The Lessons of a Square-Wheeled Trike

Nobody leaves the Metropolitan Museum of Art muttering “Okay, but where’s the art?” But something like that happens with many adult visitors to the Museum of Mathematics in New York City. They come in and see the enthusiastic middle-school kids having fun playing with the exhibits, and can’t help wondering: “Okay, but where’s the math?” The kids don’t have this problem; they just come in and play, and they don’t care so much about how grownups classify the kind of fun they’re having. But grownups are the ones who pay admission, so they have a right to ask the question, and I want to try to answer it.

But first, some disclosure is in order: I’m on the Advisory Council for the museum. This means that every year I travel to New York City and spend a day at their headquarters giving advice. In exchange, MoMath provide three meals, including a very nice dinner, and gives me a year-round discount on admission. I’m also currently helping them develop an exhibit that I hope will get built before the end of the decade; I’ll tell you about it sometime. And in early August I’ll be attending a conference at the museum called MOVES 2015, devoted to the Mathematics Of Various Entertaining Subjects. If you’re a fan of math and you’ll be in the New York area in early August, consider attending.

Back to those confused grownups. Part of the trouble stems from the name of the museum, which makes them expect a place that comprehensively treats all subjects of mathematics, both pure and applied, from antiquity to the present. One might picture — and maybe some of these visitors do — a cathedral of mathematics, where the pillars of the subject (such as the Fundamental Theorems of Arithmetic, Algebra, and Calculus) are represented by actual physical pillars, reverently decorated in chiseled marble with examples, proofs, and applications.

If MoMath called itself The Experiential Museum of Mathematics, people would understand before they even arrive that it’s not trying to be that kind of museum. Like many contemporary science museums, MoMath has chosen to focus on offering the sorts of immersive, interactive experiences that physical spaces are uniquely equipped to provide. And the iconic experience the museum offers visitors is the chance to ride on a tricycle with square wheels. Let’s examine it in detail, and see where the “missing” math is.

If you managed to steal a square-wheeled tricycle from the museum and tried to ride it home, you’d be disappointed. The vehicle would be incredibly difficult to pedal, and even if you found a way to pedal it, the ride would be jarringly bumpy. That’s because the height of the axles (and hence your seat, and hence you) would keep changing: the axle of a wheel is low when one of the sides of the square is in full contact with the road, as in the left panel of Figure 1, and high when one corner of the square points straight down and the opposite corner points straight up, as in the right panel of the figure. You’d also find the square-wheeled tricycle inclined to tip to one side, because one of the rear wheels is larger than the other (more on that later). If the police gave pursuit, there’d be no contest; even if you were a Tour de France champion, the most out-of-shape desk sergeant could chase you down on foot without breaking a sweat.

There’s only one place in New York City where you can comfortably ride the square-wheeled tricycle, and that’s at MoMath itself — on a circular track equipped with humps specifically designed to smooth out the ride.

A person might reasonably ask: Why should the museum show such an oddity as a square-wheeled tricycle, when round-wheeled vehicles are so much more common and useful? Why not give visitors a chance to ride a variety of round-wheeled vehicles under a variety of conditions, to learn about gear ratios and mechanical advantage and translational and rotational inertia? The physical and engineering principles embodied in a wheeled vehicle are inherently mathematical, so why not bring these useful principles to light in an experiential way?

Glen Whitney and the other creators of the museum turn this question around. There are lots of round wheels attached to vehicles careening through the streets of the city (vehicles all too ready to interrupt mathematical reveries with forceful reminders of the existence of the material world); why put more round wheels inside the museum? Although a well-designed round wheel is in its own way a marvel, it’s a marvel that we have become dulled to, through over-familiarity. Why not re-awaken visitors’ sense of wonder by showing them that two things that they might have thought were inextricably connected — the roundness of the wheel and the smoothness of the ride — can be separated one from the other?

This is the same tactic that museums of natural history use, when they show us dinosaurs and whales instead of animals that we are actually likely to encounter in our daily lives. “Look at what a living thing can be!”, such a science museum tells us. And in the same way, MoMath tells us, “Look at what a wheel can be!” You can disparage square-wheeled tricycles as “curiosities”, but they do encourage curiosity about the world, and that curiosity is a major motivator for scientists and mathematicians and for the kinds of kids who grow up to be them.

The idea for the square-wheeled tricycle goes back to mathematician Gerson Robison. As he explained in his 1960 article “Rockers and Rollers“: “Some years ago, while picking up my small son’s toy blocks, I became intrigued with the possibility of finding a cylindrical surface upon which a plank would roll in neutral equilibrium.” His formulation of the problem led to a surprising answer: the right sort of surface to use is one whose cross-section is a mathematical curve called a catenary, turned upside down. If you’re prone to mistake a catenary for a parabola, you’re in good company; as I learned from Anurag Agarwal and James Marengo’s article, even Galileo made this mistake!

Following up on Robison’s work, people realized that if you take a suitable arc of the catenary and repeat it over and over, you get a track-bed on which a regular polygon can roll smoothly, so that the center of the polygon stays at constant height. This discovery led the San Francisco Exploratorium to build a small model showing how a square wheel can roll over an inverted catenary road.

(The museum also created a try-it-yourself-at-home guide to making a tiny square-wheeled vehicle roll over a bumpy road composed of half-circles. The semicircular bumps make the ride smoother than it would be on a flat road.)

Around 1990, mathematician Stan Wagon saw the Exploratorium exhibit and fantasized about bringing it up to human scale. Wagon is known in the mathematical community for (among other things) a beautiful book on the mind-bending Banach-Tarski Paradox and a problem-of-the-week column he’s been running for two decades. Wagon created a full-size square-wheeled vehicle and a track for it to run on, still on display at Macalester College.

Later, when George Hart, Cindy Lawrence, Tim Nissen, and Glen Whitney (the original team that came up with most of what you’ll see at MoMath) were brainstorming what sort of math museum they wanted to create, the square-wheeled tricycle idea came up. The big problem with Wagon’s linear track design was the inconvenience of picking up the vehicle when it reached the end of the track and turning it around. Could this unfortunate feature be eliminated by devising a square-wheeled tricycle that would run on a bumpy circular track? There was some skepticism, but Hart insisted that the idea was mathematically sound: “It’s just an engineering problem.” It took a lot of good ideas and hard work by many people, but in the end Hart’s confidence proved to be correct: you can have a square-wheeled tricycle ride around in circles on a suitable circular version of the inverted catenary track. The inside wheels travel a shorter distance than the outside wheels, and the catenary curves near the center of the round track are smaller than the curves near the periphery, so the inside rear wheel needs to be smaller than the outside rear wheel. That’s why the tricycle would tip to one side if you tried to park it on a regular street.

If you have a copy of Mathematica (or the Wolfram CDF Player), you can play with wheels and roads on your own, using the Mathematica Demonstration “Roads and Wheels” and the even more versatile “Shaping a Road and Finding the Corresponding Wheel“. You can also view “Roulette: A Comfortable Ride on an N-Gon Bicycle” to see why triangular wheels are troublesome.

If you’re more into model-building, you can play with a variant of the Exploratorium design using quarter-circles rather than half-circles as humps. You’ll need a bunch of cylindrical cardboard tubes (toilet paper tubes will do nicely) to use as modules. In each module, make four evenly-spaced parallel slits part of the way along the cylinder from one circular end to the other. You can fit these modules together to form a stable structure whose upper surface is made of quarter-circle arcs. The stiffer the cardboard, the better the approximation will be. As for the vehicle that rides on the roadbed, you’ll want the side-length of each square wheel to be equal to one fourth of the perimeter of the circles at the two ends of each cylinder.

Interestingly, there’s evidence that the ancient Egyptians used humps shaped like quarter-circular arcs or catenaries or something in between as a practical solution to the problem of how to roll squares, and that they applied this to the task of transporting large blocks of stone great distances to build their monumental pyramids. The question of who built the pyramids, and how, has long occupied historians and has inspired more than a few crackpot theories. One possible clue to the “how” question is provided by the wooden “quarter-circles” that have been found near the pyramids. Engineer Gerard Fonte believes that these quarter-circles were used to create sections of track along which large blocks of stone could be rolled. Fonte’s video shows how this could have been done with approximate catenaries. Construct a bunch of humps bounded by arcs of length s (the side-length of the stone block we are trying to roll) and arrange them in a row with no gaps in between. If the humps have angles of 45 degrees on each end, then the angle between two adjoining humps will be 90 degrees, which is perfect for accommodating the right-angle corners of the block. Then one can roll the block end over end without sliding. If the arcs are mathematically exact circular arcs, then the center of mass of the block doesn’t stay at constant height; its height fluctuates by about 2% of s as the block rolls over the humps. By making the humps more like inverted catenaries (as Fonte did, using down-to-earth mathematics that the Egyptians could well have carried out), one can bring that 2% even lower, as seen in the video. The Egyptians may even have known about the catenary curve, in an imprecise but practical sense, since they constructed many arches, and seem to have been aware that circular arches were suboptimal for purposes of distributing load. The exact shape of the optimal curve, the catenary, could not have been described using Egyptian mathematics — the requisite theory of differential equations wouldn’t be invented for millennia — but the Egyptians might have derived good approximations empirically. And they could have applied the same empirical approach to designing a good road-bed for rolling blocks. If you were an ancient Egyptian engineer, or an extraterrestrial bent on building pyramids but determined to hide your extraterrestrial origins by using low-tech methods, this would have been a good way to do it.

As far as I know, nobody has built a table-top version of the MoMath square-wheeled tricycle exhibit, with a small square-wheeled tricycle or two running on a circular track made of inverted catenaries, but I’d be delighted if some reader of this blog were to devise one. I also don’t know of a really accessible explanation of why catenaries — the curves that turn up in the natural form of hanging cables and in the optimal design of arches — should turn up in the context of wheels and roads. If anyone has a good way of explaining this, please let me know!

One of the physical principles that wheels and roads obey (when skidding isn’t involved) is the no-slip condition: if you imagine a small nub of rubber at the part of the wheel where the rubber meets the road, the forward velocity of that nub of rubber, at the instant that it meets the road, is the same as the forward velocity of the small nub of roadway that it’s meeting, which is of course zero, since the road isn’t moving. This condition applies to ordinary round-wheeled vehicles as well: even when a car is speeding along at sixty miles per hour, so that the forward velocity of the axle is also sixty miles per hour, the instantaneous forward velocity of the bottom of the wheel is zero.

A curious consequence of this line of thought is that when your car is going sixty miles an hour, the top of each wheel (by which I mean, a little nub of rubber at the top) is moving forward at a hundred and twenty miles an hour. To see why, look at two nubs of rubber at opposite sides of the wheel, as in Figure 2. At every instant, the center of the wheel (one end of the axle) is halfway between nub #1 and nub #2.

So, at every instant, the forward velocity of the center of the wheel is the average of the forward velocity of nub #1 and the forward velocity of nub #2.

(If this step seems dodgy, you might find it helpful to imagine a jacked-up car, with someone inside pressing on the accelerator so that the speedometer reads 60 mph: the forward velocity of the center of the wheel is zero, which is the average of the forward velocity of nub #1 and the forward velocity of nub #2 (these velocities being equal and opposite at every moment). Now imagine a moving car. The forward motion adds 60 mph to the forward velocity of every part of the wheel, but this doesn’t affect the numerical relationship: if x is the average of y and z, then x+60 is the average of y+60 and z+60.)

What does this numerical relationship tell us, at the precise moment when nub #1 is at the bottom of the wheel and nub #2 is at the top? Since we already know two of the three velocities, this numerical relationship tells us what the third must be. Question: 60 is the average of 0 and what other number? Answer: 120. So the forward velocity of the top of the wheel is one hundred and twenty miles per hour. Points that are not at the top of the wheel, but near the top, have a slightly slower forward velocity. The MIT mathematician Dan Kleitman once tried to beat a speeding ticket by claiming that the cop’s radar gun had detected the speed of a point high up on the wheel, rather than a point on the frame of the car. The judge didn’t buy it, but it makes a good story.

Another surprising consequence of the no-slip condition involves flange-wheeled trains, in which a part of the wheel is actually lower than the top of the rail on which the wheel rides (see the Wikipedia page on train wheels if you find this hard to picture). As Martin Gardner points out in “The Perplexing Wheel” and also in his article “Wheels”, the part of the wheel below the rail is actually going backwards. To see why, use the fact that the forward velocity of a point on the wheel is a linear function of the height of the point relative to the center of the wheel. A point high above the center has positive forward velocity, the center has half as much forward velocity, and a point at the same level as the tracks has zero forward velocity; so a point on the wheel below the level of the tracks must have negative forward velocity, which is to say that it’s moving backward.

So, even though the precise catenary shape of the roadway that the square-wheeled tricycle rides on involves physical and mathematical ideas that would be beyond the typical visitor’s ken, there’s some related math with a similar flavor that can be appreciated by middle-schoolers. That’s one kind of answer to the “Where’s the math?” question.

If you enjoy no-slip motion of one object against another, you’ll probably love spirographs. But spirography is a whole topic unto itself, which I’ll talk about some other time.

Here are two fun (old) puzzles of the no-slip kind you might want to consider (I’ll post solutions in a couple of weeks).

1. Roll one dime around the periphery of another, without slipping. How many full turns does the rolling dime make as it goes around the stationary dime once? That is, if Franklin Delano Roosevelt starts with his forehead up and his chin down, how many times does he return to this orientation?

2. Cut a round hole into a piece of plastic so that the diameter of the hole is twice the diameter of a dime, and roll a dime around the inside of the hole without slipping.  What path is traveled by a point on the periphery of the dime?

Both puzzles have surprising answers, and this element of surprise is precisely what MoMath seeks to encourage. Too much math teaching stresses the predictability of math, rather than the element of surprise that many mathematicians treasure. With its square-wheeled tricycle, and with its many other offbeat exhibits, MoMath hopes to bypass math anxiety by avoiding the sorts of math some kids may have come to fear. You can learn more from an interview with Glen Whitney (the square-wheeled tricycle appears in the second half of the video), or a transcript of the interview.

A small number of adult visitors to MoMath express annoyance at the place (check out Yelp to see what I mean). One reason they feel frustrated is that the text explanations given in the video panels, and the oral explanations given by the docents, don’t always succeed in linking what these visitors see at the museum to what they already know. But I think that it goes deeper than that, and that some of these people object to the kind of museum MoMath is trying to be, with its conscious emphasis on non-core topics. People who’ve worked hard at math, and who through years of struggle have come to master the material they learned at school, might be irritated to see how little of their knowledge is applicable to what they see at MoMath. Their reaction might be paraphrased as “But … I’m good at math! And I don’t understand this at all !”

But what is “math”? One of the lessons of MoMath is that at bottom, math isn’t so much a collection of objects of thought (numbers, squares, catenaries) as it is a style of thought. And doing research in math is, for most research mathematicians, a process of playing around with ideas and figuring out new ways for them to fit together. It’s not different in spirit from what kids do naturally in a setting like MoMath, where there are so many things to play with. “Math is play!” is part of the message of MoMath, but not everybody gets it. I’m hoping that, in a few years’ time, after MoMath has expanded its explanatory materials, the only people leaving the museum disgruntled will be adults wishing they could have gone to MoMath when they were kids:  “How come none of my teachers showed me that math could be like this?!”

Part of the spirit of play is the spirit of mischief, and this is one source of the appeal of the square-wheeled tricycle. Mathematics may seem like an edifice, but it’s one that has grown over time, and sometimes the work of one era subverts or recasts the work of an earlier era. A case in point is the way the notion of number has changed over the centuries, starting from the natural numbers and over time enlarging to encompass fractions, negative numbers, irrational numbers, complex numbers, and more. These tectonic changes in the mathematical landscape were the consequences of practical or theoretical motivations, not a puerile desire to break the rules of the previous era. But many of us savor the rule-breaking side of mathematics.

Consider: Just when you’ve come to fully understand why you can’t subtract a big number from a small one — why it makes no sense to try to subtract a big number from a small one — you learn that by making a different kind of sense out of numbers, and by bringing in new numbers like “negative one”, you can subtract a big number from a small one after all. Some learners find this disorienting (and I’ll discuss this mathematical learning-style in another essay), but many find such reversals liberating; I know I did. Likewise, you might have convinced yourself that a square wheel is destined to give you a bumpy ride, but then along comes MoMath, showing you that you can have a square wheel and a smooth ride, as long as the surface you’re riding on is designed appropriately. For me, one of the chief lessons of the square-wheeled tricycle is a slogan that’s applicable throughout mathematics: If you don’t like the ride, change the road!

I’m hoping that over time MoMath can develop better ways to explain the links between its exhibits and the big wide world of math, and to show how the kind of imagination that imbues the exhibits — by turns rigorous and whimsical — played a role in the internal development of mathematics over the past several centuries. I’m also hoping MoMath can show how internal developments in mathematics have helped humanity understand the natural and human-made world, to the point of shaping what kind of human-made world humans are currently choosing to make.

I’ll wrap up this essay by comparing math with sex. This is not as bizarre a transition as you might think. A friend of mine once complained to me that her mother was very clear and complete about how babies are made, but neglected to mention the key fact that sex is pleasurable. The same could be said about too much math education: it teaches the mechanics, but forgets to teach that the activity is fun. Some middle-school kids imagine that as they get older, math will just get harder and harder, and less and less doable, until finally they hit some sort of personal limit on performance, where their brain can’t process any more math, and that’s where they’ll stop. Even kids who like math may wonder what lies in store for them as they continue along the road of mathematics. One purpose of MoMath is to reach out to kids, both “math-identified” and not, and to tell them: It gets better. And then it gets even better. And it just keeps on getting better.

Next month (August 17): MOVES 2015, combinatorial games, and surreal numbers.

This article was written with help from Matt Baker, Henry Cohn, Gerard Fonte, George Hart, Chris Hillman, David Jacobi, Cindy Lawrence, Henri Picciotto, Donna Propp, Steven Strogatz, James Tanton, Stan Wagon, and Glen Whitney. To learn more about the square-wheeled tricycle, check out some of these resources:

(1) Anurag Argarwal and James Marengo, “The Locus of the Focus of a Rolling Parabola“, College Mathematics Journal, Vol. 41, No. 2, March 2010, pp. 129–133.
(2) Gerard Fonte, “Building the Great Pyramid in a Year: An Engineer’s Report”, Perfect Paperback, 2007.
(3) Martin Gardner, “The Perplexing Wheel”, page 141 of “Aha! A Two Volume Collection: Aha! Gotcha, Aha! Insight”, published by the Mathematical Association of America.
(4) Martin Gardner, “The Great Moon Mystery”, page 49 of “Aha! A Two Volume Collection: Aha! Gotcha, Aha! Insight”, published by the Mathematical Association of America.
(5) Martin Gardner, “Wheels”, from: “Wheels, Life, and Other Mathematical Amusements”.
(6) Ivars Peterson, “Riding on Square Wheels“, Math Trek, Science News 2004.
(7) Gerson Robison, “Rockers and Rollers“, Mathematics Magazine, Vol. 33, No. 3, pp. 139–144, 1960.
(8) Stan Wagon, “The Ultimate Flat Tire“, Math Horizons, February 1999, 14-17.Next