible 3D printed curvy construction pieces and also showed examples of them in the exhibit.

DSC_9797Similarly, Henry Segerman talked about his paper with Vi Hart on creating a physical visualization of the quaternion group, and showed the resulting monkey sculpture in the exhibit.

Of course, just like 6OSME, way more awesome things happen at Bridges than I can describe in one blog post. Fortunately, everything is well documented and publicly available, so you easily read the papers, view the art exhibition, and watch the film festival from your computer at home. And, if Bridges seems like your sort of thing, then I look forward to potentially meeting you next year at Bridges 2015 in Baltimore, Maryland.

Finally, oh my gosh, tessellating bunny snub cube!



I spent the last week in Tokyo, Japan attending the 6th meeting on Origami, Science, Math, and Education (6OSME).

6OSME is an amazing gathering of researchers from all of the many disciplines relating to origami (computer science, mathematics, engineering, architecture, etc.), as well as serious origami artists. If you’ve ever doubted the validity of origami as a serious research discipline, then hearing a few talks here will surely change your mind. And if you think of origami as paper cranes made from a single folded square with no cuts, then the topics of these presentations might surprise you. There were way too many presentations for me to cover all of them, but here are a few of the topics that were covered to give you a sense of the breadth of origami research.

On the industrial engineering side, Gregory Epps kicked off the convention with a plenary talk on “Industrial Robotic Origami”, or the production of curved, folded metal by robots. Robots folding metal. ’nuff said.

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Mathematicians like to imagine that the world is made up of “ideal” paper, which is entirely rigid, bends only at fold lines and has no stretch. It’s also infinitely thin, of course. In this theoretical world, it turns out that you cannot squash paper polyhedra flat. You might have trouble achieving this world in practice, though – most paper is actually a bit stretchy. Abel, Connelly, Demaine, Demaine, Hull, Lubiw, and Tachi showed that by adding small slits it is possible to fold polyhedra flat in their work on “Rigid Flattening of Polyhedra with Slits”.

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Most people don’t think of origami as involving cuts, but 6OSME actually had an entire session devoted to “pop-up” techniques. Yoshinobu Miyamoto showed off some impressively tall and sturdy structures made from single sheets in his presentation on “Rotational erection system (RES): origami extended with cuts”. The tallest one that he demonstrated looked to be about 4 feet tall, and it seemed clear that they could get way bigger with the correct materials.
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Chris Itoh managed to pull the biological sciences into the conference with his talk on “The elusive technique of folding anatomical subjects”.

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This talk on “Curved-folding convex polyhedra through smoothing” was from a highly architectural point of view. Suryansh Chandra, Shajay Bhooshan, and Mustafa El Sayed work in the research branch of an architectural firm, where they developed this technique so as to be able to create cool, “real-world” structures.

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My own work was presented in the “modulars” session on origami made from multiple sheets of paper. I spoke on my star polyhedra series and the design process that I use to create these highly mathematical models.


The variety of origami sciences represented at OSME is amazingly impressive, and I don’t have the space to do it justice. I am missing a bunch of fantastic talks just because I was focusing more on breadth than on depth, but I haven’t even covered the full breadth here. You’ll have to check out the full papers (when they come out) to get the scoop.

OSME is held only once every four years, so if you want to attend the next one you will have to wait for a while. On the other hand, you’ll have plenty of time to prepare your presentation!