by Jonathan Kujawa
Something we all learn very early on is that things needs to be round to roll. My nieces, Abi and Sydney, are barely a year old and they already know that pyramids and cubes are terrible for rolling along the floor. The ones with more faces do better, but even a twenty sided die bumps along as you roll it across a table. Everybody knows wheels need to be round. The transportation engineers had that one figured out ages ago.
The square wheel is the archetype of an idea so obviously wrong headed that it can be rejected out of hand. After all, thousands of years of engineering have only been a refinement of Ug and Zog's original design. Even if you are open minded about the possibilities, you only need watch the Mythbusters put the idea to the test. The result is so molar shattering that surely no more needs to be said.
Like philosophers, artists, and poets, mathematicians aren't bothered with things like “practicality” and “the real world”. They are handy to have around if you want someone to challenge your assumptions and think outside the box . In the 1950s it occurred to Gerson Robison that there are actually two shapes in play here: the wheel and the road it rolls on. If you allow yourself to adjust the shape of the road's surface, then maybe, just maybe, you can put hills and dips into the surface which exactly complement the shape of your wheel.
In 1952 Robison posed the question in the puzzle section of the American Mathematical Monthly. Writing about it later he said :
Some years ago, while picking up my son's toy blocks, I became intrigued with the possibility of finding a cylindrical surface upon which a plank would roll in neutral equilibrium…. The requirement of neutral equilibrium means, first, that the center of gravity of the roller must travel in a horizontal path and, second, it must remain directly above the point above the point of contact of the two curves in all positions. In addition, the roller must actually roll into each position.
That is, center of the wheel must travel only horizontally as it rolls along so that the passengers have a smooth ride, the center must remain straight above the point of contact with the surface, and the wheel must actually, you know, roll. Robison explains that for a curve which gives a portion of the wheel's shape, you can actually calculate corresponding complementary shape for the road. It is a nice problem which turns out to only need a bit of calculus.
So what if your wheel has a straight line for a side? Like, say, the infamous square wheel? Robison does the math and throws us a surprising plot twist: the complementary shape the road should have is an already famous curve: the catenary. In Robison's paper he illustrates how the complementary road would look if you cut a circular wheel down the middle and extended it by inserting two flat sides:
The hump in the middle is the catenary. If you chain a bunch of these together with flat spots in-between for the rounded ends, this capsule shaped wheel will merrily roll along as smoothly as an ordinary tire on a flat road. Pushing this just a bit further, the math tells us that if we chain a bunch of catenary curves together we could roll a square wheel without a hitch.
In theory, a bike with square wheels will roll perfectly down a catenary road. But, as a wise man once said, “In theory there is no difference between theory and practice. In practice there is.” It is one thing to do a bunch of ivory tower, mumbo-jumbo calculations on a chalkboard. It is quite another to build an honest-to-god bike with square wheels. In the mid-90's Stan Wagon, a mathematician at Macalester College, actually built a working square wheeled bike:
If you find square wheels mundane, Robison provides a handy table of wheel shapes and their complementary road shapes. You can build whatever crazy wheels you desire!
I mentioned above that the catenary's appearance is a bit of surprise. That's because it is already a famous curve in its own right. According to Wikipedia, the catenary was given its English name by Thomas Jefferson based on the Latin cation (“chain”). It first appeared as the answer to two closely related ancient problems.
The first is the optimal shape for an arch. A semicircle, for example, makes for a weak arch. Through trial and error builders knew how to build an arch that was about as strong as you could hope for. But of course you would like to know what is optimal. In 1671 Hooke announced he had solved the problem and encrypted the solution as an anagram. A bit like how I ask my students to figure it for themselves and not immediately turn to Google. In any case, the anagram spelled out “Ut pendet continuum flexile, sic stabit contiguum rigidum inversum,” which translates as “As hangs a flexible cable so, inverted, stand the touching pieces of an arch.” This seems intuitively plausible, at least. If gravity is left to pull down the chain equally in all places, the curve it makes should evenly distribute the force due to gravity when it's flipped over and used as an arch.
This is all well and good, but Hooke is really just kicking the can down the road by giving us a new question: what is the curve made by hanging a heavy rope or chain between two poles? If you image a power line hanging between two poles, it droops a bit in the middle. If you had to guess, you might think it's a parabola. Already Galileo knew that this wasn't quite right, but the equation of the “chain curve” remained a mystery. In 1690 Jakob Bernoulli threw down the challenge to one and all to determine the chain curve. Leibniz, Huygens, and Jakob's brother (and rival) Johann all correctly determined that the catenary is given by hyperbolic cosine.
In the day of Newton, Leibniz, and the Bernoulli brothers, it was a common event for someone to issue an open challenge to the rest of the scientific world. The implication was that the challenger knew the solution and, rather than share it, he'd smugly invite others to try and solve the problem. Seventeenth century science's version of a rap battle. Hooke's anagram seems friendly in comparison!
A famous example of this is the brachistochrone problem. In this case, the question was posed by Johann Bernoulli with this tastefully understated introduction whose last lines are easily understood as an open challenge to Newton :
I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise…there are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]… have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.
The problem is this: imagine you have two points, one higher than the other, and you would like to make a wire path from the higher to the lower on which a bead can slide. Which curve should the wire follow so that the bead slides from top to bottom in the least time (brachistochrone translates to something like “quickest time”)? Once again Galileo was ahead of the curve and knew that a straight line wasn't the quickest, but without calculus he couldn't determine the optimal path. Jakob, Leibniz, Tschirnhaus, and l'Hopital all provided solutions. Newton was provoked by Johann's challenge and came out of “retirement” as the head of the Mint to solve the problem in an overnight marathon session of calculations. The story goes that he sent Johann his solution anonymously but Bernoulli recognized Newton's handwork, declaring “I recognize the lion by his paw.”
Once you've seen square wheels and their relatives, you'll find your mind has been freed and begins to wonder about all sorts of things you took for granted. For example, a natural question to ask is if there are any other shapes besides a circle which can smoothly roll between two flat surfaces. Maybe you're in ancient Egypt and wanting to move a large block from the mines to a pyramid under construction. You can, of course, use logs as circular cylinders in between the two and push the block down the road. But having nothing better to do while pushing big blocks down the road, you may get to wondering if the cylinders have to be circular, or if another shape might work.
What you are looking for are shapes of constant width. The key property of a circle in this situation is that the distance from top to bottom is always the same as it rotates. Besides the circle, does there exist any other shapes of constant width? Indeed there are! One is the Reuleaux triangle :
Notice, while it rotates it keeps in constant contact with the top and the bottom of the square as well as the two sides. Any constant width shape can rotate inside a square while keeping contact with all four sides. It turns out that there are Reuleaux polygons for any odd number of sides. Lest you think this is just a mathematical oddity, let me mention that the rotary engine is based upon such shapes. You can see a video of one in action here. And coins of several countries, including the UK and Canada, are made in the shape of Reuleaux polygons (so that they can be non-circles and still not jam up vending machines).
What about in three dimensions? We know that a sphere is a constant width solid, but are there others? There is a cheap and easy way to make such a shape: if you have a Reuleaux polygon handy you can just take the solid it makes as you spin it on an axis. With a few moments thought you'll realize that the resulting solid still has constant width. You can see some in real life in this video. This feels to me like a bit of cheat. Plus, the solid isn't quite as symmetric as it could be since the axis of rotation has a different sort of symmetry than in the other directions.
You can, however, make more truly symmetric solids of constant width. One famous example is the Meissner tetrahedron. It is an unsolved conjecture that this is the solid of constant width with smallest volume.
Last spring here at 3QD we saw that Man Ray found inspiration in mathematical solids. He did a series of painting with Shakespearean titles based on those which caught his fancy. The one entitled Hamlet is the Meissner tetrahedron:
 Which also makes them exasperating in discussions of politics, law, and the like :-).
 Quote from Robison's article “Rockers and Rollers”. Available here.
 Image from the fantastic MoMath (whose slogan really ought to be “Mo Math, Mo Problems”.
 The quote and further information about the brachistochrone problem can be found here.
 Image from wikipedia.
 Thanks to James Propp's excellent blog for some of the history and background to the square wheel problem.