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 [1]. 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 [2]:
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.