Vardi, Ilan

Infoscience - École polytechnique fédérale de Lausanne

I am often asked to explain mathematics ; is itjust about numbers and equations? The bestanswer that I’ve found is that mathematics usesnumbers and equations like a language. Howeverwhat distinguishes it from other subjects of thought– philosophy, for example – is that in maths com-plete understanding is sought, mostly by discover-ing the order in things. That is why we cannot havereal maths without formal proofs and why mathe-maticians study very simple forms to make pro-found discoveries. One good example is thetriangle, the simplest geometric shape that hasbeen studied since antiquity. Nevertheless theworld had to wait 2,000 years for Morley’s theorem,one of the few mathematical results that can beexpressed in a diagram.Horology is of interest to a mathematician becauseit enables a complete understanding of how awatch or clock works. His job is to impose asequence, just as a conductor controls an orches-tra or a computer’s real-time clock controls dataprocessing. Comprehension of a watch can becompared to a violin where science can only con-firm the preferences of its maker. Another exampleis the bicycle that has defied mathematical analysisfor more than a century. Bicycle wheels roll withoutsliding which leads to a mathematical model thathas no clear solution2. The watchmaker also needs90| watch around no 020 autumn 2015 - winter 201690LEARNINGLEARNINGLEARNINGLEARNINGMorley’s theorem (1898) states that if you trisect theangles of any triangle and extend the trisecting lines untilthey meet, the small triangle formed in the centre willalways be equilateral. Foliot clock.1 Senior Scientist, EPFL IMT Instant-Lab, Patek Philippe Chair inMicromechanical and Horological DesignIlan Vardi1foliotregulatingweightescape wheelvergepallet2011 Encyclopædia Britannica, Inc.Mathematics, the languageof watchmakingtotal understanding to make his watch work, somathematicians and watchmakers always havesomething to discuss. Since they first appeared at the end of the MiddleAges, mechanical clocks and later watches havealways been worked by the energy of a fallingweight or an unwinding spring, the rates of whichare controlled by a regulator. The original regulatorwas the foliot, a centrally pivoted arm that is swungback and forth by a gear train driven by a spring orweight. The inertia of the foliot slowed down thegear train, and with trial-and-error adjustment theclock might keep time to within a quarter of an houra day. There was little room for improvement inspring-powered timepieces because the brakingeffect of the foliot is directly linked to the torque ofthe spring. When the spring is at full tension, thefoliot swings faster and the watch gains ; con-versely the watch goes slow when the spring runsdown. These time keepers really did nothing morethan indicate the varying output of the mainspring. A craft becomes a science. Until the 17th century,clocks and watches were designed and built by arti-sans who managed, by dint of mechanical adjust-ment, to improve the precision to several minutes aday. The great leap forward in horology was theintroduction of an oscillator with its own recoilenergy. In such an oscillator the period of each oscil-lation is independent of the energy source.Isochronism – equal periods – means that the oscil-91watch around no 020 autumn 2015 - winter 2016 |LEARNINGLEARNINGLEARNINGLEARNINGThe isochronism of an oscillator became easy to describeonce Isaac Newton’s laws of motion were known in 1687.An oscillator like a blade spring obeys Hooke’s Law: theforce of a spring, F, is proportional to the distance it istensed, x. Following Newton’s laws it could be expressedby a differential equationwhere m is the mass and k, the stiffness of the spring, is aconstant; t is the time and d2x/dt2, the acceleration. Usingdifferential calculus, which we also owe to Isaac Newton, weobtain the explicit answer for the position of the blade where A is the amplitude, f, the frequency and ϕ is aphase that describes the initial position of the oscillator.You can also derive a formula for the frequencyThis formula shows that the frequency is only dependenton the mass and elasticity; it is thus independent of theamplitude, A, demonstrating the much sought-afterisochronism.2 History of bicycle steer and dynamics equations,bicycle.tudelft.nl/schwab/Bicycle/BicycleHistoryReviewlator makes its own time. Isochronism is the basicprinciple of time measurement. We owe it to Galileo,who discovered it by timing a swinging chandelier ina church by the rate of his pulse. He concluded thatthe period of the oscillation was independent of itsamplitude, and so defined isochronism.Replacing the foliot by the pendulum increasedprecision to within 15 seconds a day, almost a 100-fold improvement. It could be said that time hadbeen conquered, because clocks had becomemore regular than the sun, which can deviate by asmuch as 30 seconds a day. The accurate clock trig-gered a cultural revolution, since time measure-ment was no longer based on natural phenomena,but on an artificial and technical construct. This revolution was built on the laws of physics toformulate the problem, and of mathematics to solveit. And that is how horology became a science.The horological mathematician. Christiaan Huygens(1629-1695) is perhaps the embodiment of the horo-logical mathematician. The physicist, Galileo, haddiscovered isochronism by observing a pendulum,but Huygens the mathematician went much furtherand deeper. On December 25, 1656, he found outhow to regulate a clock with a pendulum to make itsubstantially more accurate. His calculations imme-diately showed that Galileo was wrong and that pen-dulums were not isochronous. The mathematical Huygens was not satisfied bythis and found the full answer by devising an92| watch around no 020 autumn 2015 - winter 2016The return force of a pendulum is due to gravity and doesnot obey Hooke’s Law. A pendulum is thus not a perfectoscillator. The equation for the return force is more likewhere θ is the angle of the pendulum, L is its length, gacceleration due to gravity; t is the time and d2θ/dt2, theangular acceleration. Knowing that sin θ is close to θwhen θ is small, one can abstract the difference and writeThis is the equation of a true frequency oscillatorHowever this approximation is insufficient for a properprecision clock. Increasing the amplitude of a standardpendulum by 2 to 2.5 degrees causes a loss of 3.7 sec-onds a day. Portrait of Christiaan Huygens, painted by CasparNetscher in 1671. Collection of the Historical Museum ofthe Hague. On permanent loan to the Boerhaave Museumin Leiden, the Netherlands.LEARNINGLEARNINGLEARNINGLEARNINGisochronous pendulum. This is his solution in threestages:1 The tautochrone curve. Huygens sought to findthe curve down which a ball will roll in a given time,irrespective of where it starts on the curve, asshown in the above left-hand diagram. He showedthat the curve was a cycloid. It followed that theoscillations of ball on a cycloid are isochronous. 2 Involutes and evolutes. Huygens proposed thenotion of the involute – the curve traced by theend of a taut string unwinding from a circle orother curve (the evolute). 3 The involute of a cycloid is another cycloid. Verifying these results is an easy task with the know -ledge of today, but it should be noted that Huygenshad arrived at them in 1657, meaning 30 years beforeIsaac Newton’s laws of motion and differential calcu-lus in 1687. Huygens’ methods are well explained inan excellent book by Léopold Defossez3.The result was that a pendulum on a cord thatunwinds off a cycloid, as depicted on the aboveright-hand diagram, is isochronous. Huygens’ ideawas implemented by the clockmaker, SalomonCoster, in 1657 with an immediate improvement intimekeeping precision. This story is not well known to mathematicians, butit does illustrate how abstract concepts are thoughtup to solve concrete problems, and that once thetheory is established their origins are often forgotten.Huygens’ solution is typical of the mathematician’swork: a sound and elegant theory, but of not muchpractical use. Indeed, the cycloidal pendulum didnot give very satisfactory results and was aban-doned in favour of a rigid pendulum beating at lowamplitude. But Huygens did not stop at the pendu-lum. In 1675 he introduced the sprung balance,which is theoretically isochronous according toHooke’s Law (Robert Hooke should always bementioned as the co-inventor of the balancespring). The sprung balance continues to regulatethe wristwatches of today.Other eminent mathematicians took an interest inhorology. George Biddle Airy (1801-1892) drew upthe specifications of London’s “Big Ben” clock andWilliam Thomson (1824-1907), better known asLord Kelvin of the temperature scale, built an astro-nomical clock with a unique system of maintainingpower. We can’t all be Huygens, yet advances in mechanicalhorology are still possible. Our laboratory, for instance,last year announced the first mechanical oscillator toregulate a timepiece without an escapement5.Horological theory. Mathematics continues toplay a role in the watchmaking industry. The theoryof time measurement is embodied in physics, thetechnical expression of which is the language ofmathematics. The teaching of horological theory93watch around no 020 autumn 2015 - winter 2016 |A ball rolling along a cycloid. The involute of a cycloid is also a cycloid.3 Léopold Defossez, Les savants du XVIIe siècle et la mesure dutemps, published by the Journal Suisse d’Horlogerie, Lausanne1946.LEARNINGLEARNINGLEARNINGLEARNINGand its application date back to Jules Grossmann(1829-1907), director of the horological school atLe Locle. He realised that teaching people to adjustwatches was difficult because it was a knack thatwas not easily transmitted. He thus taught the the-ory of time measurement to save hours of fiddlingabout with the adjustment. It was the equivalent oftoday’s computer-aided design that enables watch-makers to test concepts without having to make astring of prototypes. With his son, Hermann, he wrote the first Swissbook describing the theoretical basis of timekeep-ing. His successor at the Le Locle school, LéopoldDefossez, wrote the definitive book on the subject,Théorie Générale de l’Horlogerie, published in1950 and since used throughout the Swiss watchindustry. Today the textbooks for the technicalschools are The Theory of Horology of the Jouxvalley’s technical school and the Traité de con-struction horlogère by the professors of the HE-ARC technical university. •94| watch around no 020 autumn 2015 - winter 2016Clock designed by Huygens with cycloidal banking. 44 Christian Huygens, Horologium Oscillatorium, translated fromthe Latin by Ian Bruce,www.17centurymaths.com/contents/huygenscontents.html 5 S. Henein, I. Vardi, L. Rubbert, R. Bitterli, N. Ferrier, S. Fifanski,D. Lengacher, IsoSpring: vers la montre sans échappement,actes de la Journée d’Etude de la SSC 2014, 49–58.A cycloid is the curve described by a point on the rim of awheel rolling without slipping on a straight line.The involute of a curve is the line drawn by the end of astring unwinding from the curve. Gear teeth are often pro-filed according to the involute of a circle.cordLEARNINGLEARNINGLEARNINGLEARNING

Mathematics, the Language of Watchmaking

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Mathematics, the Language of Watchmaking

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Infoscience - École polytechnique fédérale de Lausanne