Category Archives: Life

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…




Mobius Napkin Rings, Spherical Video

Ever wondered what a “hands focused” tutorial video would look like if you could see the whole scene?

As you may remember, I’ve been working on live action virtual reality video with Vi Hart and Emily Eifler. Recently we picked up a few Ricoh Theta cameras, the first real consumer grade spherical camera, and have been having fun making impromptu 360 videos. The videos are pretty low resolution and not stereo, but way easier to create video with than our usual process (which is higher resolution and stereo*). This means that spherical video web logs, or sphlogs**, and impromptu videos are finally a thing.

I think “hands”-style videos and video tutorials are an interesting medium and format in ordinary rectangular video, and wondered what they would be like if you could see the whole scene. If you’re also curious then check out my video on making mobius strips out of paper napkin rings above and find out!

See more spherical videos from me, Vi Hart, and Emily Eifler on the eleVR Youtube channel, and check out to learn more about our VR exploits.

* Full disclaimer, stereo everywhere on a sphere isn’t possible, but it’s stereo in all the places and ways that people tend to look around.

** From “spherical vlog*** ”

*** From “video blog**** ”

**** From “web log***** ”

***** From an actual wooden log-type thing floated on a ship and read off of to record the speed of the vessel and people presumably asking if the “log” had been recorded

Burninating the Fractal Sand Castles

I made it to St. Croix, so it’s officially fractal sand castle time! In this edition of fractal sand castles for fractal coastlines, a sand dragon curve burninates an ordinary fractal sand castle. (It’s a TROGDOR curve!)

The dragon curve is a fun fractal that is easy to create by a variety of simple doodle games or simply by folding a strip of paper in half a bunch of times and unfolding it (my favorite construction). I’m not sure of a name for the sand castle fractal, but it’s fairly straightforward to generate – just keep adding smaller cubes off of the sides.


Finally, apologies for the picture quality. This sand castle was made at night (my camera works terribly in low lighting) and by morning it was already washed away. (or perhaps the dragon successfully burned down the castle and flew off?) Sometimes the transience of ephemeral art can be beautiful, and sometimes it’s just frustration.


An Unexpected Snow-pinski Triangle


My blog post this week was supposed to be about a fractal sand castle in St. Croix. I’m headed there for a friend’s wedding and thought that it would be a good opportunity to add yet another fractal to the series. Unfortunately, my trip from the balmy Bay Area involved a short detour in Boston. Boston, of course, promptly got hit with the brunt of a major snowstorm leaving me stranded without much appropriate snow gear (although I did manage to buy a pair of boots shortly before the blizzard shut everything down).


This snow pile on the side of the street is taller than I am.

With no sand in sight, I decided to try tromping out a fractal in the snow instead. I think most fans of math art are already familiar with the fantastic snow art of Simon Beck (if you aren’t, go click that link now, DO IT). I was most assuredly not equipped to make anything of that caliber, but that didn’t mean that I couldn’t freeze myself trying to make a smaller scale attempt. And, conveniently, just outside my hotel, a glistening flat snow bank waited for me to stomp out my very own Snow-pinski Triangle.


Tromp, sink, tromp. The snow-pinski triangle carved into the snow serves as a nice counterpoint to the sand-pinski triangle I mounded upon the ground a couple years ago. Although they share both a fractal structure and certain ephemerality, they also seem to represent opposite concepts. Summer and winter. Hill and hole. Land and water…..

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But I still wish that I was in St. Croix already.

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?