Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would show a cycloid, the curve traced out by a point on a circle as it rolls along a straight line. A light at the wheel-hub traces out a straight line. If the light is at the mid-point of a spoke, the curve it follows is a curtate cycloid. A point outside the rim traces out a prolate cycloid, with a backward loop.
Cycloids were studied by many leading mathematicians over the past 500 years. The name cycloid originates with Galileo, who studied the curve in detail. The story of Galileo dropping objects from the Leaning Tower of Pisa is well-known. Although he could not have known it, a falling object traces out an arc of an inverted cycloid. This is due to the tiny deflection caused by the Earth’s rotation. Moreover, an object thrown straight upward follows the loop of a prolate cycloid, landing slightly to the west of its launch point.
Blaise Pascal, who had abandoned mathematics for theology, found relief from a toothache by contemplating the properties of cycloids. Taking this to be a sign from above, he resumed his mathematical researches.
Pascal proposed some problems on the cycloid and one of the respondents was Christopher Wren, better known as the architect of St Paul’s Cathedral in London.
Wren proved that the length of a cycloid arch is four times the diameter of the circle that generates it. Today, this is an easy problem in integral calculus but in 1658 it was a formidable achievement.
In 1696, Johann Bernoulli posed a problem that he called the brachistochrone – or shortest time – problem: find the path along which gravity brings a mass from one point to another one not directly below it. The five mathematicians who responded included Newton, Leibniz and Johann’s brother Jakob. The desired path is a cycloid.
The story goes that Newton received the problem one evening upon returning from the Royal Mint, where he was master. He stayed up late working on it and by 4am he had obtained a solution, which he mailed that morning. Although his solution was anonymous, Bernoulli perceived its authority and brilliance, giving his reaction in the classic phrase "ex ungue leonem", the lion is recognised from his claw.
Cycloid arches have been used in some modern buildings, a notable example being the Kimbell Art Museum in Fort Worth, Texas, designed by the renowned architect Louis I Kahn.
Like many classical buildings, the museum is based on a consistent mathematical model. The basic plan is composed of cycloid vaults arranged in parallel units. These vaults have gently rising sides, giving the impression of monumentality. This geometric form is capable of supporting its own weight and can withstand heavy pressure.
In the atmosphere the rotation of the Earth generates cycloidal motion: icebergs and floating buoys have been seen to trace multiple loops of a prolate cycloid. Finally, epicycloids and hypocycloids are used in modern gear systems as they provide good contact between meshed gear teeth giving efficient energy transmission.
- Peter Lynch is emeritus professor, school of mathematics and statistics, University College Dublin. He blogs at thatsmaths.com