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A Tale of the Cycloid in Four Acts

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A Tale of the Cycloid in Four Acts A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. The History of the Cycloid: A Who’s Who of Seventeenth Century European Mathematics This intellectually rewarding, and mathematically beautiful curve led to such heated disputes, that the cycloid has been called the “Helen of geometers,” or the “Apple of discord” 10, both references to the epic poems the Iliad and the Odyssey, in which Helen’s beauty “launched a thousand ships” and led to years of war. Act 1: Early history and Galileo’s influence. There are differing opinions as to who first discovered or invented the cycloid, and some authors believe that the curve must have been known to ancient Greek geometers seeing Archimedes, among others, describes mechanically generated curves such as his spiral. Also Hipparchus used an epicycle, the curve traced by a point on a circle that rolls around the circumference of another circle, to describe the motion of the Moon. There is, however, no surviving evidence that the Greeks explicitly knew of the cycloid. Over 1700 years later, Albrecht Dürer, the German artist, also studied an epicycle5, but did not generalise this to a regular cycloid. Dürer, whose interests were mainly geometric, did not have the mathematical background required to investigate the epicycle deeply. Figure 2: (Left) Dürer’s epicycle drawing compass2. (Right) An epicycloid. Charles de Bovelles described an actual cycloid in 150110, but he considered it to be merely a circle with five-fourths the radius of the generating circle13. Galileo Galilei gave the cycloid its name, from the Greek for circle-like, and was the first to draw widespread attention to it. He wrote to Cavalieri in 1640 that he had been thinking about the curve for over fifty years. In 1599, to estimate the area of the cycloid, Galileo cut the shape of the curve out of sheet metal and compared the weight of the cycloid to the weight of the generating circle. He came to the conclusion that the cycloid area was a little less than three times the area of the generating circle13. A teaching activity in Appendices II and IV allows students to draw a cycloid using a compass and ruler, and then reproduces Galileo’s experiment with cardboard shapes. Figure 3: (Left) A mobile phone light attached to a bicycle wheel. (Right) Cycloid traced using a time-exposure photograph3. A cycloid can also be generated by taking a time exposure photograph of a light attached to the rim of a wheel. An activity for lower grades using this technique is described in Appendix III. Galileo also suggested that the cycloid would be a suitable shape for the arch of a bridge10. Galileo’s disciple, Vincenzo Viviani, used a cycloid in his design for the arch of a bridge over the River Arzana in Sardinia11. This bridge is no longer standing, but below is shown some modern day architecture using cycloids. Figure 4: (Top) Cycloidal arches at the Kimbell Art Museum, Forth Worth, Texas8. (Bottom) Stone bridge in Skopje. Two other disciples of Galileo, Bonaventura Cavalieri and Evangelista Torricelli, also studied the cycloid. Torricelli found the area under the curve and published his derivation in his 1644 book on geometry. He wrote, “One now asks what proportion the cycloidal space has to its generating circle. We demonstrate (and may thanks be given to God) that it is triple4”. His thanks perhaps were tempered somewhat when this publication led to a savage literary attack from Gille de Roberval, as we will soon see. Act 2: The French connection. Marin Mersenne, was born a peasant but trained at a Jesuit college. He likely heard of the cycloid from Galileo, with whom he corresponded. Mersenne hosted a small gathering of mathematicians twice weekly at his home, and at one such meeting in 1628, he put to Roberval the problem of finding the area under the arch of a cycloid13. Roberval accomplished this in 1634 by finding the companion curve shown in Figure 5. The companion curve AQD divides rectangle BDCA in two equal parts. Using Cavalieri’s principle, the shaded area APDQ has the same area as the half circle AEBF because horizontal cross-sections of both shapes are always of the same length. Cavalieri’s principle can be visualised in three dimensions by picturing a vertical stack of coins. The volume of the stack will be unchanged if it is skewed to one side. So looking again at Roberval’s diagram, AC is half the circle circumference because it is traversed by one half rotation. Using, as is customary, a for the radius of generating circle, the area enclosed by half of the cycloid arch is half the area of rectangle BDAC, πa2, plus half the area of the generating circle, 0.5 πa2; a total of 1.5 πa2. Therefore the whole arch has and area of 3πa2, three times the area of the generating circle. Figure 5: (Left) Roberval’s companion curve4. (Right) Visualisation of Cavalieri’s principle3. Roberval, being secretive, did not publish his result, but communicated it privately to some French colleagues. Descartes on receiving Roberval’s proof said somewhat dismissively, “Roberval has laboured overmuch to produce so small a result.” 10 As mentioned, Torricelli in 1644 published his also correct determination of the area under a cycloid, ten years after Roberval found the result. Even though Roberval had gone to great lengths to keep his method secret, he was convinced that Torricelli had stolen the proof, and Roberval savagely attacked Torricelli in a widely circulated letter. Torricelli set about collecting documents to clear his name, but unfortunately he died of typhoid in 1647 before he could publish them. His dying request was that the documents be published, but frustratingly, possibly because of the infectious disease, his wish was not honoured and many of the documents were lost. Other documents, however, came to light around 1900 that seem bear out Torricelli’s claim7. Descartes, after ridiculing Roberval, went on to find the tangent by a beautifully simple construct. Figure 6: A pentagon rolled along a line13. Descartes intriguing proof starts by showing that when rolling a pentagon on a line, the curve traced by a fixed point at a vertex, is always tangent to the line joining the vertex and the point where the pentagon touches the base line. He reasoned that by letting the number of sides approach infinity, the polygon would approach a circle, and the curve traced by the fixed point on the circle would still be tangent to the line joining this fixed point and the point at which the circle makes contact with the base line. Figure 7: Descartes tangent method13. Desarte challenged Fermat and Roberval to also find the tangent. Fermat succeeded; Roberval failed. Viviani, you may recall, also found the tangent to the cycloid, possibly using the alternative method of combining velocities. Descartes was of the opinion that the length of a cycloid could never be known.12 The cycloid, it seemed, had little more to offer, but Pascal had other ideas. Act 3: Enter fleet-footed Pascal. In 1654 Blaise Pascal had a religious vision and withdrew from secular society to Port Royal, where he made seminal contributions to theological literature, and to the ideals of serving those in need, but he largely abandoned science and mathematics. In the Iliad metaphor, Pascal is Achilles: a sometimes unwilling warrior with such lavish natural talent that when he entered the fray, the course of the battle turned. And so one evening in 1658, Pascal, suffering from a particularly bad toothache, turned his mind to the cycloid to avoid thinking of the pain. He took it as a sign of approval from God that the pain abated, and so spent the next eight days discovering nearly all the remaining geometric properties of the cycloid including its centre of mass, and volume of revolution. He circulated his results in the form of a contest with a cash prize. There were only two entrants, John Wallis and Père Lalouère, but their work was judged insufficiently complete to win the prize.
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