Figure 1: A LEGO Möbius band
I'm not quite sure how it first occurred to me to build a Möbius band out of LEGO bricks, but the thought was irresistible.
There is a large and active adult LEGO community on the internet. (If you're interested, http://www.lugnet.com, the unofficial "LEGO Central" for fans, is a good place to start looking.) Like many others among its members, I had played with LEGO as a child and forgotten about it. Decades later, and with children of my own, it slowly dawned on me that first, the little Danish bricks are still fun to play around with, and second, I could afford to buy a lot more of them than I ever had before.
There are some extremely impressive LEGO constructions on the web. There are people who build elaborate castles, people who make huge train layouts, people who make vehicle reproductions, and a few who specialize in large sculptures. Eric Harshbarger (http:/www.ericharshbarger.com) is the undisputed champion of this subgenre and has made life-size figures, a grandfather clock, and even a desk out of LEGO. Henry Lim (http://www.henrylim.org/LEGOSculptures.html) has made, among other things, a 14-foot stegosaurus and a full-size working harpsichord from LEGO bricks. Clearly, I could not compete with these masters, but for some reason it didn't seem to have occurred to anyone to make abstract sculptures from LEGO, let alone anything mathematical.
I didn't trust my artistic abilities enough to try building the Möbius band by eye. While I wasn't in any doubt that I'd be able to make one, I wanted something that would be aesthetically and mathematically pleasing. I decided to write a computer program to generate the outline of the model. The program would subdivide space into cells the shape of a 1x1 LEGO brick (the height of a brick is exactly 6/5 times the horizontal distance between studs), and produce output telling me which cells should be occupied. Of course a model built entirely of 1x1 LEGO bricks would not hold together without gluing (heresy!) so this would leave me the challenge of actually constructing it out of larger bricks in such a way that the structure would be reasonably robust. In order to make the problem more interesting, I chose a parameterization of the surface that meant that the model would balance on a single short section of the edge.
My first attempt was just over 5 inches high, and I was sufficiently pleased with it to attempt some larger versions, culminating in the model shown in Figure 1, which is about 14 inches high. There were several challenges along the way. While the makers of LEGO produce a great variety of pieces, including plates that are 1/3 of the height of a brick, I wanted to make the Möbius band using only standard rectangular bricks. The most commonly available sizes are 1xn and 2xn bricks, for n=1,2,3,4,6 and 8, with n=3,6 and 8 being significantly rarer than the others. Unfortunately I discovered that in building a curving wall which is essentially one brick thick throughout (another design goal), I kept having to dive into my stock of 1x3 bricks, which dwindled alarmingly. The overhanging "roof" of the structure, which is horizontal at the top, also constrained my options (I admit it - I cheated a bit by deviating from my computer instructions to allow the top to hold together).
If a surface, why not a knot? Shortly after completing the Möbius strips, I decided to attempt the trefoil knot 31, the simplest non-trivial embedding of the unit circle in Euclidean 3-space [1]. I was rather less sure of myself this time. I decided to parameterize the knot as a curve on the surface of a torus, thickening it to a diameter of about three LEGO studs. The result was only moderately successful, but encouraged me to attempt a more ambitious structure - the figure-eight knot 41. This time I chose a parameterization that shows off the rather nice tetrahedral skew-symmetry of this knot. Building this model was quite challenging. My experience with the trefoil indicated that I needed to thicken the curve by slightly more than the three studs I had previously used to have any hope of the structure being self-supporting. In fact, the long curving arches of the structure shown in Figure 2 required repeated backtracking as it was built. It did not become evident until close to the top of the structure which parts lower down were suffering the most stress and were likely to fall apart under the weight. The final result, however, was very satisfying.
And there it would have stopped, were it not for the involvement of a colleague. Knowing that it would provoke me to prove him wrong, he commented that most of the interesting minimal surfaces would probably be too difficult to construct in this way. How could anyone ignore such a challenge?
A minimal surface is one with mean curvature zero [2]. Locally, a small patch of such a surface has minimal area among all surfaces sharing the same boundary. They have been widely popularized as the shapes formed by bounded soap bubbles. For my purposes, of course, the interest of minimal surfaces is that they provide a small library of mathematically interesting and pleasing shapes to build, with a variety of associated construction challenges. The study of minimal surfaces has been intertwined with computation; the ability to use computers to visualize has led to new insights, and this added a small frisson given my intention to use computer programs to aid my LEGO constructions.
But which minimal surface to try? I daydreamed about producing a Costa surface - the beautiful new complete minimal embeddable surface discovered by Costa in 1984 [3]- but reluctantly decided that this would be too ambitious, at least for the time being. I settled for Enneper's surface and the Catalan surface, both of which looked easy enough to attempt.
Although I no longer claim the title, I was once a topologist. Naturally I would have to build a Klein bottle! The major difficulty here was a new one; what parameterization should I use? There were several constraints. I wanted something that would look recognizably like the classic bottle shape, with the tube feeding in through the outer wall of the bottle. I wanted the overall shape to be aesthetically pleasing, and I wanted, despite the constraints of working with LEGO bricks, for a gap genuinely to run around between the inner and outer tubes near the mouth of the bottle. I played around with Mathematica for a couple of days before coming up with something I found acceptable. When I built the bottle, I broke my usual "bricks only" rule and allowed myself a couple of LEGO hinges so that the model could be opened up. It is well known that a Klein bottle can be obtained by gluing together the edges of two Möbius strips, and the model shows this!
Well, you get the idea. One thing led to another and an initial whim turned into a hobby. Many of us have at some point enjoyed painstakingly building polyhedra or other mathematical models from paper or card. Certain areas of mathematics have a distinctly tactile pleasure in addition to their abstractly mathematical aesthetic qualities. In fact, one of the nice things about these LEGO constructions is just how accessible they seem to be to non-mathematicians.
But why LEGO? One would have thought that small rectangular bricks would be the very last thing from which one might build models of smooth surfaces! But, of course, that's part of the point. There is a very obvious contrast between the medium and the content in these models. The fun is in producing something that lets the eye shift back and forth between seeing something angular and bricky, and seeing something smooth. This is one of the reasons why I have never attempted to build a mathematical LEGO sculpture bigger that about 15 inches high; too large, and the resolution becomes too good! The fun is in picking a scale just the right size so that the intended shape can be conveyed without losing the detail of what it's made of.
There are other aspects of LEGO that pique my mathematical interest as well. I mentioned that my programs produce as output a description of which cells in a rectangular grid need to be filled. It takes very little experience to discover that filling in a specified set of cells with a given selection of LEGO bricks is distinctly non-trivial - especially when one is working under the constraint that the entire structure needs to hold together and that the only solid connections are, as with LEGO, those between vertically adjacent bricks. I have very nearly convinced myself that connected LEGO space-filling is in general an NP-hard problem, although in practice I have hardly ever found I had to backtrack by more than two or three layers while building. Of course, I'd be delighted to be shown that the problem is in fact easy. Constructive proofs preferred!
Oh, yes. And I can't end without bragging that I did eventually produce the Costa surface I had hoped for. How do you like them apples, Fred? (Figure 4)
[1] Lickorish, W.B.R. An Introduction to Knot Theory. Springer 1997
[2] Osserman, Robert. A Survey of Minimal Surfaces. Dover, 1986
[3] Costa, A. "Examples of a Complete Minimal Immersion in R3 of Genus One and Three Embedded Ends." Bol. Soc. Bras. Mat. 15, 47-54, 1984.
Andrew Lipson trained as a knot theorist and now works as a computer programmer, specializing in algorithmic optimization. His various LEGO constructions can be seen at http://www.lipsons.pwp.blueyonder.co.uk/lego.htm and he can be contacted by email at andrewlipson@blueyonder.co.uk.
LEGO® is a trademark of The Lego Group, which did not sponsor or authorize this work and with whom the author has no connection.