Twelve Card Assembly for G4G12

My gift for Gathering For Gardner 12 was a set of 12 specially marked business cards that can be used to assemble a rhombicuboctahedron. The final construction looks like this:


Apologies for the cracking when folded – I’ve never purchased business cards before and wasn’t clear about what options you needed to choose to get “optimal for folding” cards.

I first came up with this design a few years ago after I was handed a large pile of old business cards and gave myself the challenge of creating modular origami models of all of the Archimedean solids. As I noted in my post at the time, “I wouldn’t be surprised if some (or even all) of these designs were examples of parallel invention, but I haven’t seen any of them elsewhere as yet, and I certainly had a fun time coming up with and building them, which is probably the important part”.

It is possible to put this together without tape or glue, but it isn’t super easy. You may find it easier to add temporary or permanent tape as you go if you don’t mind “cheating”.

If you want to treat this construction as a puzzle, you should probably stop reading here.


How to make your own business card rhombicuboctahedron

  1. Fold all 12 cards as indicated. (If you want to use unmarked cards, see folding directions at bottom)
  2. Groups of three cards go together like this
  3. Groups of four cards go together like thisIMG_1182
  4. If you are trying to make this without tape or glue assistance, I recommend starting with a group of four cards held in your non-dominant hand
  5. Add cards around as in the next several imagesIMG_1184 IMG_1185 IMG_1186
  6. The last piece is the most difficult to add (you might have to force it a bit), but the entire structure should stay together once it has been added
  7. Don’t forget to put your pieces together in a nice color scheme!

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Folding modules from arbitrary cards (Pictorial Instructions)

These instructions show the easy way to fold modules. These modules are not centered, but the technique for centering the module if necessary to match with the card design should be straightforward.

IMG_1173 IMG_1174 IMG_1175 IMG_1176 IMG_1177

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!


Bridges Math Art 2015


Yuri Vishnevsky, Kelly Delp, Katie, and I make a human cube at Bridges 2015

Conference season is exhausting, so I’m very selective about which conferences I attend. Bridges Math Art is definitely one of my favorites and has a solid spot on my “attend” list every year.

The 2015 conference just ended, and was as amazing and exhausting as ever.

This year I presented a workshop on Fibonacci Lemonade (and other mathy layered lemonade variants). Even though Bridges is all about the junctions between math and art, the art of cooking is rarely represented. The Fibonacci Lemonade workshop diversified the conference a bit with some delicious summer math fun. You can find the full paper on the Bridges archive.

My big projects for this year’s Bridges were a couple of 4-dimensional VR art projects that were made jointly with Vi Hart, Henry Segerman, Will Segerman (monkeys only), and Marc ten Bosch. “Monkey See, Monkey Do” is a project with both 3D printed and VR monkeys arranged symmetrically in 4D space and displayed in the juried art exhibit.

Hypernom” is a sort of 4-dimensional Pacman, where you move your head around and try to “eat” the cells. We showed this in the art exhibit and also had a paper and presentation on some of the math behind the project: “Hypernom: Mapping VR Headset Orientation to S3“.

We’ve been working on these projects for a while and I’m delighted with how they turned out. Look out for upcoming blog posts about these pieces – they deserve to have more said about them than what I can fit in one paragraph in a conference summary post. (Update: Vi just made a post about Hypernom on – go check it out)

I also acted in the play, co-wrote mathy-y lyrics to Hotel California (“Hotel Hilbert”), served on the proceedings program committee, and helped jury the short movie festival. (I know, I know, I could do more conferences with less exhaustion if I just did less stuff when I went to one) Needless to say, it was a pretty hectic conference, and I definitely didn’t get to check out every presentation and workshop that I was interested in.

I did get to see the math dance performance by Karl Schaffer, Laurel Shastri, and Saki, as well as Tanya and Tim Chartier’s mime act. Both groups had some new work that I hadn’t seen before and thoroughly enjoyed.

This year’s art exhibit was quite possibly the most impressive exhibition in the whole time that I’ve been attending Bridges.

Within the mathematical theme that connects the pieces of the exhibit there is a great deal of variation both in terms of medium and focus. Here are six pieces that show some of the depth and variety of the exhibition.

Hair Band Sierpinski Tetrahedra at the MoSAIC Festival

I really enjoy running math art workshops. MoSAIC (Mathematics of Science, Art, Industry, and Culture) is a series of math art events being held around the country to get people interested in math. I was pleased to be able to participate in the very first MoSAIC Festival held at Berkeley City College last October.

Vi and I ran one of her mathematical balloon twisting workshop on the first day.


On the second day, I premiered a fractal hair band sculpture workshop inspired by Zach Abel‘s awesome rubber band workshops.

The Sierpinski tetrahedron is a natural extension of the Sierpinski triangle (which shows up in my blog quite a lot) to the third dimension. Here’s are some that we made out of cookie dough and frosting during our math cookie day last year.


The great thing about workshopping a fractal structure like the Sierpinski tetrahedron is that all of the modules are the same, and the scope of the workshop can expand and contract easily with the allocated time and number of participants – you simply make a higher or lower order final model.

The level 0 tetrahedron module that I designed is made out of 10 hair bands and looks like this when stretched out.


The level 1 tetrahedron is made out of four level 0 modules that are joined by extra white hair bands. For some reason I seem to be missing a picture of the level 1 Sierpinski tetrahedron by itself, here is a close-up shot of the larger structure that shows two such tetrahedra (one in front of the other).


Similarly, the level 2 tetrahedron is made out of four level 1 tetrahedra.


I ran two workshop sessions of an hour each and was hoping to finish a level 3 Sierpinski tetrahedron between the two of them.

We did it!

IMG_0526Like all hair tie and rubber band sculptures it has a fun bounce to it, although you still have to be careful not to boing too hard and break the hair bands. But, I don’t seem to have taken any video of us boinging it. So, instead, here is another shot of a bunch (but definitely not all) of the awesome people who came by and participated in my workshop along with our completed sculpture.

IMG_0527It turns out that the sculpture looks great stretched out on a standard tripod poster stand that was hanging around. And George Hart took it with him to showcase at future MoSAIC events.

IMG_0516I’m not sure if it’s still making the circuits, but if there is a MoSAIC festival near you soon, then you should definitely go and check it out (and let me know if you see this hair band sculpture)!




A Hilbert Curve Afghan


I’ve always been a crocheter rather than a knitter. It’s not that I can’t knit, it’s that I’m so much better at crochet that I just generally can’t be bothered with knitting. These days, I barely do either as they have sadly joined the list of activities that make my wrists and hands hurt. But, when I was younger, I spent quite a lot of time crocheting, and I’m still quite competent and knowledgeable about all of the different crochet techniques.

Crochet is a great medium for creating math art and there are some amazing mathematical crocheters out there. Even a beginner can easily create a Mobius strip. Actually, that’s probably the number one mistake that beginners make when attempting to crochet in the round for the first time. Braids, Klein bottles, Seifert surfaces, hyperbolic surfaces, and two dimensional patterns of all kinds all come very naturally to crochet.


Crocheted Seifert surface of a trefoil knot a la Matthew Wright.

I was surprised to see a crochet technique I was completely unfamiliar with at the 2012 Bridges Math Art ExhibitionKyle Calderhead exhibited an impressively large afghan created using the “interlocking mesh technique“. This two-color technique is fundamentally different from every other color change technique.

In an interlocking mesh piece two differently colored meshes are crocheted at the same time. As you crochet, the meshes go in front of and behind each other, but at no point do they actually merge or intersect. That is, you never stitch one color into the other color, you merely choose which mesh is “in front” and which is “in back” at any given time. This technique is usually done with square meshes, but Kyle Calderhead has expanded upon this too, creating this very nice Afghan using a hexagonal grid and a space filling curve.

I’ve wanted to make something using this technique myself ever since seeing Kyle’s afghan in 2012, but I only recently actually sat down and tried it, creating this level four Hilbert curve afghan.


Hilbert curves and other space filling curves (like the one that Kyle created on the hexagonal grid), are obvious choices for this technique as they have lots of color swaps or “interlocks”, where the back color shows on the front. These are necessary to keep the two meshes attached to each other. A pattern with absolutely no color changes would literally end up as two separate crocheted meshes. Regular swaps, especially at the edges are important to keep an interlock crocheted piece connected.

Regular color-changed crochet, like the Sierpinski Triangle below, has the property that the front of the piece looks more or less the same as the back. The colors that aren’t being shown are “carried through” the middle of the stitches to give this clean look on both sides.

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A crocheted Sierpinski triangle created with thread crochet and a standard color change technique

In constrast, interlock crochet has one mesh that is decidedly the front mesh and one that is the back mesh, with color changes being created by swapping which mesh is in front. This means that the back of an interlocking crochet piece can be the same as the front, but the vast majority of patterns (and any randomly chosen pattern) will be substantially different on both sides.

What does this look like? Well, for my Hilbert curve afghan, I’d say that it makes the back fascinating, but barely recognizable. The vertical pink lines on the front of my piece turn into horizontal purple lines on the back and vice versa.



You’ll notice that the back of the piece has solid lines of pink running vertically and horizontally, while the front has full mesh squares of purple. That’s because my Hilbert curve isn’t as iterated as it could be for this mesh. With the same number of mesh squares I could have created a fifth order Hilbert curve, but I ended up only creating a 4th order curve. With a fifth order curve all of the mesh holes would be filled with color, and there would be no solid pink lines on the back. As it is, my space filling curve isn’t quite iterated enough to truly fill the mesh space provided for it.

This is primarily an artifact of this being my first try at this technique. I was already a third of the way done with this square before realizing that I should have made a fifth order curve. I’m especially disappointed because the back of the fifth order curve would really be much more interesting.

So, now my project is to make a second, third, and fifth order curve on the same size mesh and then combine all of them into a larger afghan that shows one curve that iterates further in each quadrant. But, thanks to my hurting hands and limited patience, I rarely complete more than one small crochet project a year, so you can maybe look forward to that in 4 years or so…