The bicycle problem that nearly broke mathematics
Venue: Nature

[There is more to this commentary on the publisher's website. Follow the DOI. The article is open access. .Jim Papadopoulos, who is 62, has spent much of his life fascinated by bikes, often to the exclusion of everything else. He competed in amateur races while a teenager and at university, but his obsession ran deeper. He could never ride a bike without pondering the mathematical mysteries that it contained. Chief among them: What unseen forces allow a rider to balance while pedalling? Why must one initially steer right in order to lean and turn left? And how does a bike stabilize itself when propelled without a rider? He studied these questions intensely as a young engineer at Cornell University in Ithaca, New York. But he failed to publish most of his ideas and eventually drifted out of academia. By the late 1990s, he was working for a company that makes the machines that manufacture toilet paper. "In the end, if no one ever finds your work, then it was pointless," he says. But then someone did find his work. In 2003, his old friend and collaborator from Cornell, engineer Andy Ruina, called him up. A scientist from the Netherlands, Arend Schwab, had come to his lab to resurrect the team's research on bicycle stability. Together, the researchers went on to crack a century old debate about what allows a bicycle without a rider to balance itself, publishing in Proceedings of the Royal Society1 and Science2. They have sought to inject a new level of science into the US$50 billion global cycling industry, one that has relied more on intuition and experience than on hard mathematics. Their findings could spur some much needed innovation perhaps helping designers to create a new generation of pedal and electric bikes that are more stable and safer to ride. Insights from bicycles also have the potential to transfer to other fields, such as prosthetics and robotics. "Everybody knows how to ride a bike, but nobody knows how we ride bikes," says Mont Hubbard, an engineer who studies sports mechanics at the University of California, Davis. "The study of bicycles is interesting from a purely intellectual point of view, but it also has practical implications because of their ability to get people around." For a mechanician that fusty breed of engineer whose subject is defined by Newton's three laws of motion the conundrums of the bicycle hold a special allure. "We are all stuck in the nineteenth century, when there wasn't such a difference between math and physics and engineering," says Ruina. Bicycles, he says, are "a math problem that happens to relate to something you can see" The first patents for the velocipede, a two wheeled precursor to the bike, date to 1818. Bikes evolved by trial and error, and by the early twentieth century they looked much as they do today. But very few people had thought about how and why they work. William Rankine, a Scottish engineer who had analysed the steam engine, was the first to remark, in 1869, on the phenomenon of 'countersteering' whereby the rider can steer to the left only by first briefly torquing the handlebars to the right, allowing the bike to fall into a leftward lean. The link between leaning and steering gives rise to the bicycle's most curious feature: the way that it can balance while coasting on its own. Give a riderless bike a shove and it may wend and wobble, but it will usually recover its forward trajectory. In 1899, English mathematician Francis Whipple derived one of the earliest and most enduring mathematical models of a bicycle, which could be used to explore this self stability. Whipple modelled the bicycle as four rigid objects two wheels, a frame with the rider and the front fork with handlebars all connected by two axles and a hinge that are acted upon by gravity. Plugging the measurements of a particular bicycle into the model revealed its path during motion, like a frame by frame animation. An engineer could then use a technique called eigenvalue analysis to investigate the stability of the bicycle as one might do with an aeroplane design. In 1910, relying on such an analysis, the mathematicians Felix Klein and Fritz Noether along with the theoretical physicist Arnold Sommerfeld focused on the contribution of the gyroscopic effect the tendency of a spinning wheel to resist tilting. Push a bicycle over to the left and the rapidly spinning front wheel will turn left, potentially keeping the bicycle upright. In April 1970, chemist and popular science writer David Jones demolished this theory in an article for Physics Today3 in which he described riding a series of theoretically unrideable bikes. One bike that Jones built had a counter rotating wheel on its front end that would effectively cancel out the gyroscopic effect. But he had little problem riding it hands free. This discovery sent him hunting for another force that could be at play. He compared a bike's front wheel to the casters on a shopping trolley, which turn to follow the direction of motion. A bicycle's front wheel can act as a caster because the point at which the wheel contacts the ground typically sits anywhere from 5 centimetres to 10 centimetres behind the steering axis (see 'What keeps a riderless bike upright? This distance is known as the trail. Jones discovered that a bike with too much trail was so stable that it was awkward to ride, whereas one with negative trail was a death trap and would send you tumbling the moment you released the handlebars. When a bicycle starts to topple, he concluded, the caster effect steers the front end back under the falling weight, keeping the bicycle upright. To Jones, the caster trail was the sole explanation for a bike's self stability. In his memoir, published 40 years later, he counted the observation as one of his great accomplishments. "I am now hailed as the father of modern bicycle theory," he declared. .Papadopoulos's first goal was to finally understand what makes one bicycle more stable than another. He sat in his office and scrutinized 30 published attempts at writing the equations of motion for a bicycle. He was appalled by the "bad science" he says. The equations were the first step towards connecting the geometry of a bicycle frame with how it handled, but each new model made little or no reference to earlier work, many were riddled with errors and they were difficult to compare. He needed to start from scratch. [There is more to this commentary on the publisher's website. Follow the DOI. The article is open access. Language: en