Math and the Museum

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“I couldn’t help but wonder…” — Carrie Bradshaw

The best birthday party I ever had as a kid was a trip to the Museum of Natural History in New York City with half a dozen like-minded friends and my indulgent parents. The huge dinosaur skeleton in the main hall was impressive, but I was even more enchanted by the exhibits at the Hayden Planetarium. How intriguing it was to see a ball roll round and round the inside of a curvy funnel, evading its fate for what seemed like an eternity before finally falling into the hole in the middle, and how fun to wonder how the rules governing our universe not only allowed but mandated this behavior! How intriguing it was to see how much I would weigh on different planets, and how fun to wonder what that would feel like!

But as much as I enjoyed the planetarium, my childhood love of science was already secondary to my passion for math – the skeleton of the universe, you might call it. I probably would’ve enjoyed a trip to a math museum even more than a trip to a natural history museum. The trouble is, New York City didn’t have one.

That’s not true anymore. New York City now boasts one of the best mathematics museums in the world, the National Museum of Mathematics, informally called MoMath. With 19,000 square feet containing over three dozen exhibits, MoMath became a major attraction to NYC-area schoolchildren and tourists from all over the world when it opened in 2012. Sometime in 2026 it’ll be moving to a new location at 635 Sixth Avenue, where it’ll occupy either 36,000 or 46,000 square feet.

The square footage depends partly on you, as I’ll explain.

I wrote about the Museum of Mathematics in my very first Mathematical Enchantments essay “The lessons of a square-wheeled trike” (available as a WordPress document and as an audio file). This month I’ll focus on an exhibit called Beaver Run, explaining the mathematics behind the exhibit and describing how the exhibit grew out of the math. And I’ll say a bit about how the museum hopes to grow into something even bigger and better in the years ahead, with even more exhibits to enchant young minds.

CANALS AND CRYSTALS

I owe the versatile French priest Jean Sébastien Truchet (1657–1729) a debt that I didn’t discover until I was writing this essay: he was a font geek back before that was even a thing, designing fonts to a level of precision that was technologically infeasible until the modern era. One of Truchet’s creations was the ancestor of Times New Roman, one of my favorite fonts.¹ But Truchet is more famous today for a style of ornamentation that grew out of his interest in hydraulics and more specifically his study of canals. In examining the ceramic tiles decorating France’s canal-network, Truchet was especially taken with square tiles that were divided into two contrastingly-colored isosceles right triangles. Playing with such tiles on his own, he noticed that when multiple tiles were placed in various orientations as part of a tiling of the plane by squares, many pleasing patterns were formed.

The metallurgist Cyril Stanley Smith, best known for his work on fissionable materials as part of the Manhattan Project, reinvented Truchet’s tiles as an outgrowth of his interest in crystalline matter. He later learned of Truchet’s work and modified Truchet’s ideas in various ways. In Smith’s most famous variation on Truchet’s theme, each tile is decorated with two quarter-circular arcs joining the midpoints of two adjacent sides of the square. When the tiles are placed together, the endpoints of the arcs on one tile match up with the endpoints of the arcs on the four adjoining tiles, forming visually arresting collections of loops.

It’s Smith’s tiles that nowadays are called Truchet tiles, at least in the mathematical community. I first learned about the tiles from Bernd Rümmler, who used them to ingenious effect to solve a problem I’d posed about Chris Langton’s virtual ant², but that’s another story. Smith’s Truchet tiles have been popular with graphic artists because of their versatility. They’re so versatile that, in a development that Truchet-the-typographer would have found amusing, mathematician and designer David Reimann came up with a font based on them. Here’s a picture of a wall in the bathroom at MoMath’s original location. Can you find the hidden message MATH IS COOL?³

Of course, as a hydraulics expert from the early years of the Enlightenment, Truchet would also have found modern flush toilets extremely interesting.

ROTATING RAILCARS AND ROVING RODENTS

I first learned about MoMath when I heard mathematician and artist George Hart give a talk about some interesting puzzles he’d invented that he thought might be exhibited at a new museum he was founding along with mathematician-turned-stock-trader-turned-philanthropist Glen Whitney, joined by accountant-turned-math-educator Cindy Lawrence and architect-turned-designer-and-engineer Tim Nissen.

The talk took place at Microsoft’s Cambridge lab, not far from MIT’s building 20 where the hobby of recreational computing began as an outgrowth of the hobby of model train set design. The basement of building 20 was the home of the Tech Model Railroad Club, a haven for students interested not just in trains but in the switching systems that controlled them – switching systems that were influential in the design of the telephone network and the modern computer.

Hart was also working on an exhibit based on the idea of trains that run on Truchet tracks. By using a descendant of the switching systems found in the basement of Building 20, one could reconfigure the track even while the train was zipping along. (The proximity of Microsoft’s Cambridge lab to MIT’s Building 20 is irrelevant to the events of this story, but I find it satisfying.)

Consider a train, represented schematically by an arrow, about to move from one tile to another:

The train will enter the new tile along its left edge, but what happens after that depends on which kind of tile it is. The train might exit the new tile along its upper edge:

Or the train might exit the new tile along its lower edge:

If a kid operating the exhibit switches between the two kinds of tile before the train gets there, the train is diverted, and hopefully the kid who switched the track will be diverted too.

Of course, you can’t really rotate a single square within a square tiling; each tile is locked in place by its neighbors. So we need circular turntables in the middle of each square:

Now imagine twenty-four turntables, as in the image below. There are 2²⁴ or about seventeen million different ways to configure them. Sometimes twiddling a knob causes two different loops to merge; other times it causes one loop to split into two. You could twiddle the knobs all day and never see the same configuration twice.

Also notice that you can’t make the train leave the exhibit; the turntables let you reconfigure the network, but the tracks always form a collection of loops.

In the picture above, each turntable is occupied by two trains. Let’s suppose we use just two of those forty-eight trains, moving at the exact same speed, and let’s further suppose that the mechanism is set up so that a turntable can never rotate while a train is actually on it. Then (as Geoge Hart understood, and you’ll understand too by the end of this essay), no amount of track-switching will lead to a collision.

The idea of having two trains running on a dynamically reconfigurable network of Truchet tiles had almost everything a good MoMath exhibit needed: it was gripping, it had never been done before, and it could expand kids’ ideas of what math could be. But what if some young visitors would be prone to take one look at it and think “Meh, a model-trains thing, not for me”? This might especially be the case for girls. In order for the exhibit to attract interest among a broader sector of the public, it would help if the trains were replaced by some sort of critter. The animal the team chose was, appropriately, the engineer of the animal kingdom, the beaver – also, coincidentally, the MIT mascot.

Now there was a vision of an exhibit called Beaver Run, featuring two beavers traveling through a landscape of ponds and evergreens, moving along reconfigurable tracks, gracefully gliding and somehow never colliding. All that remained was to fill in the engineering details.

UNDER THE SURFACE

Implementing those details proved not to be so simple. Even while Tim Nissen was trying to get Beaver Run running, he and his team were designing and building a few dozen other exhibits, each of which was its own engineering project. The ambition of MoMath’s founders was to open a museum in which, from Day One, every exhibit would be something never seen before in any museum. As a member of the Advisory Council, I respectfully told them I thought this was crazy. I tried to convince them that this would be like an auto-maker deciding to release a car with a new kind of motor, a new kind of transmission, a new kind of suspension, a new kind of safety system, etc., all at the same time. I advised them to do what other science museums had done, mixing a few home-grown innovations with a bunch that were tried and true. I urged them to hedge their bets and license some proven winners from the San Francisco Exploratorium. Fortunately, they didn’t listen to me. The museum was a hit from the day it opened its doors in 2012, and part of its appeal was its outrageous originality.

But there were some casualties of the let’s-invent-everything ethos, and Beaver Run proved to be one of them. The engineering challenges were too great, and the leadership team reluctantly decided that it would be best not to present Beaver Run to the public until the kinks in its design had been ironed out. The ironing took four years. If that seems like a long time to you, keep in mind that MoMath ran at capacity almost from the day it opened, and most of its visitors were middle-school children, who are not exactly gentle with museum exhibits. Exhibits are certain to break, so they have to be designed with the need for periodic repair in mind.

Also keep in mind that the theorem that guarantees that the beavers will never collide requires that the beavers move at the same speed. Even tiny deviations, over the course of a day, could lead to a collision and a “Closed for repairs” banner draped over the exhibit. You need the speeds of the beavers to match up to within one part in a million. How would you engineer the exhibit to achieve that much precision without breaking the bank? You’d probably decide pretty quickly that the cars need sensors that report to a central control system that can detect when the cars are getting out of sync and can make minute speed corrections, too small for visitors to notice but crucial for keeping collisions from happening. But now the design is becoming more complicated, with increased cost of fabrication and decreased mean-time-between-failures.

Four years after MoMath had opened its doors to the public, the team felt that the exhibit was ready, and Beaver Run had its debut in 2016.

DEATH AND REBIRTH

Although the exhibit was popular, it was plagued by mechanical difficulties. In 2019, the leadership team reluctantly removed it from the Museum floor. But Executive Director Cindy Lawrence didn’t put it in the trash. As she writes, “I couldn’t bear to throw it out, so I put it in storage near where I live. It languished there for years, and every time I went to get something out of storage, I would admire it and lament its sad ending.”

Then along came a math-loving New York City dad with math-loving kids, including two daughters who had especially loved that exhibit; he wanted to know what had happened to it and what it would take to fix it. Lawrence explained that it was no simple repair job; it would be necessary to basically start from scratch and re-do the whole exhibit. It would cost a lot of money. The dad said he would help pay for the redesign/rebuild of the exhibit. The Museum hired Richard Rew, who along with his son Oliver redesigned and rebuilt the exhibit and got it to work beautifully. Beaver Run went back into the Museum in 2024, and it’s been a popular exhibit ever since.

MoMath might not have been able to redo the exhibit had two young girls not been smitten with it, and had their father not had both the vision and the resources to bring it back to life. Would the girls have loved it as much if there hadn’t been critters? Would they have identified as strongly with trains? When asked, the dad said he doubts it.

MoMath’s lease at 11 East 26th Street ran out about a year ago; the Museum has been housed in a temporary location at 225 Fifth Avenue since March 2024, pending relocation to larger quarters at 635 Sixth Avenue. The upgrade is giving the Museum the opportunity to expand and design new exhibits — one of which, Draw Your Own Conclusion, has been on leadership’s wish-we-could-build-this list for over a decade. The initial lease provided 36,000 square feet but the landlord subsequently offered to lease the Museum an additional 10,000 square feet that would bring the Museum up to 46,000 square feet in total.

MoMath signed the lease for 36,000 square feet and has been trying to raise funds for the additional 10,000 square feet. The Museum was halfway toward its capital goal when its prospective landlord at the new location made a surprising offer: the Museum could have the extra space rent-free for one year but only if the Museum signed the lease for the added space by the end of May, 2025 (just over a week from now). Here’s a link to the announcement.

Just as the future course of a beaver in the exhibit can be radically altered by a simple twist of a dial, the future of MoMath could be radically altered by what happens this month. If my essay leads to your making a donation to MoMath, tell ’em the beavers inspired you.

SEEING LIKE A MATHEMATICIAN

So, why don’t the two beavers ever collide? Since they move at the same speed, neither of them can rear-end the other, so what we’re really asking is, why won’t there ever be a head-on collision? It’s possible to imagine that, with enough patience or cleverness or pure dumb luck, you could start from a configuration that contains a tile like this

in which the beavers are passing each other, and turn it into a configuration that somewhere or other contains three tiles like this

in which the beavers are about to collide on the tile they’re poised to enter. It’s not that you can really imagine a particular way to get from the Before picture to the After picture, but you can certainly believe that you live in a universe in which something like this might happen.

But now, go back to the picture of the beavers’ world, subtract away all the distracting details of ponds and evergreens — actually, I’ve already done that for you in the preceding images — and color the terrain in your mind with the colors red and blue, so that the outer region is colored red and so that each stretch of track has red on one side and blue on the other. This isn’t something in the real world; it’s something you build in your head.

Now I can show you why the Before and After pictures can’t be stills from the same movie. In the Before picture, both beavers had Blue on their left and Red on their right or vice versa.

But in the After picture, one beaver has Blue on its left and Red on its right while the other has Red on its left and Blue on its right.

So there’s no way to get from the Before picture to the After picture, and the same is true for any collision scenario you can invent. If the beavers start with the same color on their right, that will remain true forever. There’s no way for them to get into a situation where they’re on the same loop traveling toward each other, since in that situation one beaver has red on its right and one has blue on its right.

That’s not quite the full proof. How do we know the red-blue coloring exists? That is, how do we know it’s self-consistent (unlike, say, a coloring of a Möbius strip in which one side is red and one side is blue, which is impossible)? Mathematicians might realize that there’s something to prove here, but most middle-schoolers will believe it intuitively, especially if you give them crayons and let them try some examples themselves. But we also need to understand why, in the course of the movie, our imaginary blue/red coloring of the rectangle doesn’t undergo drastic large-scale changes when a turntable gets rotated by 90 degrees. So we need one more picture, showing how, when we rotate a turntable on a tile, only the coloring of that tile changes; all the colors elsewhere remain the same, as shown in the picture below.

If neither beaver is on the tile whose turntable rotates, the imaginary colors to the right of each beaver won’t suddenly change. The imaginary blue/red coloring gives our mind’s eye a new way to look upon the scene that clarifies why collisions can’t happen.⁴

TRAINING THE MIND’S EYE

Remember the picture with the caption “Remember this picture. I’ll come back to it later”? It showed forty-eight beavers (I called them trains back then), and I wrote “Let’s suppose we use just two of those forty-eight trains.” I want you to pick two trains in that picture, and then I want you to mentally apply blue and red paint to the scene, in the manner I described. What color is to the right of the first beaver? What color is to the right of the second beaver? I’ll wait.

What you found (if you painted the scene properly) is that each beaver has red on its left and blue on its right. That’s because in this particular configuration, all the quarter-circles join up to form one big loop, and all forty-eight trains are circling it clockwise, with red to the left and blue to the right. In fact, the blue/red coloring you made in your head is exactly the one shown at the end of the previous section (the left panel of the the two-panel figure in the last paragraph).

Kids visiting the museum don’t care about the educational value of an exhibit like Beaver Run, but to parents and other adults who wonder “Where’s the math here?”, I’d say this: Seeing like a mathematician is about tackling questions through the creative process of removing irrelevant detail and replacing it by relevant detail, often of a very non-obvious kind even when it’s simple.

The coloring idea isn’t just simple; it’s also powerful. Hart initially proposed a number of variants on the scenario described above. What if there were a button that would simultaneously rotate the two turntables that the two beavers were on, or, in the case that the two beavers were passing each other on some Truchet tile, rotate just that one turntable? Once you know how to subtract irrelevant detail (ponds, trees) and add relevant structure (red regions, blue regions) in your imagination, you can show that in this variation on the MoMath exhibit, no collisions will occur. You can also analyze ways to place more than two beavers in the system and to show that with suitable rules no collisions occur.

When a mathematical idea is both simple and powerful, we call it beautiful.

Is the coloring device a mere “trick”? Maybe you’ll want to call it that. In both math and magic, surprise and wonder go hand in hand. Magicians pulling improbable things out of their headgear is such a cliché that someone seeing me introduce the red-blue coloring is likely to complain that I just pulled the idea “out of a hat”. Other magic tricks rely on miraculous disappearances: “Now you see it; now you don’t.” In both cases, knowing how the trick is done spoils most of the fun. Watching a slow-motion video of a production or a vanishing may make you admire the performer’s sleight of hand, but you also realize that it requires years of training to reach that level of skill.

But in math, seeing a trick like the coloring argument empowers you almost immediately to make the trick your own. For instance, what if we made a Truchet tiling turntable in which Smith’s two kinds of marked squares were replaced by the five kinds of marked hexagons shown below? If there were an exhibit with the Beaver Run beavers replaced by bees, would collisions occur in the honeycomb, or would the laws of math “magically” prevent them? I’ve given you the tools to figure this out for yourself.⁵

Mathematical pleasure runs on the same track as the delight we get from a magician’s vanishing effect, but in the opposite direction. Now you don’t see it; and just like that, now you do.

Thanks to George Hart, Cindy Lawrence, Tim Nissen, Beverly Tomov, and Glen Whitney. Much of the information on which this essay is based came from past and present personnel of MoMath, and many of the illustrations were either provided by MoMath or are based on materials provided by MoMath. Thanks also to Sandi Gubin.

ENDNOTES

#1. I also have a weakness for Comic Sans.

#2. See Further Travels with My Ant by David Gale, Jim Propp, Scott Sutherland, and Serge Troubetzkoy.

#3. Hint: Pivot your head by 45 degrees so that your right ear is close to your right shoulder.

#4. Here I’m omitting some technicalities that don’t get fully explained until one takes an introductory course in topology. To get a sense of the subtleties that I’m not treating, imagine a third kind of tile in the train network: an overcrossing/undercrossing junction in which one beaver goes under the other in the middle of the tile without the two critters colliding. For this kind of network, the blue/red coloring argument falls apart, and in fact the non-colliding property falls apart too.

#5. The same argument goes through using the same sort of blue/red coloring of the scene. At work here is a general principle called the Jordan Curve Theorem which asserts that any simple closed curve in the plane divides the plane into two regions. This may seem obvious, but it becomes tricky to prove when the curve is really twisty like some fractals.

REFERENCES

Kenneth Chang (New York Times), At Museum of Mathematics, Meet 2 Beavers That’ll Never Meet.

MoMath, A Truchet Tale.

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