{"version":"1.0.0","segments":[{"startTime":1.08,"endTime":4.9,"body":"- A roulette is a curve"},{"startTime":1.08,"endTime":4.9,"body":"obtained by rolling one curve"},{"startTime":4.9,"endTime":6.47,"body":"along another."},{"startTime":6.47,"endTime":9.44,"body":"This simple idea leads"},{"startTime":6.47,"endTime":9.44,"body":"to some lovely curves,"},{"startTime":9.44,"endTime":12.31,"body":"and I'm going to tell you"},{"startTime":9.44,"endTime":12.31,"body":"about some of those today."},{"startTime":12.31,"endTime":15.91,"body":"So we'll see how to construct"},{"startTime":12.31,"endTime":15.91,"body":"this beautiful design."},{"startTime":15.91,"endTime":17.99,"body":"We'll meet the curve so appealing."},{"startTime":17.99,"endTime":21.54,"body":"It was given the nickname,"},{"startTime":17.99,"endTime":21.54,"body":"the Helen of Geometry,"},{"startTime":21.54,"endTime":25.19,"body":"and we'll see examples of a"},{"startTime":21.54,"endTime":25.19,"body":"roulette curve that's so useful,"},{"startTime":25.19,"endTime":28.04,"body":"it has applications in"},{"startTime":25.19,"endTime":28.04,"body":"everything from gears"},{"startTime":28.04,"endTime":30.163,"body":"to nuclear power plants."},{"startTime":31.05,"endTime":36.05,"body":"So let's start then with the"},{"startTime":31.05,"endTime":36.05,"body":"simplest kind of curve rolling"},{"startTime":37.32,"endTime":38.153,"body":"along the curve."},{"startTime":38.153,"endTime":39.68,"body":"So what's the simplest"},{"startTime":38.153,"endTime":39.68,"body":"curve we can imagine?"},{"startTime":39.68,"endTime":42.06,"body":"Perhaps a circle, right? Simple."},{"startTime":42.06,"endTime":43.04,"body":"There's even a simpler one."},{"startTime":43.04,"endTime":44.87,"body":"If you think what actually is a curve,"},{"startTime":44.87,"endTime":48.37,"body":"it's a path made by a point"},{"startTime":44.87,"endTime":48.37,"body":"moving in space somehow."},{"startTime":48.37,"endTime":49.92,"body":"So the very simplest thing,"},{"startTime":49.92,"endTime":52.43,"body":"a point moving in space could do"},{"startTime":52.43,"endTime":53.64,"body":"is to go in a straight line."},{"startTime":53.64,"endTime":56.13,"body":"So the first curve we're going to look at,"},{"startTime":56.13,"endTime":59.38,"body":"first of these roulette"},{"startTime":56.13,"endTime":59.38,"body":"curves is the cycloid."},{"startTime":59.38,"endTime":64.38,"body":"And this is made by a circle"},{"startTime":59.38,"endTime":64.38,"body":"rolling along a straight line."},{"startTime":64.66,"endTime":68.13,"body":"So we think of a wheel"},{"startTime":64.66,"endTime":68.13,"body":"rolling along a road,"},{"startTime":68.13,"endTime":71.61,"body":"and we are following"},{"startTime":68.13,"endTime":71.61,"body":"the path of just a point"},{"startTime":71.61,"endTime":73.35,"body":"on the rim of the wheel"},{"startTime":73.35,"endTime":75.947,"body":"on the circumference of"},{"startTime":73.35,"endTime":75.947,"body":"the circle rolling along."},{"startTime":75.947,"endTime":77.983,"body":"And we can see that as it rolls,"},{"startTime":79.01,"endTime":83.18,"body":"we get these arch like"},{"startTime":79.01,"endTime":83.18,"body":"curves being produced."},{"startTime":83.18,"endTime":84.593,"body":"And that's the cycloid."},{"startTime":86.79,"endTime":90.01,"body":"Now wheels have been"},{"startTime":86.79,"endTime":90.01,"body":"around for a long time"},{"startTime":90.01,"endTime":91.47,"body":"for thousands of years."},{"startTime":91.47,"endTime":96.47,"body":"It's perhaps surprising that"},{"startTime":91.47,"endTime":96.47,"body":"the first record we have,"},{"startTime":96.78,"endTime":98.99,"body":"the first evidence we"},{"startTime":96.78,"endTime":98.99,"body":"have for someone thinking"},{"startTime":98.99,"endTime":102.97,"body":"about this curve is only in 1501"},{"startTime":103.86,"endTime":106.83,"body":"when Gilles de Roberval was working"},{"startTime":106.83,"endTime":108.25,"body":"on various mathematical problems,"},{"startTime":108.25,"endTime":110.56,"body":"including he was trying to find"},{"startTime":108.25,"endTime":110.56,"body":"a way to square the circle,"},{"startTime":110.56,"endTime":112.17,"body":"which he didn't know at the time."},{"startTime":112.17,"endTime":115.32,"body":"We do now know is impossible."},{"startTime":112.17,"endTime":115.32,"body":"There's no way to do that."},{"startTime":115.32,"endTime":117.88,"body":"And so, he was trying various things."},{"startTime":117.88,"endTime":119.82,"body":"And this was this curve,"},{"startTime":119.82,"endTime":121.11,"body":"this idea of rolling a circle"},{"startTime":121.11,"endTime":123.83,"body":"along a straight line came up in that."},{"startTime":123.83,"endTime":125.7,"body":"He didn't manage to square the circle."},{"startTime":125.7,"endTime":128.8,"body":"He also didn't get too far"},{"startTime":125.7,"endTime":128.8,"body":"with understanding this curve,"},{"startTime":128.8,"endTime":132.28,"body":"the cycloid, because he"},{"startTime":128.8,"endTime":132.28,"body":"thought that those arches,"},{"startTime":132.28,"endTime":135.9,"body":"he thought those might"},{"startTime":132.28,"endTime":135.9,"body":"be bits of a circle."},{"startTime":135.9,"endTime":138.7,"body":"And I mean, we can see in"},{"startTime":135.9,"endTime":138.7,"body":"this picture that they're not."},{"startTime":138.7,"endTime":142.3,"body":"Now, we are lucky 'cause we"},{"startTime":138.7,"endTime":142.3,"body":"can just program a computer"},{"startTime":142.3,"endTime":143.32,"body":"to draw these things."},{"startTime":143.32,"endTime":146.53,"body":"But for him, it's a reasonable guess."},{"startTime":146.53,"endTime":149.533,"body":"Later on, the mathematician,"},{"startTime":146.53,"endTime":149.533,"body":"Marin Mersenne,"},{"startTime":150.828,"endTime":154.33,"body":"and he thought that perhaps"},{"startTime":150.828,"endTime":154.33,"body":"the arches of a cycloid"},{"startTime":154.33,"endTime":157.01,"body":"were half ellipses."},{"startTime":157.01,"endTime":159.37,"body":"Now again, this is a pretty good guess,"},{"startTime":159.37,"endTime":161.61,"body":"and it's better than thinking"},{"startTime":159.37,"endTime":161.61,"body":"they're parts of circles."},{"startTime":161.61,"endTime":163.72,"body":"And if you look at the diagram there,"},{"startTime":163.72,"endTime":167.86,"body":"I've own half an ellipse"},{"startTime":163.72,"endTime":167.86,"body":"in red and the cycloid arch"},{"startTime":167.86,"endTime":169.76,"body":"in blue, and you can see,"},{"startTime":167.86,"endTime":169.76,"body":"they really are very close."},{"startTime":169.76,"endTime":172.423,"body":"So I think this is not a"},{"startTime":169.76,"endTime":172.423,"body":"bad guess from Mersenne."},{"startTime":173.72,"endTime":177.36,"body":"So when you've got the curve"},{"startTime":173.72,"endTime":177.36,"body":"and you're interested in it,"},{"startTime":177.36,"endTime":179.42,"body":"what might you ask about that curve?"},{"startTime":179.42,"endTime":180.96,"body":"Well, let's think about circles."},{"startTime":180.96,"endTime":183.39,"body":"We know, we all learn"},{"startTime":180.96,"endTime":183.39,"body":"at school the formula"},{"startTime":183.39,"endTime":184.77,"body":"for the circumference of a circle."},{"startTime":184.77,"endTime":187.77,"body":"If the radius is R, the"},{"startTime":184.77,"endTime":187.77,"body":"circumference is two two pie R,"},{"startTime":187.77,"endTime":191.38,"body":"and we know the area of"},{"startTime":187.77,"endTime":191.38,"body":"a circle, pie R squared."},{"startTime":191.38,"endTime":192.77,"body":"So we know those formulae."},{"startTime":192.77,"endTime":195.04,"body":"And so you might ask about cycloids."},{"startTime":195.04,"endTime":197.58,"body":"So I mean a cycloid goes on forever."},{"startTime":197.58,"endTime":200.49,"body":"So if the whole thing is infinitely long,"},{"startTime":200.49,"endTime":203.33,"body":"but if we just look at one"},{"startTime":200.49,"endTime":203.33,"body":"of the arches, we could ask,"},{"startTime":203.33,"endTime":205.86,"body":"how long is that arch?"},{"startTime":205.86,"endTime":207.67,"body":"How long is the length there?"},{"startTime":207.67,"endTime":211.34,"body":"And perhaps what's the"},{"startTime":207.67,"endTime":211.34,"body":"area underneath the arch?"},{"startTime":211.34,"endTime":215.31,"body":"Now, Galileo was thinking"},{"startTime":211.34,"endTime":215.31,"body":"about these questions"},{"startTime":215.31,"endTime":218.24,"body":"and it's actually Galileo"},{"startTime":215.31,"endTime":218.24,"body":"who gave this curve its name,"},{"startTime":218.24,"endTime":219.93,"body":"the cycloid."},{"startTime":219.93,"endTime":223.53,"body":"He found a very practical way to try"},{"startTime":223.53,"endTime":227.51,"body":"and estimate the area"},{"startTime":223.53,"endTime":227.51,"body":"under a cycloid arch."},{"startTime":227.51,"endTime":232.51,"body":"He actually got a bit of"},{"startTime":227.51,"endTime":232.51,"body":"metal, he made a circle,"},{"startTime":233.35,"endTime":236.09,"body":"then he rolled it along"},{"startTime":233.35,"endTime":236.09,"body":"another bit of the same metal,"},{"startTime":236.09,"endTime":239.04,"body":"the same thickness, and"},{"startTime":236.09,"endTime":239.04,"body":"he marked out a cycloid"},{"startTime":239.04,"endTime":243.22,"body":"and then he cut up them out"},{"startTime":239.04,"endTime":243.22,"body":"and he weighed them, right?"},{"startTime":243.22,"endTime":244.99,"body":"Very practical."},{"startTime":244.99,"endTime":247.76,"body":"This is a applied maths"},{"startTime":244.99,"endTime":247.76,"body":"or something, right?"},{"startTime":247.76,"endTime":248.75,"body":"So he weighed them."},{"startTime":248.75,"endTime":251.07,"body":"And of course, if you've got"},{"startTime":248.75,"endTime":251.07,"body":"two bits of the same material,"},{"startTime":251.07,"endTime":253.49,"body":"the same thickness, then"},{"startTime":251.07,"endTime":253.49,"body":"the weights of those,"},{"startTime":253.49,"endTime":256.94,"body":"what they weigh will be"},{"startTime":253.49,"endTime":256.94,"body":"proportional to their areas."},{"startTime":256.94,"endTime":261.33,"body":"And he found that the"},{"startTime":256.94,"endTime":261.33,"body":"area of that cycloid arch,"},{"startTime":261.33,"endTime":264.69,"body":"the area underneath it"},{"startTime":261.33,"endTime":264.69,"body":"was about a subject,"},{"startTime":264.69,"endTime":266.41,"body":"bit of an accuracy in the"},{"startTime":264.69,"endTime":266.41,"body":"cutting outs and everything."},{"startTime":266.41,"endTime":269.21,"body":"It seemed to be roughly three-ish times"},{"startTime":269.21,"endTime":271.52,"body":"the area of the circle that made it."},{"startTime":271.52,"endTime":274.299,"body":"Now, if you are trying to"},{"startTime":271.52,"endTime":274.299,"body":"guess making a conjecture,"},{"startTime":274.299,"endTime":277.54,"body":"some number around three-ish"},{"startTime":274.299,"endTime":277.54,"body":"that's got something"},{"startTime":277.54,"endTime":282.54,"body":"to do with circles then"},{"startTime":277.54,"endTime":282.54,"body":"something pops into your mind."},{"startTime":283.19,"endTime":285.93,"body":"So he thought maybe it could be pie times,"},{"startTime":285.93,"endTime":288.1,"body":"but he couldn't quite resolve it."},{"startTime":288.1,"endTime":289.92,"body":"So that was a puzzle."},{"startTime":289.92,"endTime":291.297,"body":"And we want to know what this is,"},{"startTime":291.297,"endTime":294.07,"body":"and we're interested in perhaps"},{"startTime":291.297,"endTime":294.07,"body":"the length of this thing."},{"startTime":294.07,"endTime":297.14,"body":"So what's the area"},{"startTime":294.07,"endTime":297.14,"body":"under this cycloid arch?"},{"startTime":297.14,"endTime":299.84,"body":"Galileo tried it by experiment."},{"startTime":299.84,"endTime":301.43,"body":"And he told Mersenne about this."},{"startTime":301.43,"endTime":304.85,"body":"So Mersenne hears about"},{"startTime":301.43,"endTime":304.85,"body":"cycloids from Galileo,"},{"startTime":304.85,"endTime":308.46,"body":"and Mersenne is a very"},{"startTime":304.85,"endTime":308.46,"body":"important figure in Europe"},{"startTime":308.46,"endTime":311.02,"body":"at the time in terms of mathematics,"},{"startTime":311.02,"endTime":315.21,"body":"because he's sometimes called"},{"startTime":311.02,"endTime":315.21,"body":"the post-box of Europe."},{"startTime":315.21,"endTime":318.69,"body":"He corresponded with dozens"},{"startTime":315.21,"endTime":318.69,"body":"and dozens of people."},{"startTime":318.69,"endTime":320.324,"body":"Lots of mathematicians all around Europe"},{"startTime":320.324,"endTime":322.18,"body":"were in correspond with,"},{"startTime":322.18,"endTime":323.754,"body":"or if they happen to be living in France,"},{"startTime":323.754,"endTime":325.83,"body":"actually meeting up with Mersenne"},{"startTime":325.83,"endTime":327.536,"body":"and telling him about"},{"startTime":325.83,"endTime":327.536,"body":"what they were doing."},{"startTime":327.536,"endTime":329.11,"body":"And he would say, \"Oh, did you know?\""},{"startTime":329.11,"endTime":331.157,"body":"So he told Gilles de Roberval,"},{"startTime":331.157,"endTime":334.75,"body":"\"Hey, Galileo has been"},{"startTime":331.157,"endTime":334.75,"body":"working on this curve,"},{"startTime":334.75,"endTime":336.47,"body":"this thing called a cycloid,"},{"startTime":336.47,"endTime":338.29,"body":"and he's been working out"},{"startTime":336.47,"endTime":338.29,"body":"wondering about the area"},{"startTime":338.29,"endTime":339.24,"body":"under the arches."},{"startTime":339.24,"endTime":341.8,"body":"And so, Mersenne would"},{"startTime":339.24,"endTime":341.8,"body":"pass on all this kind"},{"startTime":341.8,"endTime":343.718,"body":"of mathematic gossip, I guess,"},{"startTime":343.718,"endTime":346.78,"body":"to various people all over Europe."},{"startTime":346.78,"endTime":348.41,"body":"And this is a really useful thing."},{"startTime":348.41,"endTime":350.46,"body":"It's precursor, I suppose,"},{"startTime":350.46,"endTime":353.52,"body":"of what we have perhaps"},{"startTime":350.46,"endTime":353.52,"body":"now with academic journals,"},{"startTime":353.52,"endTime":355.02,"body":"where you say, \"Look, I'm doing this.\""},{"startTime":355.02,"endTime":356.27,"body":"And then everyone else has a look,"},{"startTime":356.27,"endTime":359.25,"body":"and then they might try"},{"startTime":356.27,"endTime":359.25,"body":"and extend those ideas."},{"startTime":359.25,"endTime":360.35,"body":"So Mersenne is very useful."},{"startTime":360.35,"endTime":363.76,"body":"He's also, I have to say,"},{"startTime":360.35,"endTime":363.76,"body":"he's a good mathematician"},{"startTime":363.76,"endTime":364.92,"body":"in his own, right?"},{"startTime":364.92,"endTime":367.46,"body":"So you may well know the name, Mersenne"},{"startTime":367.46,"endTime":369.07,"body":"from at least two contexts."},{"startTime":369.07,"endTime":371.8,"body":"So if you happen to see any"},{"startTime":369.07,"endTime":371.8,"body":"of my lectures on mathematics"},{"startTime":371.8,"endTime":375.03,"body":"in music last year, you'll"},{"startTime":371.8,"endTime":375.03,"body":"have heard of Mersenne laws,"},{"startTime":375.03,"endTime":379.86,"body":"which tells you how the"},{"startTime":375.03,"endTime":379.86,"body":"frequency of the sound made"},{"startTime":379.86,"endTime":383.97,"body":"by a vibrating string"},{"startTime":379.86,"endTime":383.97,"body":"varies with its length,"},{"startTime":383.97,"endTime":387.07,"body":"its tension and its linear density."},{"startTime":387.07,"endTime":388.16,"body":"Yes, got that right."},{"startTime":388.16,"endTime":391.15,"body":"So Mersenne laws tell you"},{"startTime":388.16,"endTime":391.15,"body":"something about maths in music"},{"startTime":391.15,"endTime":393.91,"body":"and he investigated that area."},{"startTime":393.91,"endTime":396.62,"body":"We also may have heard of Mersenne primes."},{"startTime":396.62,"endTime":398.45,"body":"So Mersenne studied prime numbers."},{"startTime":398.45,"endTime":401.11,"body":"He was interested actually"},{"startTime":398.45,"endTime":401.11,"body":"in perfect numbers."},{"startTime":401.11,"endTime":403.724,"body":"And he studied primes"},{"startTime":401.11,"endTime":403.724,"body":"that are of the form,"},{"startTime":403.724,"endTime":406.57,"body":"power of two minus one."},{"startTime":406.57,"endTime":409.05,"body":"So seven, for example,"},{"startTime":406.57,"endTime":409.05,"body":"two cubed minus one."},{"startTime":409.05,"endTime":411.12,"body":"So we perhaps have"},{"startTime":409.05,"endTime":411.12,"body":"heard of Mersenne prime,"},{"startTime":411.12,"endTime":413.46,"body":"so he did a lot of his own mathematics,"},{"startTime":413.46,"endTime":414.949,"body":"but he also had this very important role"},{"startTime":414.949,"endTime":418.36,"body":"of passing along"},{"startTime":414.949,"endTime":418.36,"body":"information between people."},{"startTime":418.36,"endTime":422.67,"body":"So Gilles de Roberval,"},{"startTime":418.36,"endTime":422.67,"body":"he actually did find"},{"startTime":422.67,"endTime":426.84,"body":"a formula for the area"},{"startTime":422.67,"endTime":426.84,"body":"under a cycloid arch."},{"startTime":426.84,"endTime":430.57,"body":"And it's a really lovely thing,"},{"startTime":426.84,"endTime":430.57,"body":"perhaps slightly unexpected,"},{"startTime":430.57,"endTime":433.14,"body":"especially since I led"},{"startTime":430.57,"endTime":433.14,"body":"you down the garden path."},{"startTime":433.14,"endTime":436.13,"body":"The area under a cycloid"},{"startTime":433.14,"endTime":436.13,"body":"arch is exactly three times,"},{"startTime":436.13,"endTime":439.09,"body":"not pie two, three times"},{"startTime":436.13,"endTime":439.09,"body":"the area of the circle"},{"startTime":439.09,"endTime":439.993,"body":"that generates it."},{"startTime":441.08,"endTime":444.38,"body":"To me, surprisingly, it's"},{"startTime":441.08,"endTime":444.38,"body":"just a whole number multiple"},{"startTime":445.793,"endTime":447.65,"body":"of the circle area."},{"startTime":447.65,"endTime":450.76,"body":"So Gilles de Roberval found that out."},{"startTime":450.76,"endTime":454.687,"body":"Now, he was a curious character, Roberval."},{"startTime":456.58,"endTime":459.531,"body":"He held his information quite jealously"},{"startTime":459.531,"endTime":462.79,"body":"because he wouldn't actually reveal,"},{"startTime":462.79,"endTime":465.14,"body":"he wouldn't often even tell"},{"startTime":462.79,"endTime":465.14,"body":"people what he'd managed"},{"startTime":465.14,"endTime":466.37,"body":"to prove."},{"startTime":466.37,"endTime":468.34,"body":"And if he did tell them he'd"},{"startTime":466.37,"endTime":468.34,"body":"managed to prove something,"},{"startTime":468.34,"endTime":471.43,"body":"he wouldn't necessarily"},{"startTime":468.34,"endTime":471.43,"body":"share the proof with them,"},{"startTime":471.43,"endTime":474.47,"body":"and this was a rather"},{"startTime":471.43,"endTime":474.47,"body":"idiosyncratic thing, of course,"},{"startTime":474.47,"endTime":477.583,"body":"but there was method in the madness."},{"startTime":478.69,"endTime":480.97,"body":"Roberval's job, he had a professorship,"},{"startTime":480.97,"endTime":484.66,"body":"and this professorship was"},{"startTime":480.97,"endTime":484.66,"body":"renewable every three years."},{"startTime":484.66,"endTime":488.86,"body":"And the way they decided who"},{"startTime":484.66,"endTime":488.86,"body":"would be the next professor"},{"startTime":488.86,"endTime":491.55,"body":"for the next three years"},{"startTime":488.86,"endTime":491.55,"body":"was there was a competition,"},{"startTime":491.55,"endTime":494.53,"body":"an open competition with"},{"startTime":491.55,"endTime":494.53,"body":"some mathematical problems,"},{"startTime":494.53,"endTime":497.27,"body":"and you could submit your"},{"startTime":494.53,"endTime":497.27,"body":"answers to those problems,"},{"startTime":497.27,"endTime":499.22,"body":"and if your solutions were the best,"},{"startTime":499.22,"endTime":502.05,"body":"you got to be that professor"},{"startTime":499.22,"endTime":502.05,"body":"for the next three years."},{"startTime":502.05,"endTime":506.0,"body":"So the twist is that the"},{"startTime":502.05,"endTime":506.0,"body":"problems in that competition"},{"startTime":506.0,"endTime":508.898,"body":"were set by the incumbent professor."},{"startTime":508.898,"endTime":512.04,"body":"So there is a great incentive"},{"startTime":512.04,"endTime":515.49,"body":"if you have a particular"},{"startTime":512.04,"endTime":515.49,"body":"thing that you know how to do."},{"startTime":515.49,"endTime":518.04,"body":"You don't actually want to"},{"startTime":515.49,"endTime":518.04,"body":"tell anyone else how to do it,"},{"startTime":518.04,"endTime":521.44,"body":"because then you can set that"},{"startTime":518.04,"endTime":521.44,"body":"problem every three years,"},{"startTime":521.44,"endTime":523.13,"body":"and you are the only one"},{"startTime":521.44,"endTime":523.13,"body":"who knows how to do it,"},{"startTime":523.13,"endTime":525.26,"body":"so you win the competition"},{"startTime":523.13,"endTime":525.26,"body":"that you have designed,"},{"startTime":525.26,"endTime":527.88,"body":"and you are entering every single time."},{"startTime":527.88,"endTime":530.7,"body":"So there is a bit of an"},{"startTime":527.88,"endTime":530.7,"body":"incentive for Roberval"},{"startTime":530.7,"endTime":532.56,"body":"to have some secret techniques,"},{"startTime":532.56,"endTime":536.16,"body":"but even so, he was especially"},{"startTime":532.56,"endTime":536.16,"body":"secretive about his work"},{"startTime":536.16,"endTime":538.41,"body":"and a lot of what he did wasn't published"},{"startTime":538.41,"endTime":541.014,"body":"until a long time later after his death."},{"startTime":541.014,"endTime":544.22,"body":"But we do know that he"},{"startTime":541.014,"endTime":544.22,"body":"was very likely the first"},{"startTime":544.22,"endTime":547.293,"body":"to find the area under the cycloid arch."},{"startTime":548.21,"endTime":551.763,"body":"Now, it wasn't just Roberval,"},{"startTime":548.21,"endTime":551.763,"body":"and Mersenne, and Galileo."},{"startTime":553.19,"endTime":554.84,"body":"Anyone who is anyone,"},{"startTime":554.84,"endTime":558.26,"body":"think of a mathematician you've heard of"},{"startTime":558.26,"endTime":562.54,"body":"from this kind of time, and"},{"startTime":558.26,"endTime":562.54,"body":"they almost certainly worked"},{"startTime":562.54,"endTime":564.15,"body":"on cycloid at some point."},{"startTime":564.15,"endTime":566.443,"body":"They are so appealing to people."},{"startTime":567.44,"endTime":568.96,"body":"The statue there, you can see,"},{"startTime":568.96,"endTime":572.04,"body":"that's from the Louvre, that's of Pascal."},{"startTime":572.04,"endTime":574.15,"body":"We may have heard of"},{"startTime":572.04,"endTime":574.15,"body":"Pascal's trying on the things"},{"startTime":574.15,"endTime":575.586,"body":"he more or less invented with the Fermat,"},{"startTime":575.586,"endTime":577.96,"body":"the science of probability theory,"},{"startTime":577.96,"endTime":580.42,"body":"but he also, he worked on the cycloid,"},{"startTime":580.42,"endTime":582.4,"body":"and it happened like this."},{"startTime":582.4,"endTime":584.543,"body":"He decided actually at one point,"},{"startTime":585.483,"endTime":586.91,"body":"\"I'm not going to do maths anymore."},{"startTime":586.91,"endTime":589.06,"body":"I'm going to focus on theology."},{"startTime":589.06,"endTime":590.89,"body":"I think that's what I should be doing.\""},{"startTime":590.89,"endTime":593.09,"body":"And then one night he"},{"startTime":590.89,"endTime":593.09,"body":"had a terrible toothache"},{"startTime":593.09,"endTime":594.44,"body":"and he couldn't sleep."},{"startTime":594.44,"endTime":596.83,"body":"So he did what sensible person would do."},{"startTime":596.83,"endTime":599.56,"body":"He thought about lovely"},{"startTime":596.83,"endTime":599.56,"body":"mathematics to distract him."},{"startTime":599.56,"endTime":603.0,"body":"And the lovely mathematics he"},{"startTime":599.56,"endTime":603.0,"body":"thought about was the cycloid."},{"startTime":603.0,"endTime":605.47,"body":"And actually, this statue of him,"},{"startTime":605.47,"endTime":607.21,"body":"depending on where you are sitting,"},{"startTime":607.21,"endTime":608.58,"body":"you maybe just be able to see"},{"startTime":608.58,"endTime":613.2,"body":"that the stone tablet he's"},{"startTime":608.58,"endTime":613.2,"body":"looking at has a cycloid on it."},{"startTime":613.2,"endTime":617.22,"body":"So this is the Descartes, so"},{"startTime":613.2,"endTime":617.22,"body":"pleased when I us to show you."},{"startTime":617.22,"endTime":618.705,"body":"This is him studying the cycloid."},{"startTime":618.705,"endTime":623.09,"body":"That night, what he was"},{"startTime":618.705,"endTime":623.09,"body":"thinking about, the cycloid,"},{"startTime":623.09,"endTime":626.42,"body":"it was so distracting"},{"startTime":623.09,"endTime":626.42,"body":"that the pain went away."},{"startTime":626.42,"endTime":629.66,"body":"And so, he took this as"},{"startTime":626.42,"endTime":629.66,"body":"a sin that God was okay"},{"startTime":629.66,"endTime":632.19,"body":"with him doing mathematics,"},{"startTime":629.66,"endTime":632.19,"body":"and we're glad he did"},{"startTime":632.19,"endTime":634.15,"body":"because he did some lovely"},{"startTime":632.19,"endTime":634.15,"body":"mathematics after that."},{"startTime":634.15,"endTime":637.58,"body":"He went on to think about"},{"startTime":634.15,"endTime":637.58,"body":"cycloids for eight more days,"},{"startTime":637.58,"endTime":639.91,"body":"and he proved a lot of"},{"startTime":637.58,"endTime":639.91,"body":"things during that time."},{"startTime":639.91,"endTime":642.81,"body":"So this is a very distracting"},{"startTime":639.91,"endTime":642.81,"body":"and appealing curve."},{"startTime":642.81,"endTime":645.22,"body":"That's one reason why it's"},{"startTime":642.81,"endTime":645.22,"body":"called the Helen of Geometry,"},{"startTime":645.22,"endTime":646.6,"body":"Helen of Troy,"},{"startTime":646.6,"endTime":648.35,"body":"the face that launched a thousand ships,"},{"startTime":648.35,"endTime":652.1,"body":"famous beauty, but not"},{"startTime":648.35,"endTime":652.1,"body":"just for being beautiful,"},{"startTime":652.1,"endTime":654.63,"body":"for causing, let's say, squabbles."},{"startTime":654.63,"endTime":657.53,"body":"And this is another"},{"startTime":654.63,"endTime":657.53,"body":"reason why this cycloid"},{"startTime":657.53,"endTime":661.15,"body":"has got this nickname because"},{"startTime":657.53,"endTime":661.15,"body":"so many people studied it,"},{"startTime":661.15,"endTime":663.75,"body":"and then they argued"},{"startTime":661.15,"endTime":663.75,"body":"about who done what first"},{"startTime":663.75,"endTime":665.56,"body":"and whose ideas were better."},{"startTime":665.56,"endTime":670.1,"body":"So Descartes and Fermat"},{"startTime":665.56,"endTime":670.1,"body":"both studied the cycloid"},{"startTime":670.1,"endTime":672.54,"body":"and they were trying to"},{"startTime":670.1,"endTime":672.54,"body":"understand properties"},{"startTime":672.54,"endTime":674.01,"body":"of tangents to the cycloid."},{"startTime":674.01,"endTime":676.04,"body":"So the tangent to a curve at any point"},{"startTime":676.04,"endTime":678.189,"body":"is a line that just touch it."},{"startTime":678.189,"endTime":680.45,"body":"It doesn't cross it, it just touches it."},{"startTime":680.45,"endTime":682.42,"body":"They both had studied these things."},{"startTime":682.42,"endTime":684.08,"body":"They didn't like each other's methods."},{"startTime":684.08,"endTime":686.27,"body":"Descartes said that what Fermat had done"},{"startTime":686.27,"endTime":688.29,"body":"was ridiculous gibberish,"},{"startTime":688.29,"endTime":691.4,"body":"which isn't absolutely"},{"startTime":688.29,"endTime":691.4,"body":"the polite way to speak"},{"startTime":691.4,"endTime":693.313,"body":"of a fellow mathematician."},{"startTime":694.936,"endTime":697.06,"body":"Descartes didn't like"},{"startTime":694.936,"endTime":697.06,"body":"what Fermat had done."},{"startTime":697.06,"endTime":698.24,"body":"Roberval, here he is again."},{"startTime":698.24,"endTime":701.5,"body":"Now, remember, he didn't"},{"startTime":698.24,"endTime":701.5,"body":"tell anyone what he'd done"},{"startTime":701.5,"endTime":703.5,"body":"or he didn't like to"},{"startTime":701.5,"endTime":703.5,"body":"explain what he'd done."},{"startTime":703.5,"endTime":706.36,"body":"So when the Italian"},{"startTime":703.5,"endTime":706.36,"body":"mathematician, Torrichelli,"},{"startTime":706.36,"endTime":710.2,"body":"independently came up with"},{"startTime":706.36,"endTime":710.2,"body":"a way to find the area under"},{"startTime":710.2,"endTime":712.97,"body":"the cycloid arch, Roberval"},{"startTime":710.2,"endTime":712.97,"body":"got really crossed."},{"startTime":712.97,"endTime":715.56,"body":"He went around saying,"},{"startTime":712.97,"endTime":715.56,"body":"\"Torrichelli's stolen this from me."},{"startTime":715.56,"endTime":717.49,"body":"He must have stolen it somehow.\""},{"startTime":717.49,"endTime":721.06,"body":"We don't know, Roberval didn't"},{"startTime":717.49,"endTime":721.06,"body":"tell anyone what he'd done."},{"startTime":721.06,"endTime":724.35,"body":"And then the rumor went"},{"startTime":721.06,"endTime":724.35,"body":"round that Torrichelli"},{"startTime":724.35,"endTime":727.76,"body":"was so ashamed when he'd"},{"startTime":724.35,"endTime":727.76,"body":"been caught plagiarizing"},{"startTime":727.76,"endTime":728.593,"body":"in this way."},{"startTime":728.593,"endTime":729.89,"body":"Don't think he did,"},{"startTime":729.89,"endTime":733.29,"body":"but he was so ashamed that"},{"startTime":729.89,"endTime":733.29,"body":"he'd actually died of shame."},{"startTime":733.29,"endTime":736.69,"body":"Now you can judge for yourself"},{"startTime":733.29,"endTime":736.69,"body":"whether this is true or not."},{"startTime":736.69,"endTime":740.78,"body":"I do have to say, at the"},{"startTime":736.69,"endTime":740.78,"body":"time when he died of shame,"},{"startTime":740.78,"endTime":742.88,"body":"he also happened to have typhoid,"},{"startTime":742.88,"endTime":746.53,"body":"but we know, let's not"},{"startTime":742.88,"endTime":746.53,"body":"make any assumptions."},{"startTime":746.53,"endTime":749.645,"body":"So all these arguments and many others"},{"startTime":749.645,"endTime":753.2,"body":"were how this beautiful curve"},{"startTime":749.645,"endTime":753.2,"body":"got to get it's nickname"},{"startTime":753.2,"endTime":754.853,"body":"as the Helen of Geometry."},{"startTime":756.8,"endTime":759.283,"body":"But the story of the"},{"startTime":756.8,"endTime":759.283,"body":"cycloid doesn't end there."},{"startTime":760.18,"endTime":761.84,"body":"I want to tell you something about clocks."},{"startTime":761.84,"endTime":766.84,"body":"So in 1656, there was a"},{"startTime":761.84,"endTime":766.84,"body":"breakthrough in clock design."},{"startTime":766.86,"endTime":771.84,"body":"Before 1656, the best, most"},{"startTime":766.86,"endTime":771.84,"body":"accurate time keeping device"},{"startTime":771.84,"endTime":773.023,"body":"was a sundial."},{"startTime":774.88,"endTime":778.9,"body":"Not very useful if you are"},{"startTime":774.88,"endTime":778.9,"body":"indoors or if it's nighttime,"},{"startTime":778.9,"endTime":783.06,"body":"or if you live in Britain,"},{"startTime":778.9,"endTime":783.06,"body":"where it's often cloudy,"},{"startTime":783.06,"endTime":786.56,"body":"but sundials were the most"},{"startTime":783.06,"endTime":786.56,"body":"accurate things that we had."},{"startTime":786.56,"endTime":788.29,"body":"They were mechanical clocks,"},{"startTime":788.29,"endTime":790.04,"body":"but they were, I don't want to be rude,"},{"startTime":790.04,"endTime":793.46,"body":"but they were rubbish"},{"startTime":790.04,"endTime":793.46,"body":"because they lost up to,"},{"startTime":793.46,"endTime":797.85,"body":"or inaccurate up to about 15"},{"startTime":793.46,"endTime":797.85,"body":"minutes a day they could lose."},{"startTime":797.85,"endTime":800.0,"body":"And so, this wasn't good enough,"},{"startTime":800.0,"endTime":802.1,"body":"and people were trying to"},{"startTime":800.0,"endTime":802.1,"body":"think of mechanical clocks"},{"startTime":802.1,"endTime":804.36,"body":"that would be better at time keeping."},{"startTime":804.36,"endTime":807.7,"body":"And 1656 was the year"},{"startTime":804.36,"endTime":807.7,"body":"when Christiaan Huygens,"},{"startTime":807.7,"endTime":811.377,"body":"the Dutch scientist and"},{"startTime":807.7,"endTime":811.377,"body":"mathematician produced"},{"startTime":811.377,"endTime":815.1,"body":"and built the very first pendulum clock."},{"startTime":815.1,"endTime":818.463,"body":"So this was a great leap forward."},{"startTime":819.35,"endTime":820.38,"body":"What's good about pendulums?"},{"startTime":820.38,"endTime":822.8,"body":"Why would you use a pendulum in a clock?"},{"startTime":822.8,"endTime":825.53,"body":"Well, I'm not going to go into"},{"startTime":822.8,"endTime":825.53,"body":"like the full details of it,"},{"startTime":825.53,"endTime":830.53,"body":"but I want to just show"},{"startTime":825.53,"endTime":830.53,"body":"you a bit of what's going"},{"startTime":830.72,"endTime":831.553,"body":"on mathematically,"},{"startTime":831.553,"endTime":834.53,"body":"because then I want to explain"},{"startTime":831.553,"endTime":834.53,"body":"why there might be a way"},{"startTime":834.53,"endTime":836.91,"body":"to improve upon it even"},{"startTime":834.53,"endTime":836.91,"body":"when you've got a pendulum."},{"startTime":836.91,"endTime":839.76,"body":"So what happens to the pendulum?"},{"startTime":839.76,"endTime":842.92,"body":"So you've got a rod or a wire or a string,"},{"startTime":842.92,"endTime":845.41,"body":"something like that fixed at the top."},{"startTime":845.41,"endTime":847.44,"body":"And then at the air"},{"startTime":845.41,"endTime":847.44,"body":"bottom, it's got a weight."},{"startTime":847.44,"endTime":849.27,"body":"We call it the bob usually,"},{"startTime":849.27,"endTime":851.47,"body":"and then it will swing"},{"startTime":849.27,"endTime":851.47,"body":"backwards and forward."},{"startTime":851.47,"endTime":854.883,"body":"And if you just let it go"},{"startTime":851.47,"endTime":854.883,"body":"and it falls under gravity,"},{"startTime":855.87,"endTime":858.63,"body":"the bob will move in archs of circles,"},{"startTime":858.63,"endTime":860.79,"body":"and it move backwards and forwards."},{"startTime":860.79,"endTime":862.26,"body":"It'll tick and then it'll tock,"},{"startTime":862.26,"endTime":863.703,"body":"come backwards and forwards."},{"startTime":864.712,"endTime":866.523,"body":"And what's great about pendulums"},{"startTime":867.54,"endTime":870.163,"body":"is that this movement,"},{"startTime":867.54,"endTime":870.163,"body":"backwards and forwards,"},{"startTime":871.54,"endTime":873.62,"body":"is very regular, right?"},{"startTime":873.62,"endTime":875.72,"body":"So you can literally set your watch by it."},{"startTime":875.72,"endTime":877.44,"body":"Now, why is it regular?"},{"startTime":877.44,"endTime":880.06,"body":"Well, again, I won't go"},{"startTime":877.44,"endTime":880.06,"body":"into all of the details"},{"startTime":880.06,"endTime":882.47,"body":"because to do every line"},{"startTime":880.06,"endTime":882.47,"body":"of what I'm about to say"},{"startTime":882.47,"endTime":884.56,"body":"would need a bit of calculus."},{"startTime":884.56,"endTime":885.71,"body":"So we won't do that,"},{"startTime":885.71,"endTime":887.005,"body":"but I just want to give you an indication."},{"startTime":887.005,"endTime":889.147,"body":"So what's happening in a pendulum?"},{"startTime":889.147,"endTime":890.0,"body":"When you've got a pendulum"},{"startTime":890.0,"endTime":891.78,"body":"and you're going to hold it here"},{"startTime":891.78,"endTime":892.613,"body":"and then you're going to release it"},{"startTime":892.613,"endTime":893.8,"body":"and let it move just with gravity."},{"startTime":893.8,"endTime":896.43,"body":"So maybe it's an angle to the vertical."},{"startTime":896.43,"endTime":899.58,"body":"When you set it going, that"},{"startTime":896.43,"endTime":899.58,"body":"angle could be called theater."},{"startTime":899.58,"endTime":901.47,"body":"And then there are some forces acting."},{"startTime":901.47,"endTime":903.57,"body":"If there's no forces, there's no movement."},{"startTime":904.41,"endTime":907.943,"body":"So what are the forces?"},{"startTime":904.41,"endTime":907.943,"body":"Well, you've got gravity."},{"startTime":909.12,"endTime":911.01,"body":"So force is mass times acceleration."},{"startTime":911.01,"endTime":911.94,"body":"What's the acceleration here?"},{"startTime":911.94,"endTime":913.62,"body":"It's the acceleration due to gravity."},{"startTime":913.62,"endTime":915.02,"body":"That's what that little G is."},{"startTime":915.02,"endTime":917.98,"body":"And then the M is just the mass"},{"startTime":915.02,"endTime":917.98,"body":"of the bob, whatever it is."},{"startTime":917.98,"endTime":922.04,"body":"So that's the force downwards,"},{"startTime":917.98,"endTime":922.04,"body":"vertically downwards."},{"startTime":922.04,"endTime":924.19,"body":"And this thing, it's joined on,"},{"startTime":924.19,"endTime":926.76,"body":"the bob is joined to"},{"startTime":924.19,"endTime":926.76,"body":"the top of the pendulum."},{"startTime":926.76,"endTime":929.262,"body":"And so it can't get any"},{"startTime":926.76,"endTime":929.262,"body":"further away from the center"},{"startTime":929.262,"endTime":930.5,"body":"of that circle."},{"startTime":930.5,"endTime":932.13,"body":"So it's got to be something pulling it"},{"startTime":932.13,"endTime":933.21,"body":"towards the center of the circle."},{"startTime":933.21,"endTime":934.34,"body":"That's what I've called T there,"},{"startTime":934.34,"endTime":937.76,"body":"the tension in that string or rod."},{"startTime":937.76,"endTime":939.43,"body":"So that's pulling you towards"},{"startTime":937.76,"endTime":939.43,"body":"the center of the circle"},{"startTime":939.43,"endTime":942.76,"body":"and exactly counteracting"},{"startTime":939.43,"endTime":942.76,"body":"the bit of the downward force"},{"startTime":942.76,"endTime":945.59,"body":"that's trying to move you away."},{"startTime":945.59,"endTime":948.203,"body":"So what is left? So those balanced out."},{"startTime":948.203,"endTime":950.314,"body":"Then the only force that"},{"startTime":948.203,"endTime":950.314,"body":"they're sort of left over"},{"startTime":950.314,"endTime":953.06,"body":"is the force that kind of pulling you"},{"startTime":953.06,"endTime":955.46,"body":"along the arch of the circle."},{"startTime":955.46,"endTime":958.273,"body":"And if you use a bit of trigonometry,"},{"startTime":959.89,"endTime":962.14,"body":"you find that actually"},{"startTime":959.89,"endTime":962.14,"body":"that's related to the sin"},{"startTime":962.14,"endTime":964.74,"body":"of theater, the sin of that angle."},{"startTime":964.74,"endTime":965.92,"body":"So at this point,"},{"startTime":965.92,"endTime":970.92,"body":"and if you've ever did"},{"startTime":965.92,"endTime":970.92,"body":"study earlier on pendulums,"},{"startTime":973.67,"endTime":975.07,"body":"you will have made an approximation."},{"startTime":975.07,"endTime":977.72,"body":"And it's this, when theater is very small,"},{"startTime":977.72,"endTime":980.583,"body":"sin of theater is very"},{"startTime":977.72,"endTime":980.583,"body":"close to being theater."},{"startTime":981.5,"endTime":983.95,"body":"So you approximate, you"},{"startTime":981.5,"endTime":983.95,"body":"make an approximation."},{"startTime":983.95,"endTime":986.19,"body":"And the reason you do that"},{"startTime":983.95,"endTime":986.19,"body":"is because you can't solve"},{"startTime":986.19,"endTime":988.01,"body":"the equations if it has sin theater,"},{"startTime":988.01,"endTime":990.01,"body":"but you can't do it if"},{"startTime":988.01,"endTime":990.01,"body":"it has theater, okay?"},{"startTime":990.01,"endTime":994.15,"body":"So then you do a bit of equation"},{"startTime":990.01,"endTime":994.15,"body":"solving, a bit of algebra,"},{"startTime":994.15,"endTime":995.154,"body":"a bit of calculus,"},{"startTime":995.154,"endTime":998.83,"body":"and you can get an"},{"startTime":995.154,"endTime":998.83,"body":"expression for the period"},{"startTime":998.83,"endTime":999.663,"body":"of the pendulum."},{"startTime":999.663,"endTime":1000.496,"body":"What's the period?"},{"startTime":1000.496,"endTime":1002.2,"body":"It's the amount of time it takes to get"},{"startTime":1002.2,"endTime":1005.07,"body":"from your starting point"},{"startTime":1002.2,"endTime":1005.07,"body":"over there and back again."},{"startTime":1005.07,"endTime":1007.49,"body":"That's the period. Back again."},{"startTime":1007.49,"endTime":1010.85,"body":"And when you do that with"},{"startTime":1007.49,"endTime":1010.85,"body":"this little approximation,"},{"startTime":1010.85,"endTime":1013.59,"body":"this is the expression"},{"startTime":1010.85,"endTime":1013.59,"body":"you get for the period."},{"startTime":1013.59,"endTime":1015.51,"body":"Doesn't matter exactly what"},{"startTime":1013.59,"endTime":1015.51,"body":"it is for these purposes."},{"startTime":1015.51,"endTime":1018.44,"body":"But what I'm want you to notice"},{"startTime":1015.51,"endTime":1018.44,"body":"is, so it's got a pie in it."},{"startTime":1018.44,"endTime":1020.88,"body":"It's got an L, that's"},{"startTime":1018.44,"endTime":1020.88,"body":"the length of the string."},{"startTime":1020.88,"endTime":1023.35,"body":"I didn't say that. And it's got gravity."},{"startTime":1023.35,"endTime":1026.64,"body":"This does not depend on the"},{"startTime":1023.35,"endTime":1026.64,"body":"angle that you started with."},{"startTime":1026.64,"endTime":1027.54,"body":"It doesn't depend."},{"startTime":1027.54,"endTime":1029.35,"body":"So what this tells you is,"},{"startTime":1029.35,"endTime":1032.37,"body":"whatever height you are,"},{"startTime":1029.35,"endTime":1032.37,"body":"whatever angle you're at,"},{"startTime":1032.37,"endTime":1033.644,"body":"when you release the pendulum,"},{"startTime":1033.644,"endTime":1036.32,"body":"it takes a same amount"},{"startTime":1033.644,"endTime":1036.32,"body":"of time to get back."},{"startTime":1036.32,"endTime":1039.15,"body":"And that means if you set it going,"},{"startTime":1039.15,"endTime":1041.19,"body":"then over time, the system"},{"startTime":1039.15,"endTime":1041.19,"body":"loses a bit of energy."},{"startTime":1041.19,"endTime":1044.73,"body":"Maybe it's moving to a smaller angle,"},{"startTime":1044.73,"endTime":1046.74,"body":"but that doesn't affect the period."},{"startTime":1046.74,"endTime":1049.44,"body":"So it still keeps ticking"},{"startTime":1046.74,"endTime":1049.44,"body":"and tocking at the same rate"},{"startTime":1051.02,"endTime":1053.86,"body":"whatever the angle is."},{"startTime":1053.86,"endTime":1057.42,"body":"That's great, except for"},{"startTime":1053.86,"endTime":1057.42,"body":"this fact that I mentioned,"},{"startTime":1057.42,"endTime":1060.16,"body":"all of this depends on an approximation."},{"startTime":1060.16,"endTime":1065.16,"body":"So it only is truly"},{"startTime":1060.16,"endTime":1065.16,"body":"accurate for small angles."},{"startTime":1066.01,"endTime":1068.67,"body":"And as soon as the"},{"startTime":1066.01,"endTime":1068.67,"body":"angle gets a bit bigger,"},{"startTime":1068.67,"endTime":1070.5,"body":"it becomes less accurate."},{"startTime":1070.5,"endTime":1075.25,"body":"So these pendulum clocks were"},{"startTime":1070.5,"endTime":1075.25,"body":"great, fantastic innovation."},{"startTime":1075.25,"endTime":1077.53,"body":"They now meant that you"},{"startTime":1075.25,"endTime":1077.53,"body":"could produce clocks"},{"startTime":1077.53,"endTime":1080.31,"body":"though accurate to 15 seconds a day,"},{"startTime":1080.31,"endTime":1081.223,"body":"not 15 minutes a day, right?"},{"startTime":1081.223,"endTime":1085.55,"body":"This is a huge improvement"},{"startTime":1081.223,"endTime":1085.55,"body":"in clock design."},{"startTime":1085.55,"endTime":1087.4,"body":"Huygens wasn't satisfied with this."},{"startTime":1087.4,"endTime":1089.31,"body":"I mean, it's great, but he thought,"},{"startTime":1089.31,"endTime":1092.64,"body":"he was aware of this"},{"startTime":1089.31,"endTime":1092.64,"body":"approximation that had to be made."},{"startTime":1092.64,"endTime":1094.76,"body":"He thought, what would be great is,"},{"startTime":1094.76,"endTime":1097.387,"body":"this pendulum is moving"},{"startTime":1094.76,"endTime":1097.387,"body":"along the arch of a circle,"},{"startTime":1097.387,"endTime":1100.91,"body":"and approximately has a constant period."},{"startTime":1100.91,"endTime":1102.97,"body":"It doesn't matter where"},{"startTime":1100.91,"endTime":1102.97,"body":"you start it going from,"},{"startTime":1102.97,"endTime":1106.59,"body":"it reaches the bottom"},{"startTime":1102.97,"endTime":1106.59,"body":"in the same time, right?"},{"startTime":1106.59,"endTime":1108.973,"body":"If only there was some sort of curve"},{"startTime":1108.973,"endTime":1111.4,"body":"that I could get my"},{"startTime":1108.973,"endTime":1111.4,"body":"pendulum to move along,"},{"startTime":1111.4,"endTime":1113.46,"body":"which really did have that property,"},{"startTime":1113.46,"endTime":1116.32,"body":"that it really did always reach the bottom"},{"startTime":1116.32,"endTime":1117.973,"body":"in the same amount of time."},{"startTime":1117.973,"endTime":1121.293,"body":"And if you take the Greek"},{"startTime":1117.973,"endTime":1121.293,"body":"for the same amount of time,"},{"startTime":1122.156,"endTime":1122.989,"body":"that's tautochrone."},{"startTime":1122.989,"endTime":1125.43,"body":"And so this thing is known"},{"startTime":1122.989,"endTime":1125.43,"body":"as the tautochrone problem."},{"startTime":1125.43,"endTime":1128.68,"body":"Find a curve, such that"},{"startTime":1125.43,"endTime":1128.68,"body":"wherever a particle"},{"startTime":1128.68,"endTime":1132.75,"body":"is release on that curve and"},{"startTime":1128.68,"endTime":1132.75,"body":"it's just moving under gravity,"},{"startTime":1132.75,"endTime":1134.3,"body":"it will reach the bottom in the same time,"},{"startTime":1134.3,"endTime":1135.45,"body":"and then sort of bisymmetry,"},{"startTime":1135.45,"endTime":1137.13,"body":"it'll go back up to the opposite point"},{"startTime":1137.13,"endTime":1138.19,"body":"and then come back again."},{"startTime":1138.19,"endTime":1140.483,"body":"So with such a curve, if it exists,"},{"startTime":1141.41,"endTime":1144.19,"body":"if you could get a pendulum"},{"startTime":1141.41,"endTime":1144.19,"body":"to move along that path,"},{"startTime":1144.19,"endTime":1146.16,"body":"then you really would"},{"startTime":1144.19,"endTime":1146.16,"body":"have a constant period,"},{"startTime":1146.16,"endTime":1148.15,"body":"not just an approximation to one."},{"startTime":1148.15,"endTime":1150.4,"body":"So Huygens saw, wouldn't it be lovely"},{"startTime":1150.4,"endTime":1152.048,"body":"if there was a curve like that?"},{"startTime":1152.048,"endTime":1155.54,"body":"Can you guess what curve that might be?"},{"startTime":1155.54,"endTime":1157.36,"body":"I mean, you can see it here."},{"startTime":1157.36,"endTime":1159.5,"body":"It's actually a cycloid."},{"startTime":1159.5,"endTime":1160.333,"body":"I mean, it's what you know."},{"startTime":1160.333,"endTime":1161.57,"body":"I've turned it upside down,"},{"startTime":1161.57,"endTime":1163.46,"body":"but it's exactly the"},{"startTime":1161.57,"endTime":1163.46,"body":"cycloid that we've seen,"},{"startTime":1163.46,"endTime":1166.24,"body":"which is like astonishing"},{"startTime":1163.46,"endTime":1166.24,"body":"the first time you see this,"},{"startTime":1166.24,"endTime":1169.49,"body":"because the cycloid just"},{"startTime":1166.24,"endTime":1169.49,"body":"is like a circle rolling"},{"startTime":1169.49,"endTime":1170.323,"body":"along a straight."},{"startTime":1170.323,"endTime":1172.408,"body":"Why that be the answer to"},{"startTime":1170.323,"endTime":1172.408,"body":"the tautochrone problem?"},{"startTime":1172.408,"endTime":1174.11,"body":"That's one of the wonderful"},{"startTime":1172.408,"endTime":1174.11,"body":"things about mathematics"},{"startTime":1174.11,"endTime":1177.22,"body":"that you sometimes have"},{"startTime":1174.11,"endTime":1177.22,"body":"these beautiful coincidences."},{"startTime":1177.22,"endTime":1179.51,"body":"So this is the solution to the problem,"},{"startTime":1179.51,"endTime":1182.388,"body":"but we are not done because"},{"startTime":1179.51,"endTime":1182.388,"body":"that's all very nice."},{"startTime":1182.388,"endTime":1185.03,"body":"How on earth do you get a pendulum"},{"startTime":1185.03,"endTime":1187.6,"body":"to move along the cycloid, right?"},{"startTime":1187.6,"endTime":1190.0,"body":"If you just set it going,"},{"startTime":1187.6,"endTime":1190.0,"body":"it'll make an arch of a circle."},{"startTime":1190.0,"endTime":1191.85,"body":"How would you get it to do this?"},{"startTime":1191.85,"endTime":1195.37,"body":"So for that, we need to"},{"startTime":1191.85,"endTime":1195.37,"body":"think about the next kind"},{"startTime":1195.37,"endTime":1196.79,"body":"of roulette curve."},{"startTime":1196.79,"endTime":1198.64,"body":"So roulette curves, remember,"},{"startTime":1196.79,"endTime":1198.64,"body":"are curves that are made"},{"startTime":1198.64,"endTime":1200.823,"body":"by rolling one curve along another."},{"startTime":1201.78,"endTime":1205.01,"body":"And this kind of roulette"},{"startTime":1201.78,"endTime":1205.01,"body":"curve is called an involute,"},{"startTime":1205.01,"endTime":1209.38,"body":"an involutes of what you get when you roll"},{"startTime":1209.38,"endTime":1212.543,"body":"a straight line along curve."},{"startTime":1214.71,"endTime":1216.38,"body":"This gift was going a bit faster earlier,"},{"startTime":1216.38,"endTime":1218.613,"body":"but this is an example here."},{"startTime":1219.56,"endTime":1222.72,"body":"We're rolling a straight"},{"startTime":1219.56,"endTime":1222.72,"body":"line along a circle."},{"startTime":1222.72,"endTime":1224.83,"body":"So on the graph here,"},{"startTime":1224.83,"endTime":1226.9,"body":"you've got this red straight line,"},{"startTime":1226.9,"endTime":1229.7,"body":"which you can think of"},{"startTime":1226.9,"endTime":1229.7,"body":"as being like a thread"},{"startTime":1229.7,"endTime":1232.09,"body":"that's unwinding gradually from a roll"},{"startTime":1232.09,"endTime":1233.51,"body":"or threat or something like that."},{"startTime":1233.51,"endTime":1236.81,"body":"And you're following what"},{"startTime":1233.51,"endTime":1236.81,"body":"that straight line does,"},{"startTime":1236.81,"endTime":1238.61,"body":"a point on the end of the straight line,"},{"startTime":1238.61,"endTime":1239.96,"body":"a line segment, I guess."},{"startTime":1239.96,"endTime":1242.523,"body":"You're just following its"},{"startTime":1239.96,"endTime":1242.523,"body":"path as it moves around."},{"startTime":1244.42,"endTime":1247.94,"body":"So these lines that are"},{"startTime":1244.42,"endTime":1247.94,"body":"rolling along the curves,"},{"startTime":1247.94,"endTime":1250.93,"body":"they're going to be tangents"},{"startTime":1247.94,"endTime":1250.93,"body":"to the curve, I guess."},{"startTime":1250.93,"endTime":1252.95,"body":"So you can either think of it"},{"startTime":1250.93,"endTime":1252.95,"body":"as that rolling a line along,"},{"startTime":1252.95,"endTime":1255.5,"body":"or you can think of it as"},{"startTime":1252.95,"endTime":1255.5,"body":"unwinding a taut thread"},{"startTime":1255.5,"endTime":1256.5,"body":"from the curve."},{"startTime":1256.5,"endTime":1258.56,"body":"So this is the involute of a circle"},{"startTime":1258.56,"endTime":1261.333,"body":"that I'm showing you here,"},{"startTime":1258.56,"endTime":1261.333,"body":"looks a bit like a spiral."},{"startTime":1262.25,"endTime":1265.81,"body":"So what's this got to do"},{"startTime":1262.25,"endTime":1265.81,"body":"with the pendulum problem?"},{"startTime":1265.81,"endTime":1268.26,"body":"Well, there's an involute"},{"startTime":1265.81,"endTime":1268.26,"body":"of circle just there."},{"startTime":1268.26,"endTime":1270.63,"body":"So you'd have to gradually"},{"startTime":1268.26,"endTime":1270.63,"body":"watch it going round."},{"startTime":1270.63,"endTime":1271.76,"body":"So we've produced these curves."},{"startTime":1271.76,"endTime":1274.47,"body":"Now, if you imagine, just"},{"startTime":1271.76,"endTime":1274.47,"body":"turn that around a little bit."},{"startTime":1274.47,"endTime":1276.23,"body":"So here's a circle."},{"startTime":1276.23,"endTime":1279.793,"body":"If I had just a circular"},{"startTime":1276.23,"endTime":1279.793,"body":"thing in my clock somewhere,"},{"startTime":1279.793,"endTime":1284.02,"body":"and I wrapped my pendulum around it,"},{"startTime":1284.02,"endTime":1285.84,"body":"so you fix the pendulum"},{"startTime":1284.02,"endTime":1285.84,"body":"and I guess, you know,"},{"startTime":1285.84,"endTime":1288.04,"body":"it can't just be a rigid"},{"startTime":1285.84,"endTime":1288.04,"body":"rod that's holding,"},{"startTime":1288.04,"endTime":1292.04,"body":"it'll have to be a string"},{"startTime":1288.04,"endTime":1292.04,"body":"or a wire that can bend."},{"startTime":1292.04,"endTime":1294.547,"body":"So you wrap it around the circle."},{"startTime":1294.547,"endTime":1297.63,"body":"And then when you release,"},{"startTime":1294.547,"endTime":1297.63,"body":"the bobs at the bottom,"},{"startTime":1297.63,"endTime":1301.83,"body":"when you release that, it'll"},{"startTime":1297.63,"endTime":1301.83,"body":"unwrap itself from the circle"},{"startTime":1301.83,"endTime":1305.34,"body":"as it goes down until I"},{"startTime":1301.83,"endTime":1305.34,"body":"guess in this diagram,"},{"startTime":1305.34,"endTime":1307.01,"body":"until it reaches the vertical,"},{"startTime":1307.01,"endTime":1309.561,"body":"and then it'll start"},{"startTime":1307.01,"endTime":1309.561,"body":"moving in some freeway."},{"startTime":1309.561,"endTime":1312.72,"body":"But for that first bit of the motion,"},{"startTime":1312.72,"endTime":1314.65,"body":"it's unwrapping itself from the circle,"},{"startTime":1314.65,"endTime":1316.75,"body":"and it will produce exactly this involute."},{"startTime":1317.67,"endTime":1321.34,"body":"So if you wrap your"},{"startTime":1317.67,"endTime":1321.34,"body":"pendulum around a curve"},{"startTime":1321.34,"endTime":1324.29,"body":"and then release it, you'll"},{"startTime":1321.34,"endTime":1324.29,"body":"get the involute of that curve."},{"startTime":1325.16,"endTime":1329.757,"body":"So now what we need to ask"},{"startTime":1325.16,"endTime":1329.757,"body":"is, is there a curve, right?"},{"startTime":1330.99,"endTime":1335.447,"body":"X, some mysterious curve"},{"startTime":1330.99,"endTime":1335.447,"body":"whose involute is a cycloid"},{"startTime":1335.447,"endTime":1338.95,"body":"'cause then instead of a"},{"startTime":1335.447,"endTime":1338.95,"body":"circle, we could put that there,"},{"startTime":1338.95,"endTime":1340.92,"body":"wrap the pendulum around it, let it go,"},{"startTime":1340.92,"endTime":1344.39,"body":"and it would form a cycloid path."},{"startTime":1344.39,"endTime":1346.57,"body":"So guess what?"},{"startTime":1346.57,"endTime":1350.75,"body":"Guess what curve has an"},{"startTime":1346.57,"endTime":1350.75,"body":"involute that's a cycloid."},{"startTime":1350.75,"endTime":1352.117,"body":"It's a cycloid."},{"startTime":1352.117,"endTime":1353.99,"body":"Cycloid is down to everything."},{"startTime":1353.99,"endTime":1357.1,"body":"Yeah. The involute of"},{"startTime":1353.99,"endTime":1357.1,"body":"a cycloid is a cycloid."},{"startTime":1357.1,"endTime":1358.62,"body":"So what you can do is,"},{"startTime":1358.62,"endTime":1362.33,"body":"you can get two cycloids,"},{"startTime":1358.62,"endTime":1362.33,"body":"put them next to each other,"},{"startTime":1362.33,"endTime":1364.49,"body":"then wrap the pendulum around,"},{"startTime":1364.49,"endTime":1367.22,"body":"and you have to have to make"},{"startTime":1364.49,"endTime":1367.22,"body":"sure the length are right."},{"startTime":1367.22,"endTime":1369.46,"body":"It has to be half the length of this arch."},{"startTime":1369.46,"endTime":1370.9,"body":"And then, when you release it,"},{"startTime":1370.9,"endTime":1374.25,"body":"it will make a nice cycloid path."},{"startTime":1374.25,"endTime":1377.4,"body":"So this is exactly what"},{"startTime":1374.25,"endTime":1377.4,"body":"you need to produce"},{"startTime":1377.4,"endTime":1379.803,"body":"at the perfect pendulum clock."},{"startTime":1380.871,"endTime":1382.66,"body":"Okay. Slight hitch."},{"startTime":1382.66,"endTime":1385.29,"body":"With real world, when you actually try"},{"startTime":1385.29,"endTime":1386.123,"body":"and build these things,"},{"startTime":1386.123,"endTime":1389.12,"body":"you find that the amount of"},{"startTime":1386.123,"endTime":1389.12,"body":"extra friction they introduce"},{"startTime":1389.12,"endTime":1391.67,"body":"into the system balances"},{"startTime":1389.12,"endTime":1391.67,"body":"out the improvement"},{"startTime":1391.67,"endTime":1394.37,"body":"that you made in the theoretical design,"},{"startTime":1394.37,"endTime":1395.72,"body":"but, okay, nevermind."},{"startTime":1395.72,"endTime":1397.83,"body":"It's a lovely bit of mathematics."},{"startTime":1397.83,"endTime":1399.43,"body":"So these, cycloid are great."},{"startTime":1399.43,"endTime":1401.543,"body":"The involute of a cycloid is a cycloid."},{"startTime":1402.55,"endTime":1406.45,"body":"Well, since we are talking"},{"startTime":1402.55,"endTime":1406.45,"body":"about the Torrichelli problem,"},{"startTime":1406.45,"endTime":1408.73,"body":"I want to mention Brachistochrone problem,"},{"startTime":1408.73,"endTime":1411.58,"body":"which depending on how good your Greek is,"},{"startTime":1411.58,"endTime":1412.413,"body":"mine isn't very good."},{"startTime":1412.413,"endTime":1415.31,"body":"But I happen to know that"},{"startTime":1412.413,"endTime":1415.31,"body":"Brachistochrone means kind"},{"startTime":1415.31,"endTime":1417.186,"body":"of shortest time."},{"startTime":1417.186,"endTime":1422.186,"body":"And this problem was posed"},{"startTime":1417.186,"endTime":1422.186,"body":"by Johann Bernouilli in 1696."},{"startTime":1422.46,"endTime":1425.1,"body":"And he said, \"Okay, what about,"},{"startTime":1425.1,"endTime":1426.53,"body":"if we have a slightly different situation,"},{"startTime":1426.53,"endTime":1428.5,"body":"we're not interested in the curve"},{"startTime":1428.5,"endTime":1432.22,"body":"that has the same time"},{"startTime":1428.5,"endTime":1432.22,"body":"to get to the bottom"},{"startTime":1432.22,"endTime":1433.92,"body":"wherever you start from."},{"startTime":1433.92,"endTime":1436.89,"body":"What we're interested in"},{"startTime":1433.92,"endTime":1436.89,"body":"is if you have two points,"},{"startTime":1436.89,"endTime":1440.48,"body":"top on a bottom point"},{"startTime":1436.89,"endTime":1440.48,"body":"and you want a particle"},{"startTime":1440.48,"endTime":1443.02,"body":"that's falling along a wire between them,"},{"startTime":1443.02,"endTime":1445.36,"body":"you want it to get to the"},{"startTime":1443.02,"endTime":1445.36,"body":"bottom in the quickest time."},{"startTime":1445.36,"endTime":1446.62,"body":"What shape should that Y be?"},{"startTime":1446.62,"endTime":1448.24,"body":"What curve should that be?\""},{"startTime":1448.24,"endTime":1452.4,"body":"So you could think of various"},{"startTime":1448.24,"endTime":1452.4,"body":"guesses like parabola, maybe,"},{"startTime":1452.4,"endTime":1454.88,"body":"I dunno, lips, something like that."},{"startTime":1454.88,"endTime":1457.76,"body":"And Johann Bernouilli posed this problem,"},{"startTime":1457.76,"endTime":1461.8,"body":"sort of an open problem to mathematicians."},{"startTime":1461.8,"endTime":1465.19,"body":"What's the answer to this?"},{"startTime":1461.8,"endTime":1465.19,"body":"What curve should I pick?"},{"startTime":1465.19,"endTime":1466.68,"body":"And several people managed"},{"startTime":1466.68,"endTime":1469.763,"body":"to solve this problem quite quickly."},{"startTime":1471.85,"endTime":1475.19,"body":"So Johann himself solved it."},{"startTime":1475.19,"endTime":1477.435,"body":"Johann's brother, Jacob solved it."},{"startTime":1477.435,"endTime":1479.27,"body":"Leibniz, his girlfriend Leibniz,"},{"startTime":1479.27,"endTime":1481.5,"body":"one of the inventors of"},{"startTime":1479.27,"endTime":1481.5,"body":"the calculus solved it."},{"startTime":1481.5,"endTime":1484.51,"body":"And there was a mysterious anonymous entry"},{"startTime":1485.504,"endTime":1487.26,"body":"to this problem."},{"startTime":1487.26,"endTime":1489.37,"body":"And when Bernoulli saw the entry"},{"startTime":1489.37,"endTime":1493.337,"body":"and he saw the lovely mathematics"},{"startTime":1489.37,"endTime":1493.337,"body":"that it had used, he said,"},{"startTime":1493.337,"endTime":1494.61,"body":"\"I know."},{"startTime":1494.61,"endTime":1496.657,"body":"I know who did this (speaking"},{"startTime":1494.61,"endTime":1496.657,"body":"in foreign language)."},{"startTime":1499.39,"endTime":1501.6,"body":"We know the lion by his claw."},{"startTime":1501.6,"endTime":1502.7,"body":"The mathematics are so good,"},{"startTime":1502.7,"endTime":1504.23,"body":"it could only have been one person"},{"startTime":1504.23,"endTime":1506.45,"body":"who did this, Isaac Newton.\""},{"startTime":1506.45,"endTime":1507.93,"body":"And it was."},{"startTime":1507.93,"endTime":1510.1,"body":"So Newton submitted an anonymous entry,"},{"startTime":1510.1,"endTime":1513.033,"body":"and so all of these people"},{"startTime":1510.1,"endTime":1513.033,"body":"solved this problem."},{"startTime":1513.94,"endTime":1517.05,"body":"Guess what curve is the solution"},{"startTime":1517.05,"endTime":1518.7,"body":"of the Brachistochrone problem."},{"startTime":1518.7,"endTime":1520.39,"body":"It's a cycloid."},{"startTime":1520.39,"endTime":1521.85,"body":"Again, it's amazing."},{"startTime":1521.85,"endTime":1524.6,"body":"So a cycloid is wonderful."},{"startTime":1521.85,"endTime":1524.6,"body":"We all love the cycloid now."},{"startTime":1525.499,"endTime":1529.27,"body":"It is also, not only does it"},{"startTime":1525.499,"endTime":1529.27,"body":"reach the same amount of time,"},{"startTime":1529.27,"endTime":1530.68,"body":"but it's the best possible."},{"startTime":1530.68,"endTime":1532.873,"body":"It's the quickest time that"},{"startTime":1530.68,"endTime":1532.873,"body":"you could possibly have"},{"startTime":1532.873,"endTime":1535.25,"body":"of all the possible poles you could take"},{"startTime":1535.25,"endTime":1536.93,"body":"from one point to the bottom."},{"startTime":1536.93,"endTime":1539.323,"body":"So the cycloid, there"},{"startTime":1536.93,"endTime":1539.323,"body":"it is. It strikes again."},{"startTime":1540.3,"endTime":1545.3,"body":"Now I mentioned before that"},{"startTime":1540.3,"endTime":1545.3,"body":"Gilles de Roberval managed"},{"startTime":1546.36,"endTime":1550.1,"body":"to work out the area under a cycloid arch."},{"startTime":1550.1,"endTime":1552.21,"body":"And I want to show you his argument."},{"startTime":1552.21,"endTime":1556.06,"body":"And I will say that this"},{"startTime":1552.21,"endTime":1556.06,"body":"was before calculus,"},{"startTime":1556.06,"endTime":1558.613,"body":"it was before a lot of modern techniques."},{"startTime":1559.58,"endTime":1562.77,"body":"This argument though, it has good bones."},{"startTime":1562.77,"endTime":1565.23,"body":"It's basically the right idea."},{"startTime":1565.23,"endTime":1567.033,"body":"It's a really neat little argument."},{"startTime":1568.64,"endTime":1571.58,"body":"Nowadays we wouldn't say"},{"startTime":1568.64,"endTime":1571.58,"body":"it's totally rigorous,"},{"startTime":1571.58,"endTime":1573.27,"body":"but it can be made rigorous,"},{"startTime":1573.27,"endTime":1575.38,"body":"and that's what I mean by"},{"startTime":1573.27,"endTime":1575.38,"body":"saying, it's got good bones."},{"startTime":1575.38,"endTime":1578.24,"body":"You can make it rigorous"},{"startTime":1575.38,"endTime":1578.24,"body":"now with calculus,"},{"startTime":1578.24,"endTime":1579.51,"body":"and it still works,"},{"startTime":1579.51,"endTime":1580.89,"body":"but I wanted to show you it"},{"startTime":1580.89,"endTime":1584.33,"body":"'cause I think it's a very"},{"startTime":1580.89,"endTime":1584.33,"body":"lovely argument, really clever."},{"startTime":1584.33,"endTime":1586.19,"body":"So remember with a cycloid,"},{"startTime":1586.19,"endTime":1589.76,"body":"we've got a circle rolling along."},{"startTime":1589.76,"endTime":1592.98,"body":"And if you start with"},{"startTime":1589.76,"endTime":1592.98,"body":"a point at the bottom,"},{"startTime":1592.98,"endTime":1595.31,"body":"and you start rolling"},{"startTime":1592.98,"endTime":1595.31,"body":"along, then at that point,"},{"startTime":1595.31,"endTime":1597.26,"body":"at some point it will"},{"startTime":1595.31,"endTime":1597.26,"body":"reach the highest place"},{"startTime":1597.26,"endTime":1598.55,"body":"it could possibly go."},{"startTime":1598.55,"endTime":1603.21,"body":"And that's after the circle has"},{"startTime":1598.55,"endTime":1603.21,"body":"kind of gone half way round."},{"startTime":1603.21,"endTime":1605.5,"body":"So in the distance between the point being"},{"startTime":1605.5,"endTime":1608.516,"body":"at the bottom and the"},{"startTime":1605.5,"endTime":1608.516,"body":"point being at the top,"},{"startTime":1608.516,"endTime":1613.36,"body":"the circle has rolled half the"},{"startTime":1608.516,"endTime":1613.36,"body":"circumference along the road."},{"startTime":1613.36,"endTime":1615.79,"body":"So if the circle has radius R,"},{"startTime":1615.79,"endTime":1619.86,"body":"that point at the top is two"},{"startTime":1615.79,"endTime":1619.86,"body":"R the diameter above the road."},{"startTime":1619.86,"endTime":1621.77,"body":"And how far has the circle traveled?"},{"startTime":1621.77,"endTime":1623.32,"body":"It's traveled half the circumference."},{"startTime":1623.32,"endTime":1625.06,"body":"So it's half of two pie R,"},{"startTime":1625.06,"endTime":1626.73,"body":"so it's traveled a distance of pie R,"},{"startTime":1626.73,"endTime":1628.63,"body":"so just remember that for later."},{"startTime":1628.63,"endTime":1631.6,"body":"So what Roberval did is quite ingenious."},{"startTime":1631.6,"endTime":1633.07,"body":"He created something."},{"startTime":1633.07,"endTime":1635.9,"body":"So he took half the cycloid arch,"},{"startTime":1635.9,"endTime":1637.89,"body":"And then of course, if you"},{"startTime":1635.9,"endTime":1637.89,"body":"know the area under that,"},{"startTime":1637.89,"endTime":1640.62,"body":"you can double it and find the total area."},{"startTime":1640.62,"endTime":1642.29,"body":"So once he got here,"},{"startTime":1642.29,"endTime":1645.36,"body":"he's created this thing"},{"startTime":1642.29,"endTime":1645.36,"body":"called the companion curve."},{"startTime":1645.36,"endTime":1650.36,"body":"So you take a semicircle,"},{"startTime":1645.36,"endTime":1650.36,"body":"so your original circle,"},{"startTime":1650.84,"endTime":1652.87,"body":"we just take half of that"},{"startTime":1650.84,"endTime":1652.87,"body":"semicircle sitting there"},{"startTime":1652.87,"endTime":1657.48,"body":"at the beginning and a semicircle."},{"startTime":1657.48,"endTime":1660.81,"body":"It's made up of a whole"},{"startTime":1657.48,"endTime":1660.81,"body":"bunch of lines stacked"},{"startTime":1660.81,"endTime":1663.37,"body":"on top of each other, like any shape."},{"startTime":1663.37,"endTime":1665.42,"body":"And of course, if you want"},{"startTime":1663.37,"endTime":1665.42,"body":"to know the area of that,"},{"startTime":1665.42,"endTime":1667.68,"body":"you add up all the little lines"},{"startTime":1667.68,"endTime":1670.653,"body":"in some kind of infinitesimal"},{"startTime":1667.68,"endTime":1670.653,"body":"way that we won't go into."},{"startTime":1671.493,"endTime":1674.23,"body":"But this is the same shape we take."},{"startTime":1674.23,"endTime":1676.24,"body":"So then what we've got in that picture,"},{"startTime":1676.24,"endTime":1680.1,"body":"you've got the blue line,"},{"startTime":1676.24,"endTime":1680.1,"body":"that's your half cycloid arch,"},{"startTime":1680.1,"endTime":1681.71,"body":"and you're creating this new curve called"},{"startTime":1681.71,"endTime":1682.59,"body":"the companion curve."},{"startTime":1682.59,"endTime":1684.47,"body":"And what you do is,"},{"startTime":1684.47,"endTime":1687.266,"body":"you take at any point as you're going up,"},{"startTime":1687.266,"endTime":1690.4,"body":"you take the point on the cycloid arch"},{"startTime":1690.4,"endTime":1695.112,"body":"and you shift it to the"},{"startTime":1690.4,"endTime":1695.112,"body":"right by exactly the width"},{"startTime":1695.112,"endTime":1698.87,"body":"that is the width of the"},{"startTime":1695.112,"endTime":1698.87,"body":"semi circle at that point."},{"startTime":1698.87,"endTime":1700.33,"body":"So you can see all of these green lines"},{"startTime":1700.33,"endTime":1702.07,"body":"are just being shifted along,"},{"startTime":1702.07,"endTime":1704.87,"body":"and that's how far you move to right."},{"startTime":1704.87,"endTime":1708.71,"body":"And when you do this, you"},{"startTime":1704.87,"endTime":1708.71,"body":"create this companion curve."},{"startTime":1708.71,"endTime":1710.62,"body":"Now, it's actually a sin curve."},{"startTime":1710.62,"endTime":1712.93,"body":"We don't think Roberval"},{"startTime":1710.62,"endTime":1712.93,"body":"necessarily knew that,"},{"startTime":1712.93,"endTime":1714.68,"body":"but you can prove it nowadays"},{"startTime":1714.68,"endTime":1716.67,"body":"that that thing you produce"},{"startTime":1714.68,"endTime":1716.67,"body":"turns out to be a sin curve,"},{"startTime":1716.67,"endTime":1719.15,"body":"which is already quite cute."},{"startTime":1719.15,"endTime":1722.11,"body":"So remember, we're trying to"},{"startTime":1719.15,"endTime":1722.11,"body":"find something about the area."},{"startTime":1722.11,"endTime":1726.21,"body":"The area off this new shape,"},{"startTime":1726.21,"endTime":1728.14,"body":"we've made this kind"},{"startTime":1726.21,"endTime":1728.14,"body":"of little curvy shape,"},{"startTime":1728.14,"endTime":1731.87,"body":"the bit that's bounded by the cycloid arch"},{"startTime":1731.87,"endTime":1733.253,"body":"and this companion curve."},{"startTime":1734.87,"endTime":1737.12,"body":"There's something called"},{"startTime":1734.87,"endTime":1737.12,"body":"Cavalieri principle,"},{"startTime":1737.12,"endTime":1739.65,"body":"which says, if you've got two shapes"},{"startTime":1739.65,"endTime":1741.95,"body":"and whenever you take"},{"startTime":1739.65,"endTime":1741.95,"body":"a slice through them,"},{"startTime":1741.95,"endTime":1745.27,"body":"the line you get has exactly"},{"startTime":1741.95,"endTime":1745.27,"body":"the same length in each shape,"},{"startTime":1745.27,"endTime":1747.27,"body":"then areas of those shapes are the same."},{"startTime":1748.23,"endTime":1752.95,"body":"And that's something you"},{"startTime":1748.23,"endTime":1752.95,"body":"can prove with calculus,"},{"startTime":1752.95,"endTime":1755.79,"body":"but this was in use before that,"},{"startTime":1755.79,"endTime":1759.08,"body":"and it's plausible, but also happily true."},{"startTime":1759.08,"endTime":1763.3,"body":"So with that, given the way"},{"startTime":1759.08,"endTime":1763.3,"body":"we've constructed these shapes,"},{"startTime":1763.3,"endTime":1765.78,"body":"it's clear that the semicircle"},{"startTime":1763.3,"endTime":1765.78,"body":"has got the same area"},{"startTime":1765.78,"endTime":1770.73,"body":"as this new strange companion curve shape"},{"startTime":1770.73,"endTime":1772.49,"body":"because we have just got exactly,"},{"startTime":1772.49,"endTime":1774.8,"body":"by creation, we've done this,"},{"startTime":1774.8,"endTime":1776.88,"body":"we've made those lines"},{"startTime":1774.8,"endTime":1776.88,"body":"be all the same length."},{"startTime":1776.88,"endTime":1778.36,"body":"That's how we constructed it."},{"startTime":1778.36,"endTime":1780.52,"body":"So those two things are the same area."},{"startTime":1780.52,"endTime":1782.87,"body":"So I've labeled them a for the area"},{"startTime":1782.87,"endTime":1784.89,"body":"of the companion kill bit"},{"startTime":1784.89,"endTime":1787.51,"body":"and S for the area of the"},{"startTime":1784.89,"endTime":1787.51,"body":"semicircle, they are the same."},{"startTime":1787.51,"endTime":1789.29,"body":"And if the circle has radius R,"},{"startTime":1789.29,"endTime":1791.68,"body":"the total area would be pie R squared."},{"startTime":1791.68,"endTime":1795.72,"body":"So the area of the semicircle"},{"startTime":1791.68,"endTime":1795.72,"body":"is half pie R squared."},{"startTime":1795.72,"endTime":1798.97,"body":"So that's one bit of the"},{"startTime":1795.72,"endTime":1798.97,"body":"area we're interested in."},{"startTime":1798.97,"endTime":1801.06,"body":"We'll just move that over"},{"startTime":1798.97,"endTime":1801.06,"body":"to the side for a moment."},{"startTime":1801.06,"endTime":1802.38,"body":"Let's think about now,"},{"startTime":1802.38,"endTime":1807.15,"body":"I've enclosed that half cycloid"},{"startTime":1802.38,"endTime":1807.15,"body":"as you see in a rectangle."},{"startTime":1807.15,"endTime":1810.84,"body":"And here's a rectangle,"},{"startTime":1807.15,"endTime":1810.84,"body":"that same rectangle,"},{"startTime":1810.84,"endTime":1814.09,"body":"I've got the companion"},{"startTime":1810.84,"endTime":1814.09,"body":"curve in the middle of it,"},{"startTime":1814.09,"endTime":1817.41,"body":"and lovely blue and yellow colors"},{"startTime":1817.41,"endTime":1821.07,"body":"'cause they're best colors, and"},{"startTime":1817.41,"endTime":1821.07,"body":"there's the companion curve."},{"startTime":1821.07,"endTime":1825.41,"body":"And you can sort of plausibly"},{"startTime":1821.07,"endTime":1825.41,"body":"see that bisymmetry, right,"},{"startTime":1825.41,"endTime":1829.22,"body":"we made this thing by rolling"},{"startTime":1825.41,"endTime":1829.22,"body":"a circle from left to right."},{"startTime":1829.22,"endTime":1831.71,"body":"If you turn it the other way"},{"startTime":1829.22,"endTime":1831.71,"body":"up and roll it from right"},{"startTime":1831.71,"endTime":1834.3,"body":"to left, you get the same curve."},{"startTime":1834.3,"endTime":1836.78,"body":"And by the symmetry of the situation,"},{"startTime":1836.78,"endTime":1839.83,"body":"that means that this rectangle"},{"startTime":1836.78,"endTime":1839.83,"body":"has a rotational symmetry."},{"startTime":1839.83,"endTime":1841.96,"body":"That when you turn it around,"},{"startTime":1839.83,"endTime":1841.96,"body":"you get the same picture."},{"startTime":1841.96,"endTime":1844.67,"body":"And that means that this"},{"startTime":1841.96,"endTime":1844.67,"body":"curve divides the area"},{"startTime":1844.67,"endTime":1847.81,"body":"of the rectangle exactly in half."},{"startTime":1847.81,"endTime":1851.633,"body":"So the blue area is the"},{"startTime":1847.81,"endTime":1851.633,"body":"same as the yellow area."},{"startTime":1853.01,"endTime":1854.96,"body":"What is the area of them then?"},{"startTime":1854.96,"endTime":1856.853,"body":"Well, we talked about this rectangle."},{"startTime":1857.92,"endTime":1860.63,"body":"The base of it is pie"},{"startTime":1857.92,"endTime":1860.63,"body":"R because it's how far"},{"startTime":1860.63,"endTime":1863.51,"body":"the circle rolls when it's"},{"startTime":1860.63,"endTime":1863.51,"body":"gone half the way round."},{"startTime":1863.51,"endTime":1865.71,"body":"And the height of it"},{"startTime":1863.51,"endTime":1865.71,"body":"is two R, the diameter."},{"startTime":1866.62,"endTime":1868.57,"body":"And so, the area of the whole rectangle"},{"startTime":1868.57,"endTime":1871.67,"body":"would be base times"},{"startTime":1868.57,"endTime":1871.67,"body":"height, two R times pie R,"},{"startTime":1871.67,"endTime":1874.21,"body":"which is two pie R squared."},{"startTime":1874.21,"endTime":1875.23,"body":"And we want half of that."},{"startTime":1875.23,"endTime":1878.2,"body":"So the bits I've labeled"},{"startTime":1875.23,"endTime":1878.2,"body":"B and C have equal area."},{"startTime":1878.2,"endTime":1879.95,"body":"Each of them is half of that,"},{"startTime":1879.95,"endTime":1881.933,"body":"so each of them is pie R squared."},{"startTime":1882.82,"endTime":1884.15,"body":"Okay. We're nearly there."},{"startTime":1884.15,"endTime":1886.453,"body":"All we need now is to"},{"startTime":1884.15,"endTime":1886.453,"body":"add together A and B."},{"startTime":1887.79,"endTime":1890.34,"body":"So what we want for our half cycloid,"},{"startTime":1890.34,"endTime":1892.7,"body":"we've got the bit under"},{"startTime":1890.34,"endTime":1892.7,"body":"the companion curve,"},{"startTime":1892.7,"endTime":1894.323,"body":"and then we've got the bit"},{"startTime":1892.7,"endTime":1894.323,"body":"between the companion curve"},{"startTime":1894.323,"endTime":1895.7,"body":"and the cycloid."},{"startTime":1895.7,"endTime":1897.28,"body":"So that's A plus B."},{"startTime":1897.28,"endTime":1900.45,"body":"A is half pie R squared,"},{"startTime":1897.28,"endTime":1900.45,"body":"B is pie R squared."},{"startTime":1900.45,"endTime":1901.44,"body":"Add them together."},{"startTime":1901.44,"endTime":1903.8,"body":"You get one and a half pie R squared,"},{"startTime":1903.8,"endTime":1905.1,"body":"and that's half the answer we want"},{"startTime":1905.1,"endTime":1907.07,"body":"because we want the whole arch."},{"startTime":1907.07,"endTime":1909.51,"body":"We know half of it is one and a half."},{"startTime":1909.51,"endTime":1913.01,"body":"So all of it is going to"},{"startTime":1909.51,"endTime":1913.01,"body":"be three pie R squared."},{"startTime":1913.01,"endTime":1915.76,"body":"Hey, so that's what Roberval did."},{"startTime":1915.76,"endTime":1917.71,"body":"And I really like that little argument."},{"startTime":1920.81,"endTime":1922.17,"body":"Many other people worked on cycloids."},{"startTime":1922.17,"endTime":1924.69,"body":"If I want to talk about"},{"startTime":1922.17,"endTime":1924.69,"body":"any other roulettes,"},{"startTime":1924.69,"endTime":1927.08,"body":"this will be my last slide on cycloids."},{"startTime":1927.08,"endTime":1928.45,"body":"Many other people worked on them."},{"startTime":1928.45,"endTime":1930.83,"body":"So you might recognize"},{"startTime":1928.45,"endTime":1930.83,"body":"the chap on the left there"},{"startTime":1930.83,"endTime":1932.12,"body":"with St. Paul's Cathedral behind him,"},{"startTime":1932.12,"endTime":1935.68,"body":"that's sir, Christopher"},{"startTime":1932.12,"endTime":1935.68,"body":"Wren, he worked on cycloids,"},{"startTime":1935.68,"endTime":1938.92,"body":"and he managed to prove that"},{"startTime":1935.68,"endTime":1938.92,"body":"the arch length of a cycloid"},{"startTime":1938.92,"endTime":1942.47,"body":"is exactly another lovely whole"},{"startTime":1938.92,"endTime":1942.47,"body":"number multiple eight times"},{"startTime":1942.47,"endTime":1945.09,"body":"the radius of the circle that creates it."},{"startTime":1945.09,"endTime":1947.098,"body":"So that's very nice."},{"startTime":1947.098,"endTime":1949.08,"body":"Cycloids didn't just catch the imagination"},{"startTime":1949.08,"endTime":1950.67,"body":"of mathematicians either."},{"startTime":1950.67,"endTime":1955.436,"body":"They appear in several very"},{"startTime":1950.67,"endTime":1955.436,"body":"famous works of literature,"},{"startTime":1955.436,"endTime":1958.683,"body":"which we all have either"},{"startTime":1955.436,"endTime":1958.683,"body":"read or think we've read."},{"startTime":1960.03,"endTime":1964.29,"body":"So \"Moby Dick\", there's a"},{"startTime":1960.03,"endTime":1964.29,"body":"whole bit about cycloids"},{"startTime":1964.29,"endTime":1967.23,"body":"in \"Moby Dick\", which is very interesting."},{"startTime":1967.23,"endTime":1968.98,"body":"They're mentioned in \"Tristram Shandy\"?"},{"startTime":1968.98,"endTime":1970.75,"body":"They're mentioned in \"Gulliver Travels\"."},{"startTime":1970.75,"endTime":1973.1,"body":"So obviously, these are things that are"},{"startTime":1973.1,"endTime":1975.67,"body":"in people's minds this time."},{"startTime":1975.67,"endTime":1978.23,"body":"And so, there's this advert here."},{"startTime":1978.23,"endTime":1981.84,"body":"Next year, it's going to"},{"startTime":1978.23,"endTime":1981.84,"body":"be the 300th anniversary"},{"startTime":1981.84,"endTime":1984.22,"body":"of Christopher Wren's death."},{"startTime":1984.22,"endTime":1987.3,"body":"And he was a Gresham"},{"startTime":1984.22,"endTime":1987.3,"body":"professor, not of geometry."},{"startTime":1987.3,"endTime":1988.3,"body":"He could easily have been that,"},{"startTime":1988.3,"endTime":1990.9,"body":"but he was Gresham professor of astronomy."},{"startTime":1990.9,"endTime":1994.22,"body":"And so Gresham College are"},{"startTime":1990.9,"endTime":1994.22,"body":"going to be doing some events"},{"startTime":1994.22,"endTime":1995.53,"body":"around Christopher Wren."},{"startTime":1995.53,"endTime":1998.16,"body":"And so if you're around"},{"startTime":1995.53,"endTime":1998.16,"body":"in 2023 here in London,"},{"startTime":1998.16,"endTime":1999.47,"body":"or even if you want to watch online,"},{"startTime":1999.47,"endTime":2001.57,"body":"there'll be some events"},{"startTime":1999.47,"endTime":2001.57,"body":"around Christopher Wren"},{"startTime":2001.57,"endTime":2004.71,"body":"to mark that 300 years."},{"startTime":2004.71,"endTime":2006.72,"body":"And the other thing next year is,"},{"startTime":2006.72,"endTime":2008.227,"body":"obviously you are thinking now,"},{"startTime":2008.227,"endTime":2009.38,"body":"\"My goodness, these links"},{"startTime":2009.38,"endTime":2010.518,"body":"between mathematics and literature,"},{"startTime":2010.518,"endTime":2013.33,"body":"if only someone would write a book"},{"startTime":2013.33,"endTime":2014.23,"body":"about the wonderful links"},{"startTime":2014.23,"endTime":2015.9,"body":"between mathematics and literature."},{"startTime":2015.9,"endTime":2018.67,"body":"And if you are thinking"},{"startTime":2015.9,"endTime":2018.67,"body":"that, surely you are,"},{"startTime":2018.67,"endTime":2021.4,"body":"then you will delighted"},{"startTime":2018.67,"endTime":2021.4,"body":"to know that in 2023,"},{"startTime":2021.4,"endTime":2023.433,"body":"my book on mathematics and literature,"},{"startTime":2024.46,"endTime":2025.9,"body":"which is called \"Once Upon a Prime\","},{"startTime":2025.9,"endTime":2028.05,"body":"already it's great"},{"startTime":2025.9,"endTime":2028.05,"body":"title, right, is coming."},{"startTime":2028.05,"endTime":2030.35,"body":"So that's for next year."},{"startTime":2030.35,"endTime":2031.62,"body":"Okay, cycloids."},{"startTime":2031.62,"endTime":2034.73,"body":"We've talked about"},{"startTime":2031.62,"endTime":2034.73,"body":"cycloids, they're brilliant."},{"startTime":2034.73,"endTime":2036.45,"body":"We roll a circle along the line."},{"startTime":2036.45,"endTime":2039.1,"body":"What else can you roll a circle along?"},{"startTime":2039.1,"endTime":2042.03,"body":"You can roll it along another circle."},{"startTime":2042.03,"endTime":2044.68,"body":"And when you do that,"},{"startTime":2042.03,"endTime":2044.68,"body":"you've got to decide,"},{"startTime":2044.68,"endTime":2047.42,"body":"am I going to roll it along"},{"startTime":2044.68,"endTime":2047.42,"body":"the outside of the circle?"},{"startTime":2047.42,"endTime":2050.81,"body":"In which case you might get"},{"startTime":2047.42,"endTime":2050.81,"body":"something like this called"},{"startTime":2050.81,"endTime":2053.08,"body":"a cardioid 'cause it's"},{"startTime":2050.81,"endTime":2053.08,"body":"sort of heart shaped,"},{"startTime":2053.08,"endTime":2055.86,"body":"or am I going to roll it"},{"startTime":2053.08,"endTime":2055.86,"body":"along the inside of a circle?"},{"startTime":2055.86,"endTime":2056.693,"body":"And those things."},{"startTime":2056.693,"endTime":2059.55,"body":"So outside of a circle, epicycloid,"},{"startTime":2059.55,"endTime":2061.823,"body":"inside of a circle, hypocycloid."},{"startTime":2062.95,"endTime":2065.13,"body":"So you've got some decisions to make then."},{"startTime":2065.13,"endTime":2069.8,"body":"Do I roll my circle along"},{"startTime":2065.13,"endTime":2069.8,"body":"a circle of the same size?"},{"startTime":2069.8,"endTime":2072.0,"body":"If you do that, you get"},{"startTime":2069.8,"endTime":2072.0,"body":"this thing, cardiod,"},{"startTime":2072.94,"endTime":2076.66,"body":"or shall I make the thing"},{"startTime":2072.94,"endTime":2076.66,"body":"I'm rolling it along bigger?"},{"startTime":2076.66,"endTime":2078.23,"body":"And if you take sort of"},{"startTime":2076.66,"endTime":2078.23,"body":"whole number multiples"},{"startTime":2078.23,"endTime":2080.81,"body":"of the radius, you get"},{"startTime":2078.23,"endTime":2080.81,"body":"some quite good things."},{"startTime":2080.81,"endTime":2082.01,"body":"So there's your cardiod."},{"startTime":2083.359,"endTime":2084.81,"body":"Perhaps you could draw your circle"},{"startTime":2084.81,"endTime":2087.08,"body":"along one that has twice the radius."},{"startTime":2087.08,"endTime":2089.29,"body":"Then you get what's called a nephroid."},{"startTime":2089.29,"endTime":2091.883,"body":"So that's from the word"},{"startTime":2089.29,"endTime":2091.883,"body":"nephroid from kidney."},{"startTime":2092.97,"endTime":2095.61,"body":"Well, it looks a bit"},{"startTime":2092.97,"endTime":2095.61,"body":"like a kidney, I suppose."},{"startTime":2095.61,"endTime":2098.273,"body":"So you'd find that if you do this,"},{"startTime":2099.12,"endTime":2101.91,"body":"with the nephroid, you"},{"startTime":2099.12,"endTime":2101.91,"body":"get two arches with this."},{"startTime":2101.91,"endTime":2103.67,"body":"Cardiod, you get one arch,"},{"startTime":2103.67,"endTime":2106.95,"body":"and you can, if you take"},{"startTime":2103.67,"endTime":2106.95,"body":"higher and higher multiples"},{"startTime":2106.95,"endTime":2110.343,"body":"of your radius to roll along,"},{"startTime":2106.95,"endTime":2110.343,"body":"you get more and more arches."},{"startTime":2112.898,"endTime":2114.41,"body":"What about if you roll your"},{"startTime":2112.898,"endTime":2114.41,"body":"circle along the inside"},{"startTime":2114.41,"endTime":2115.89,"body":"of a circle?"},{"startTime":2115.89,"endTime":2119.53,"body":"So I'm not going to talk in"},{"startTime":2115.89,"endTime":2119.53,"body":"detail about all these curves,"},{"startTime":2119.53,"endTime":2124.53,"body":"but if you roll your circle"},{"startTime":2119.53,"endTime":2124.53,"body":"on the inside of a circle,"},{"startTime":2124.69,"endTime":2127.97,"body":"then you start to get things"},{"startTime":2124.69,"endTime":2127.97,"body":"that look a bit like if anyone"},{"startTime":2127.97,"endTime":2130.45,"body":"has a kid, had a toy like a spirograph"},{"startTime":2130.45,"endTime":2132.86,"body":"or various other toys like this,"},{"startTime":2132.86,"endTime":2136.18,"body":"what you do is you you get a little circle"},{"startTime":2136.18,"endTime":2139.74,"body":"and you put your pencil in to a small hole"},{"startTime":2139.74,"endTime":2142.42,"body":"just inside the rim of that little circle,"},{"startTime":2142.42,"endTime":2143.935,"body":"and then you kind of"},{"startTime":2142.42,"endTime":2143.935,"body":"roll it round and round"},{"startTime":2143.935,"endTime":2145.81,"body":"and round the inside and you"},{"startTime":2143.935,"endTime":2145.81,"body":"get these beautiful pictures."},{"startTime":2145.81,"endTime":2147.68,"body":"I've got some on my top today."},{"startTime":2147.68,"endTime":2149.21,"body":"And so, these things, well now,"},{"startTime":2149.21,"endTime":2151.897,"body":"they're not quite hypocycloids."},{"startTime":2152.97,"endTime":2155.72,"body":"The reason is that the point"},{"startTime":2152.97,"endTime":2155.72,"body":"who's part of your tracing"},{"startTime":2155.72,"endTime":2159.63,"body":"is not on the circumference,"},{"startTime":2155.72,"endTime":2159.63,"body":"it's a little inside."},{"startTime":2159.63,"endTime":2162.16,"body":"So there's a word for when you do that."},{"startTime":2162.16,"endTime":2164.87,"body":"And it's this word? Well,"},{"startTime":2162.16,"endTime":2164.87,"body":"trocoid is involved."},{"startTime":2164.87,"endTime":2167.397,"body":"So when we did cycloids,"},{"startTime":2164.87,"endTime":2167.397,"body":"circles rolling along the line,"},{"startTime":2167.397,"endTime":2171.8,"body":"if that involved a point"},{"startTime":2167.397,"endTime":2171.8,"body":"on the circumference,"},{"startTime":2171.8,"endTime":2173.75,"body":"if you take a point not"},{"startTime":2171.8,"endTime":2173.75,"body":"quite on the circumference,"},{"startTime":2173.75,"endTime":2175.55,"body":"a little bit inside, maybe,"},{"startTime":2175.55,"endTime":2177.83,"body":"or sometimes a little"},{"startTime":2175.55,"endTime":2177.83,"body":"bit outside, if you want,"},{"startTime":2177.83,"endTime":2179.52,"body":"what you get isn't a"},{"startTime":2177.83,"endTime":2179.52,"body":"cycloid, but a trocoid."},{"startTime":2179.52,"endTime":2181.96,"body":"There are various possibilities"},{"startTime":2179.52,"endTime":2181.96,"body":"for what you get based"},{"startTime":2181.96,"endTime":2185.74,"body":"on how far away from the"},{"startTime":2181.96,"endTime":2185.74,"body":"circumference your point is."},{"startTime":2185.74,"endTime":2189.12,"body":"So by an analogy, you've"},{"startTime":2185.74,"endTime":2189.12,"body":"got epicycloids rolling"},{"startTime":2189.12,"endTime":2190.823,"body":"along the outside of a circle."},{"startTime":2192.05,"endTime":2194.36,"body":"Again, if you have the point not quite"},{"startTime":2194.36,"endTime":2196.41,"body":"on the circumference"},{"startTime":2194.36,"endTime":2196.41,"body":"of the rolling circle,"},{"startTime":2196.41,"endTime":2198.39,"body":"then it becomes an epitrocoid."},{"startTime":2198.39,"endTime":2202.2,"body":"And so, what spirographs produce"},{"startTime":2198.39,"endTime":2202.2,"body":"is not quite hypocycloids,"},{"startTime":2202.2,"endTime":2203.88,"body":"we're not following the"},{"startTime":2202.2,"endTime":2203.88,"body":"rim of this little circle"},{"startTime":2203.88,"endTime":2206.307,"body":"that's rolling around"},{"startTime":2203.88,"endTime":2206.307,"body":"with somewhere inside."},{"startTime":2206.307,"endTime":2208.93,"body":"Just inside, and that's where"},{"startTime":2206.307,"endTime":2208.93,"body":"you put your pencil through."},{"startTime":2208.93,"endTime":2211.23,"body":"And so, actually what"},{"startTime":2208.93,"endTime":2211.23,"body":"these spirographs produce"},{"startTime":2211.23,"endTime":2213.11,"body":"are hypotrochoids."},{"startTime":2213.11,"endTime":2215.7,"body":"So you can go and tell people that"},{"startTime":2215.7,"endTime":2219.26,"body":"because then you sound"},{"startTime":2215.7,"endTime":2219.26,"body":"awfully knowledgeable."},{"startTime":2219.26,"endTime":2222.1,"body":"Actually I think we'll"},{"startTime":2219.26,"endTime":2222.1,"body":"find it's a hypotrochoid,"},{"startTime":2222.1,"endTime":2223.98,"body":"that's how to make friends, right?"},{"startTime":2223.98,"endTime":2226.44,"body":"So these are lovely toys,"},{"startTime":2226.44,"endTime":2229.492,"body":"and you can produce some"},{"startTime":2226.44,"endTime":2229.492,"body":"very pretty pictures."},{"startTime":2229.492,"endTime":2232.403,"body":"What I want to talk about"},{"startTime":2229.492,"endTime":2232.403,"body":"though is epicycloids,"},{"startTime":2233.67,"endTime":2237.53,"body":"especially, so the picture at"},{"startTime":2233.67,"endTime":2237.53,"body":"the very start of the lecture"},{"startTime":2237.53,"endTime":2240.263,"body":"was an example of a cardioid."},{"startTime":2242.63,"endTime":2246.35,"body":"And I just wanted to give"},{"startTime":2242.63,"endTime":2246.35,"body":"you a nice couple of facts"},{"startTime":2246.35,"endTime":2248.78,"body":"about these arches 'cause"},{"startTime":2246.35,"endTime":2248.78,"body":"we talked about what happens"},{"startTime":2248.78,"endTime":2250.216,"body":"with the cycloid."},{"startTime":2250.216,"endTime":2253.93,"body":"And know that the area"},{"startTime":2250.216,"endTime":2253.93,"body":"under the cycloid arch"},{"startTime":2253.93,"endTime":2257.35,"body":"is exactly three times that of"},{"startTime":2253.93,"endTime":2257.35,"body":"the circle that produces it."},{"startTime":2257.35,"endTime":2259.0,"body":"And Christopher Wren proved"},{"startTime":2259.0,"endTime":2260.75,"body":"that the length of a cycloid arch"},{"startTime":2260.75,"endTime":2263.03,"body":"is exactly eight times the radius."},{"startTime":2263.03,"endTime":2265.47,"body":"And you can actually derive formulae"},{"startTime":2265.47,"endTime":2267.44,"body":"for these epicycloids as well."},{"startTime":2267.44,"endTime":2269.52,"body":"So here, this is the nephroid, right?"},{"startTime":2269.52,"endTime":2273.21,"body":"It's produced by rolling a"},{"startTime":2269.52,"endTime":2273.21,"body":"circle of radius R, let's say"},{"startTime":2273.21,"endTime":2276.383,"body":"along one of radius two R."},{"startTime":2276.383,"endTime":2280.71,"body":"In general, if you have a"},{"startTime":2276.383,"endTime":2280.71,"body":"rolling circle of radius R,"},{"startTime":2280.71,"endTime":2282.83,"body":"and let's think about what"},{"startTime":2280.71,"endTime":2282.83,"body":"happens when you roll it"},{"startTime":2282.83,"endTime":2287.83,"body":"along a bigger one that has"},{"startTime":2282.83,"endTime":2287.83,"body":"radius K times R for sum K."},{"startTime":2288.15,"endTime":2290.32,"body":"So for the nephroid, K equals two,"},{"startTime":2290.32,"endTime":2292.013,"body":"for the cardioid, K equals one."},{"startTime":2293.51,"endTime":2297.23,"body":"For this guy, which I"},{"startTime":2293.51,"endTime":2297.23,"body":"don't think it has a name,"},{"startTime":2297.23,"endTime":2299.86,"body":"but it is an epicycloid with five arches,"},{"startTime":2299.86,"endTime":2301.3,"body":"K equals five in that case."},{"startTime":2301.3,"endTime":2303.97,"body":"So the number of arches is the amount,"},{"startTime":2303.97,"endTime":2307.32,"body":"the multiple of the radius"},{"startTime":2303.97,"endTime":2307.32,"body":"with the biggest circle."},{"startTime":2307.32,"endTime":2312.32,"body":"So there are some facts about"},{"startTime":2307.32,"endTime":2312.32,"body":"the area under the arches"},{"startTime":2313.47,"endTime":2314.38,"body":"and the length."},{"startTime":2314.38,"endTime":2316.9,"body":"So for a cardiod, so we"},{"startTime":2314.38,"endTime":2316.9,"body":"are looking at the area"},{"startTime":2316.9,"endTime":2321.89,"body":"kind of between the fixed"},{"startTime":2316.9,"endTime":2321.89,"body":"inside circle and the curve."},{"startTime":2321.89,"endTime":2322.9,"body":"And the cardioid, I guess,"},{"startTime":2322.9,"endTime":2325.163,"body":"is kind of one arch really wrapped around."},{"startTime":2326.17,"endTime":2328.901,"body":"And it can be shown that"},{"startTime":2326.17,"endTime":2328.901,"body":"the length of the cardioid"},{"startTime":2328.901,"endTime":2332.61,"body":"is 16 R, 16 times the original radius,"},{"startTime":2332.61,"endTime":2335.41,"body":"and there are nice whole"},{"startTime":2332.61,"endTime":2335.41,"body":"numbers and the shaded area,"},{"startTime":2335.41,"endTime":2337.047,"body":"so the area between the fixed circle"},{"startTime":2337.047,"endTime":2341.47,"body":"and the the cardioid turned"},{"startTime":2337.047,"endTime":2341.47,"body":"out to be five pie R squared,"},{"startTime":2341.47,"endTime":2344.01,"body":"five times the area of the rolling circle."},{"startTime":2344.01,"endTime":2345.32,"body":"If you want the whole cardiod,"},{"startTime":2345.32,"endTime":2346.917,"body":"if you want to fill in that bit"},{"startTime":2346.917,"endTime":2348.11,"body":"and just get the whole cardiod,"},{"startTime":2348.11,"endTime":2350.81,"body":"you could add in another pie"},{"startTime":2348.11,"endTime":2350.81,"body":"R squared if you want it."},{"startTime":2351.84,"endTime":2356.21,"body":"Alright, so there's a"},{"startTime":2351.84,"endTime":2356.21,"body":"result in general that says,"},{"startTime":2356.21,"endTime":2357.44,"body":"if you are rolling around something"},{"startTime":2357.44,"endTime":2360.71,"body":"of radius K times the rolling thing,"},{"startTime":2360.71,"endTime":2363.04,"body":"then there's this formula for arch length,"},{"startTime":2363.04,"endTime":2366.765,"body":"eight R times something,"},{"startTime":2363.04,"endTime":2366.765,"body":"one plus one over K,"},{"startTime":2366.765,"endTime":2370.31,"body":"and C for the cardioid,"},{"startTime":2366.765,"endTime":2370.31,"body":"K equal one, right?"},{"startTime":2370.31,"endTime":2374.197,"body":"So one plus one is two. So"},{"startTime":2370.31,"endTime":2374.197,"body":"that gives you your 16 R, yeah?"},{"startTime":2374.197,"endTime":2378.54,"body":"And the area, so the bit that's"},{"startTime":2374.197,"endTime":2378.54,"body":"in blue on that slide there,"},{"startTime":2378.54,"endTime":2382.45,"body":"the area is pie R squared times something."},{"startTime":2382.45,"endTime":2384.76,"body":"And the something is"},{"startTime":2382.45,"endTime":2384.76,"body":"three plus two over K."},{"startTime":2384.76,"endTime":2387.79,"body":"So again, cardioid K equals"},{"startTime":2384.76,"endTime":2387.79,"body":"one, three plus two is five."},{"startTime":2387.79,"endTime":2390.81,"body":"So we retrieve the thing we already know."},{"startTime":2390.81,"endTime":2391.643,"body":"But you can work these out,"},{"startTime":2391.643,"endTime":2394.95,"body":"and this is true for any these epicycloids"},{"startTime":2394.95,"endTime":2396.189,"body":"where you're rolling along a circle"},{"startTime":2396.189,"endTime":2399.553,"body":"a whole number multiple"},{"startTime":2396.189,"endTime":2399.553,"body":"K times the radius."},{"startTime":2400.75,"endTime":2403.53,"body":"So we could work this out if"},{"startTime":2400.75,"endTime":2403.53,"body":"we wanted to for this five."},{"startTime":2403.53,"endTime":2405.56,"body":"This thing looks a bit like a flower,"},{"startTime":2405.56,"endTime":2408.64,"body":"and K is getting to be a bit bigger there."},{"startTime":2408.64,"endTime":2410.43,"body":"And so, if we were to work this out,"},{"startTime":2410.43,"endTime":2413.16,"body":"one plus one over five,"},{"startTime":2410.43,"endTime":2413.16,"body":"it's a bit more than one."},{"startTime":2413.16,"endTime":2417.16,"body":"So the arch length for each"},{"startTime":2413.16,"endTime":2417.16,"body":"arch, it'd be eight R and a bit."},{"startTime":2417.16,"endTime":2420.95,"body":"And the area under each"},{"startTime":2417.16,"endTime":2420.95,"body":"R would be K is big."},{"startTime":2420.95,"endTime":2422.42,"body":"So two over K is small,"},{"startTime":2422.42,"endTime":2427.01,"body":"so it's three pie R squared and"},{"startTime":2422.42,"endTime":2427.01,"body":"a bit, area under each arch."},{"startTime":2427.01,"endTime":2430.83,"body":"Now, what happens when K gets"},{"startTime":2427.01,"endTime":2430.83,"body":"bigger and bigger and bigger?"},{"startTime":2430.83,"endTime":2431.67,"body":"Well, then, what you're looking at"},{"startTime":2431.67,"endTime":2434.85,"body":"is you're rolling a circle"},{"startTime":2431.67,"endTime":2434.85,"body":"along a circle whose radius"},{"startTime":2434.85,"endTime":2436.45,"body":"is getting larger and"},{"startTime":2434.85,"endTime":2436.45,"body":"larger and larger and larger"},{"startTime":2436.45,"endTime":2439.28,"body":"and larger and larger and"},{"startTime":2436.45,"endTime":2439.28,"body":"larger until at the limit,"},{"startTime":2439.28,"endTime":2442.94,"body":"it's a straight line, a circle"},{"startTime":2439.28,"endTime":2442.94,"body":"of infinite radius, right?"},{"startTime":2442.94,"endTime":2444.13,"body":"A straight line."},{"startTime":2444.13,"endTime":2448.68,"body":"So we ought to, it ought"},{"startTime":2444.13,"endTime":2448.68,"body":"to be that we might hope"},{"startTime":2448.68,"endTime":2452.26,"body":"when K gets very large"},{"startTime":2448.68,"endTime":2452.26,"body":"and tends to infinity,"},{"startTime":2452.26,"endTime":2457.26,"body":"one over K becomes very small"},{"startTime":2452.26,"endTime":2457.26,"body":"and gets closer to zero."},{"startTime":2457.27,"endTime":2460.936,"body":"If we actually do that, the"},{"startTime":2457.27,"endTime":2460.936,"body":"limiting case of these things"},{"startTime":2460.936,"endTime":2463.623,"body":"should give us what we"},{"startTime":2460.936,"endTime":2463.623,"body":"got for the cycloid."},{"startTime":2463.623,"endTime":2465.16,"body":"And that exactly does happen."},{"startTime":2465.16,"endTime":2467.49,"body":"If you put K, whose is infinity,"},{"startTime":2467.49,"endTime":2468.52,"body":"or let K go to infinity,"},{"startTime":2468.52,"endTime":2471.84,"body":"then one over K becomes essentially zero,"},{"startTime":2471.84,"endTime":2474.25,"body":"and you retrieve eight"},{"startTime":2471.84,"endTime":2474.25,"body":"R for the arch length,"},{"startTime":2474.25,"endTime":2475.76,"body":"exactly what Wren found."},{"startTime":2475.76,"endTime":2478.12,"body":"If you, in the area formula,"},{"startTime":2478.12,"endTime":2481.48,"body":"if you let K get very big,"},{"startTime":2478.12,"endTime":2481.48,"body":"two over K becomes close"},{"startTime":2481.48,"endTime":2483.76,"body":"and close to zero, and in the limit,"},{"startTime":2483.76,"endTime":2486.75,"body":"that formula becomes three pie R squared."},{"startTime":2486.75,"endTime":2488.35,"body":"That's exactly what we've"},{"startTime":2486.75,"endTime":2488.35,"body":"got for the cycloid."},{"startTime":2488.35,"endTime":2489.29,"body":"That's really nice."},{"startTime":2489.29,"endTime":2491.62,"body":"In the limiting case of an epicycloid,"},{"startTime":2491.62,"endTime":2493.32,"body":"you get back to being a cycloid."},{"startTime":2494.345,"endTime":2496.87,"body":"There's a really nice way to produce,"},{"startTime":2496.87,"endTime":2500.08,"body":"oh, there it is, a nice"},{"startTime":2496.87,"endTime":2500.08,"body":"way to produce a cardiod."},{"startTime":2500.08,"endTime":2502.32,"body":"I won't go into the details,"},{"startTime":2500.08,"endTime":2502.32,"body":"but you can try this."},{"startTime":2502.32,"endTime":2504.35,"body":"You can try this at home."},{"startTime":2504.35,"endTime":2505.577,"body":"What you do is you get a circle,"},{"startTime":2505.577,"endTime":2508.75,"body":"you mark off some points equally"},{"startTime":2505.577,"endTime":2508.75,"body":"spaced around the circle."},{"startTime":2508.75,"endTime":2510.93,"body":"So I've got 12 points"},{"startTime":2508.75,"endTime":2510.93,"body":"cut out like a clock."},{"startTime":2510.93,"endTime":2513.48,"body":"This is just as a starter."},{"startTime":2513.48,"endTime":2518.48,"body":"And what you do is you join"},{"startTime":2513.48,"endTime":2518.48,"body":"each point to its double."},{"startTime":2518.55,"endTime":2521.56,"body":"So one goes to two, two goes"},{"startTime":2518.55,"endTime":2521.56,"body":"to four, three goes to six,"},{"startTime":2521.56,"endTime":2523.77,"body":"four goes to eight and"},{"startTime":2521.56,"endTime":2523.77,"body":"you're sort of thinking,"},{"startTime":2523.77,"endTime":2525.72,"body":"hang on, what does seven go to?"},{"startTime":2525.72,"endTime":2526.89,"body":"There's no 14 on here,"},{"startTime":2526.89,"endTime":2528.713,"body":"but it's kind of like the 24 hour clock."},{"startTime":2528.713,"endTime":2532.42,"body":"If you go around seven more"},{"startTime":2528.713,"endTime":2532.42,"body":"from seven on an actual clock,"},{"startTime":2532.42,"endTime":2534.97,"body":"you don't get 14.00 o'clock,"},{"startTime":2532.42,"endTime":2534.97,"body":"you get to two o'clock."},{"startTime":2534.97,"endTime":2538.02,"body":"Right? So you just do that"},{"startTime":2534.97,"endTime":2538.02,"body":"same thing in this situation."},{"startTime":2538.02,"endTime":2540.14,"body":"So every point's going to its double."},{"startTime":2540.14,"endTime":2541.74,"body":"When you do that with 12 points,"},{"startTime":2541.74,"endTime":2543.88,"body":"you get something that"},{"startTime":2541.74,"endTime":2543.88,"body":"looks not very nice,"},{"startTime":2543.88,"endTime":2546.05,"body":"but let's double the number of points,"},{"startTime":2546.05,"endTime":2547.58,"body":"and let's do it again."},{"startTime":2547.58,"endTime":2550.353,"body":"And let's do it again,"},{"startTime":2547.58,"endTime":2550.353,"body":"and let's do it again,"},{"startTime":2551.19,"endTime":2552.805,"body":"and let's do it one more time."},{"startTime":2552.805,"endTime":2557.13,"body":"And you can see, look, that's"},{"startTime":2552.805,"endTime":2557.13,"body":"not a mathematical proof,"},{"startTime":2557.13,"endTime":2558.85,"body":"but you can do a mathematical proof"},{"startTime":2558.85,"endTime":2561.03,"body":"that what you do then to get here,"},{"startTime":2561.03,"endTime":2563.73,"body":"kind of the involute of"},{"startTime":2561.03,"endTime":2563.73,"body":"all these straight lines"},{"startTime":2563.73,"endTime":2565.68,"body":"is a cardiod, and that's really pretty."},{"startTime":2567.53,"endTime":2570.43,"body":"Another place you might see"},{"startTime":2567.53,"endTime":2570.43,"body":"what looks like a cardiod"},{"startTime":2570.43,"endTime":2572.823,"body":"is in a cup of tea."},{"startTime":2573.896,"endTime":2576.693,"body":"You see on that picture,"},{"startTime":2573.896,"endTime":2576.693,"body":"look at the light."},{"startTime":2576.693,"endTime":2581.03,"body":"It's sort of making a curved shape"},{"startTime":2581.03,"endTime":2583.561,"body":"that looks like a cardiod."},{"startTime":2583.561,"endTime":2588.561,"body":"And well, that's interesting."},{"startTime":2589.12,"endTime":2589.953,"body":"Is it? Is it a cardioid?"},{"startTime":2589.953,"endTime":2591.7,"body":"And what is this curve actually?"},{"startTime":2591.7,"endTime":2594.92,"body":"So I've got this word"},{"startTime":2591.7,"endTime":2594.92,"body":"on the slide, a caustic."},{"startTime":2594.92,"endTime":2599.92,"body":"If light hits a curved"},{"startTime":2594.92,"endTime":2599.92,"body":"surface and reflects off,"},{"startTime":2599.92,"endTime":2603.28,"body":"the curve produced by"},{"startTime":2599.92,"endTime":2603.28,"body":"the kind of the boundary"},{"startTime":2603.28,"endTime":2605.99,"body":"of the reflected rays is called a caustic."},{"startTime":2605.99,"endTime":2609.51,"body":"So we were asking, what is this caustic?"},{"startTime":2609.51,"endTime":2611.97,"body":"And let's see what's going on."},{"startTime":2611.97,"endTime":2613.21,"body":"So imagine first,"},{"startTime":2613.21,"endTime":2614.963,"body":"if you have a light actually"},{"startTime":2613.21,"endTime":2614.963,"body":"on the circumference"},{"startTime":2614.963,"endTime":2618.0,"body":"of a circle, and the"},{"startTime":2614.963,"endTime":2618.0,"body":"rays are coming off it,"},{"startTime":2618.0,"endTime":2621.5,"body":"and we know in physics"},{"startTime":2618.0,"endTime":2621.5,"body":"that when light hits"},{"startTime":2621.5,"endTime":2624.979,"body":"and reflects off a surface,"},{"startTime":2621.5,"endTime":2624.979,"body":"the angle that it comes in at"},{"startTime":2624.979,"endTime":2626.63,"body":"is equal to the angle"},{"startTime":2624.979,"endTime":2626.63,"body":"that it comes out at."},{"startTime":2626.63,"endTime":2628.607,"body":"Angle of incidence equals"},{"startTime":2626.63,"endTime":2628.607,"body":"angle of reflection."},{"startTime":2628.607,"endTime":2629.96,"body":"You might remember that."},{"startTime":2629.96,"endTime":2632.7,"body":"So we've got these two equal angles here."},{"startTime":2632.7,"endTime":2635.62,"body":"And imagine now, I'm interested"},{"startTime":2632.7,"endTime":2635.62,"body":"in what's the reflected ray,"},{"startTime":2635.62,"endTime":2637.66,"body":"what are all those reflected rays doing?"},{"startTime":2637.66,"endTime":2642.24,"body":"So imagine that the lights"},{"startTime":2637.66,"endTime":2642.24,"body":"is at zero or the point 0,"},{"startTime":2642.24,"endTime":2644.31,"body":"and it hits the circle at point p."},{"startTime":2644.31,"endTime":2648.03,"body":"So I've just marked off some"},{"startTime":2644.31,"endTime":2648.03,"body":"end points around the circle."},{"startTime":2648.03,"endTime":2651.11,"body":"So as it hits at the"},{"startTime":2648.03,"endTime":2651.11,"body":"point p, what is this q?"},{"startTime":2651.11,"endTime":2652.21,"body":"What's this q?"},{"startTime":2652.21,"endTime":2654.88,"body":"Well, there's a lot of"},{"startTime":2652.21,"endTime":2654.88,"body":"symmetry in the situation."},{"startTime":2654.88,"endTime":2657.55,"body":"And actually, because as"},{"startTime":2654.88,"endTime":2657.55,"body":"these angles are the same,"},{"startTime":2657.55,"endTime":2659.63,"body":"it means that chords have the same length."},{"startTime":2659.63,"endTime":2662.166,"body":"And then the bit of the"},{"startTime":2659.63,"endTime":2662.166,"body":"circle they mark off"},{"startTime":2662.166,"endTime":2663.29,"body":"is the same length."},{"startTime":2663.29,"endTime":2666.05,"body":"So the distance from zero"},{"startTime":2663.29,"endTime":2666.05,"body":"to p must equal the distance"},{"startTime":2666.05,"endTime":2667.99,"body":"around the circle from p to q."},{"startTime":2667.99,"endTime":2670.32,"body":"So actually that's got to be p again."},{"startTime":2670.32,"endTime":2672.73,"body":"So q must equal to p."},{"startTime":2672.73,"endTime":2675.09,"body":"So this ray is going from p to two p."},{"startTime":2675.09,"endTime":2677.06,"body":"It's exactly this doubling thing"},{"startTime":2677.06,"endTime":2680.224,"body":"that we saw gave us a cardioid."},{"startTime":2680.224,"endTime":2682.14,"body":"The problem is, the light source probably"},{"startTime":2682.14,"endTime":2684.052,"body":"isn't on the rim of your cup of tea."},{"startTime":2684.052,"endTime":2685.816,"body":"It's probably somewhere in the distance."},{"startTime":2685.816,"endTime":2688.73,"body":"So maybe we want to think"},{"startTime":2685.816,"endTime":2688.73,"body":"about when light is coming"},{"startTime":2688.73,"endTime":2692.74,"body":"from a far distance and"},{"startTime":2688.73,"endTime":2692.74,"body":"bouncing off then the circle."},{"startTime":2692.74,"endTime":2693.96,"body":"And normally if that's happening,"},{"startTime":2693.96,"endTime":2696.15,"body":"we sort of assume that the"},{"startTime":2693.96,"endTime":2696.15,"body":"light rays are then parallel"},{"startTime":2696.15,"endTime":2698.34,"body":"'cause they're coming from infinity."},{"startTime":2698.34,"endTime":2700.9,"body":"So if you solve that problem,"},{"startTime":2700.9,"endTime":2702.26,"body":"it's a bit of a complicated diagram."},{"startTime":2702.26,"endTime":2703.5,"body":"There's too much going on in this diagram."},{"startTime":2703.5,"endTime":2705.81,"body":"Let's get rid of these parallel lines."},{"startTime":2705.81,"endTime":2708.41,"body":"You just get this. It looks all right."},{"startTime":2708.41,"endTime":2710.43,"body":"Let's add a few more points,"},{"startTime":2710.43,"endTime":2711.967,"body":"and then let's clear things up a bit"},{"startTime":2711.967,"endTime":2715.29,"body":"and just show you very"},{"startTime":2711.967,"endTime":2715.29,"body":"clearly what this curve is."},{"startTime":2715.29,"endTime":2717.743,"body":"Is this a cardioid? Looks cardioidy."},{"startTime":2719.43,"endTime":2720.67,"body":"I don't even quite see this,"},{"startTime":2720.67,"endTime":2723.13,"body":"but here is a cardioid superimposed."},{"startTime":2723.13,"endTime":2723.963,"body":"It's all right in the middle,"},{"startTime":2723.963,"endTime":2724.867,"body":"but at the bottom and the top there"},{"startTime":2724.867,"endTime":2726.22,"body":"are these little tiny gaps."},{"startTime":2726.22,"endTime":2727.403,"body":"It's not quite right."},{"startTime":2728.51,"endTime":2731.7,"body":"The thing that is quite"},{"startTime":2728.51,"endTime":2731.7,"body":"right is an nephroid."},{"startTime":2731.7,"endTime":2734.02,"body":"That is in fact what you get"},{"startTime":2734.02,"endTime":2736.18,"body":"if you have light coming,"},{"startTime":2734.02,"endTime":2736.18,"body":"parallel beams reflecting"},{"startTime":2736.18,"endTime":2738.89,"body":"for circle, you get part of an nephroid."},{"startTime":2738.89,"endTime":2740.27,"body":"So in practice,"},{"startTime":2740.27,"endTime":2742.65,"body":"what's happening in your cup"},{"startTime":2740.27,"endTime":2742.65,"body":"of tea is that the light source"},{"startTime":2742.65,"endTime":2746.62,"body":"is neither on the rim of the"},{"startTime":2742.65,"endTime":2746.62,"body":"cup, neither is it at infinity."},{"startTime":2746.62,"endTime":2747.99,"body":"It's somewhere in between."},{"startTime":2747.99,"endTime":2749.759,"body":"And so, the curve you get is sort of"},{"startTime":2749.759,"endTime":2754.12,"body":"between a cardioid and"},{"startTime":2749.759,"endTime":2754.12,"body":"a nephroid very slightly"},{"startTime":2754.12,"endTime":2756.91,"body":"depending on where the lighting"},{"startTime":2754.12,"endTime":2756.91,"body":"is in the room you are in."},{"startTime":2756.91,"endTime":2758.96,"body":"So that's a nice way to see these things."},{"startTime":2760.31,"endTime":2762.56,"body":"Where else can you see cardioids?"},{"startTime":2762.56,"endTime":2764.237,"body":"If you're a sound engineer,"},{"startTime":2762.56,"endTime":2764.237,"body":"you'll know what that is."},{"startTime":2764.237,"endTime":2766.13,"body":"It's a cardioid microphone."},{"startTime":2766.13,"endTime":2767.858,"body":"You don't see that thing,"},{"startTime":2767.858,"endTime":2771.77,"body":"but that's the pattern of"},{"startTime":2767.858,"endTime":2771.77,"body":"where sound is picked up"},{"startTime":2771.77,"endTime":2772.76,"body":"by such a microphone."},{"startTime":2772.76,"endTime":2774.75,"body":"It's very useful if you want to pick up"},{"startTime":2774.75,"endTime":2776.97,"body":"just the stuff that's in"},{"startTime":2774.75,"endTime":2776.97,"body":"front of the microphone"},{"startTime":2776.97,"endTime":2778.62,"body":"and not the extraneous noise behind."},{"startTime":2778.62,"endTime":2780.53,"body":"If you're recording"},{"startTime":2778.62,"endTime":2780.53,"body":"live music or something,"},{"startTime":2780.53,"endTime":2781.86,"body":"you want to hear the singer's voice,"},{"startTime":2781.86,"endTime":2784.31,"body":"but you don't want to pick"},{"startTime":2781.86,"endTime":2784.31,"body":"up the audience sound."},{"startTime":2786.8,"endTime":2789.39,"body":"The shape there is"},{"startTime":2786.8,"endTime":2789.39,"body":"cardioid sort of rotated"},{"startTime":2789.39,"endTime":2790.37,"body":"around that point"},{"startTime":2791.41,"endTime":2795.17,"body":"One other place, the Manga on set."},{"startTime":2795.17,"endTime":2796.6,"body":"So this is probably the most famous flat"},{"startTime":2796.6,"endTime":2798.36,"body":"in the world, right?"},{"startTime":2798.36,"endTime":2801.81,"body":"This largest black area is a cardioid."},{"startTime":2801.81,"endTime":2804.94,"body":"Now, without going into the"},{"startTime":2801.81,"endTime":2804.94,"body":"whole geometry of fractals,"},{"startTime":2804.94,"endTime":2806.49,"body":"which we don't have time to do,"},{"startTime":2807.57,"endTime":2810.79,"body":"I could at least maybe convince"},{"startTime":2807.57,"endTime":2810.79,"body":"you that this is plausible."},{"startTime":2810.79,"endTime":2815.123,"body":"This picture is drawn on a plane,"},{"startTime":2815.123,"endTime":2818.58,"body":"but it's using the geometry of"},{"startTime":2815.123,"endTime":2818.58,"body":"what called complex numbers,"},{"startTime":2818.58,"endTime":2819.55,"body":"which if you've heard of them, fine,"},{"startTime":2819.55,"endTime":2821.76,"body":"and if you haven't, also fine."},{"startTime":2821.76,"endTime":2823.26,"body":"But what you need to know about this is,"},{"startTime":2823.26,"endTime":2825.467,"body":"when you produce this"},{"startTime":2823.26,"endTime":2825.467,"body":"design, the way you do it,"},{"startTime":2825.467,"endTime":2827.79,"body":"it's a sort of iterative process."},{"startTime":2827.79,"endTime":2832.79,"body":"You take a number, you"},{"startTime":2827.79,"endTime":2832.79,"body":"square it at a constant,"},{"startTime":2832.81,"endTime":2834.36,"body":"and then you repeat that"},{"startTime":2832.81,"endTime":2834.36,"body":"square at a constant,"},{"startTime":2834.36,"endTime":2835.396,"body":"repeat, repeat, repeat."},{"startTime":2835.396,"endTime":2837.55,"body":"If when you do that,"},{"startTime":2837.55,"endTime":2839.537,"body":"the answer gets bigger"},{"startTime":2837.55,"endTime":2839.537,"body":"and bigger and bigger,"},{"startTime":2839.537,"endTime":2843.34,"body":"and sort of zooms off to"},{"startTime":2839.537,"endTime":2843.34,"body":"infinity, that's the blue bit."},{"startTime":2843.34,"endTime":2845.65,"body":"If instead it stays kind of where it is"},{"startTime":2845.65,"endTime":2850.59,"body":"and converges to a finite"},{"startTime":2845.65,"endTime":2850.59,"body":"amount, that's the black bit."},{"startTime":2850.59,"endTime":2852.887,"body":"So the other bit you need to know for this"},{"startTime":2852.887,"endTime":2854.76,"body":"is that these kind of"},{"startTime":2852.887,"endTime":2854.76,"body":"numbers we're working"},{"startTime":2854.76,"endTime":2857.02,"body":"with complex numbers, you can put them,"},{"startTime":2857.02,"endTime":2859.76,"body":"you can draw 'em on a picture"},{"startTime":2857.02,"endTime":2859.76,"body":"by saying how far away they"},{"startTime":2859.76,"endTime":2862.98,"body":"are from the origin"},{"startTime":2859.76,"endTime":2862.98,"body":"and what angle the line"},{"startTime":2862.98,"endTime":2866.43,"body":"to them makes with the axis."},{"startTime":2866.43,"endTime":2868.58,"body":"When you square one of these things,"},{"startTime":2868.58,"endTime":2870.41,"body":"you square the distance from the origin"},{"startTime":2870.41,"endTime":2872.76,"body":"and you double the angle."},{"startTime":2872.76,"endTime":2875.33,"body":"And so, it's at least plausible, I hope,"},{"startTime":2875.33,"endTime":2878.38,"body":"that something which"},{"startTime":2875.33,"endTime":2878.38,"body":"involves angle doubling"},{"startTime":2878.38,"endTime":2880.76,"body":"might be related to the cardioid"},{"startTime":2878.38,"endTime":2880.76,"body":"'cause we've already seen,"},{"startTime":2880.76,"endTime":2883.01,"body":"you can make that with doublings."},{"startTime":2883.01,"endTime":2885.58,"body":"Okay, so that's that's cardioids."},{"startTime":2885.58,"endTime":2889.655,"body":"I want to spend five minutes"},{"startTime":2885.58,"endTime":2889.655,"body":"now on talking about involutes,"},{"startTime":2889.655,"endTime":2891.85,"body":"in particular, circle involutes,"},{"startTime":2891.85,"endTime":2893.81,"body":"because they are some nice applications."},{"startTime":2893.81,"endTime":2896.39,"body":"And the first one is to do with gears."},{"startTime":2896.39,"endTime":2899.83,"body":"So as in, clockwork toys"},{"startTime":2899.83,"endTime":2902.29,"body":"or other kind of mechanical devices."},{"startTime":2902.29,"endTime":2907.29,"body":"So when you have a system"},{"startTime":2902.29,"endTime":2907.29,"body":"of gears or cog wheels,"},{"startTime":2907.66,"endTime":2910.01,"body":"like you might find in a clockwork toy,"},{"startTime":2910.01,"endTime":2914.66,"body":"what you want is you have gears"},{"startTime":2910.01,"endTime":2914.66,"body":"that are transferring energy"},{"startTime":2915.57,"endTime":2916.95,"body":"from one to another,"},{"startTime":2916.95,"endTime":2919.22,"body":"and the input gear is kind of rotating"},{"startTime":2919.22,"endTime":2921.46,"body":"at, usually, at some constant speed."},{"startTime":2921.46,"endTime":2924.24,"body":"And it's interlocking with another gear,"},{"startTime":2924.24,"endTime":2926.87,"body":"and it's transferring the energy across,"},{"startTime":2926.87,"endTime":2929.1,"body":"and what you want is, you want the teeth"},{"startTime":2929.1,"endTime":2931.57,"body":"to be in constant contact"},{"startTime":2929.1,"endTime":2931.57,"body":"so that the energy"},{"startTime":2931.57,"endTime":2933.93,"body":"is transferred efficiently."},{"startTime":2933.93,"endTime":2936.55,"body":"And if the first gear is"},{"startTime":2933.93,"endTime":2936.55,"body":"moving at a constant speed,"},{"startTime":2936.55,"endTime":2938.05,"body":"then for most applications, what you want"},{"startTime":2938.05,"endTime":2940.42,"body":"is the second gear, assuming"},{"startTime":2938.05,"endTime":2940.42,"body":"they're both circular gears."},{"startTime":2940.42,"endTime":2942.76,"body":"You want that to be moving"},{"startTime":2940.42,"endTime":2942.76,"body":"at a constant speed,"},{"startTime":2942.76,"endTime":2944.77,"body":"not the same constant speed"},{"startTime":2942.76,"endTime":2944.77,"body":"if they're different sizes,"},{"startTime":2944.77,"endTime":2946.373,"body":"but you want to preserve that,"},{"startTime":2947.72,"endTime":2950.17,"body":"and transfer the energy as"},{"startTime":2947.72,"endTime":2950.17,"body":"efficiently as possible."},{"startTime":2950.17,"endTime":2953.053,"body":"Now, if it turns out, on"},{"startTime":2950.17,"endTime":2953.053,"body":"the left in this slide,"},{"startTime":2953.053,"endTime":2954.853,"body":"there's a picture of an"},{"startTime":2953.053,"endTime":2954.853,"body":"involute of a circle,"},{"startTime":2954.853,"endTime":2957.56,"body":"just to remind you, it"},{"startTime":2954.853,"endTime":2957.56,"body":"looks a bit spiraly."},{"startTime":2957.56,"endTime":2962.56,"body":"And on the right, is a"},{"startTime":2957.56,"endTime":2962.56,"body":"picture of two gear wheels"},{"startTime":2962.95,"endTime":2964.287,"body":"with their interlocking teeth,"},{"startTime":2964.287,"endTime":2967.2,"body":"and one is transferring"},{"startTime":2964.287,"endTime":2967.2,"body":"energy to the other."},{"startTime":2967.2,"endTime":2970.02,"body":"Now, if the profile of those gear teeth"},{"startTime":2971.2,"endTime":2974.315,"body":"is an involute of a circle,"},{"startTime":2974.315,"endTime":2977.35,"body":"then it turns out that at the"},{"startTime":2974.315,"endTime":2977.35,"body":"point where they're touching,"},{"startTime":2977.35,"endTime":2978.332,"body":"they're constantly touching,"},{"startTime":2978.332,"endTime":2980.643,"body":"the point where they're"},{"startTime":2978.332,"endTime":2980.643,"body":"touching at any time,"},{"startTime":2981.6,"endTime":2984.65,"body":"if we think how we make"},{"startTime":2981.6,"endTime":2984.65,"body":"these circle involutes,"},{"startTime":2984.65,"endTime":2987.88,"body":"we think about unwinding"},{"startTime":2984.65,"endTime":2987.88,"body":"this line to a thread,"},{"startTime":2987.88,"endTime":2989.49,"body":"unwinding it from a circle,"},{"startTime":2989.49,"endTime":2992.46,"body":"that is basically a tangent"},{"startTime":2989.49,"endTime":2992.46,"body":"to the circle the whole time."},{"startTime":2992.46,"endTime":2995.52,"body":"And so, if both of these things"},{"startTime":2992.46,"endTime":2995.52,"body":"are involutes of the circle,"},{"startTime":2995.52,"endTime":2996.868,"body":"then whenever they touch,"},{"startTime":2996.868,"endTime":2998.15,"body":"the force, the energy"},{"startTime":2998.15,"endTime":3000.65,"body":"is transferred along"},{"startTime":2998.15,"endTime":3000.65,"body":"a line that is tangent"},{"startTime":3000.65,"endTime":3001.7,"body":"to both of the circles."},{"startTime":3001.7,"endTime":3006.7,"body":"And that makes for a very"},{"startTime":3001.7,"endTime":3006.7,"body":"smooth transfer of the energy."},{"startTime":3006.82,"endTime":3008.99,"body":"So these are really great for gears,"},{"startTime":3008.99,"endTime":3011.42,"body":"for the profiles of those gear teeth."},{"startTime":3011.42,"endTime":3013.68,"body":"Another brilliant advantage"},{"startTime":3011.42,"endTime":3013.68,"body":"of circle involutes"},{"startTime":3013.68,"endTime":3014.86,"body":"in this context,"},{"startTime":3014.86,"endTime":3018.45,"body":"and they used to use cycloids"},{"startTime":3014.86,"endTime":3018.45,"body":"actually, cycloid profiles,"},{"startTime":3018.45,"endTime":3021.7,"body":"but involutes are better"},{"startTime":3018.45,"endTime":3021.7,"body":"for most applications"},{"startTime":3021.7,"endTime":3025.68,"body":"because of a practical reason."},{"startTime":3025.68,"endTime":3030.23,"body":"It turns out that as long as"},{"startTime":3025.68,"endTime":3030.23,"body":"long as you keep the same,"},{"startTime":3030.23,"endTime":3031.95,"body":"what's called diametrical pitch,"},{"startTime":3031.95,"endTime":3035.98,"body":"which is the number of"},{"startTime":3031.95,"endTime":3035.98,"body":"teeth per inch on the wheel,"},{"startTime":3035.98,"endTime":3039.5,"body":"on the gear, then it turns out"},{"startTime":3035.98,"endTime":3039.5,"body":"you can use identical teeth"},{"startTime":3039.5,"endTime":3040.96,"body":"on any size gear."},{"startTime":3040.96,"endTime":3041.87,"body":"So that means,"},{"startTime":3041.87,"endTime":3044.63,"body":"if you're producing these"},{"startTime":3041.87,"endTime":3044.63,"body":"things industrially,"},{"startTime":3044.63,"endTime":3047.69,"body":"you don't need to have multiple"},{"startTime":3044.63,"endTime":3047.69,"body":"different possibilities."},{"startTime":3047.69,"endTime":3049.29,"body":"You can use the same on every gear."},{"startTime":3049.29,"endTime":3052.71,"body":"So that's a very useful"},{"startTime":3049.29,"endTime":3052.71,"body":"property for them to have."},{"startTime":3052.71,"endTime":3055.593,"body":"So, yes, you see archs of involutes,"},{"startTime":3055.593,"endTime":3059.823,"body":"bits of involutes of circles in gears."},{"startTime":3060.93,"endTime":3064.87,"body":"The other application is"},{"startTime":3060.93,"endTime":3064.87,"body":"inside a nuclear reactor."},{"startTime":3064.87,"endTime":3066.63,"body":"So actually what I'm talking about is,"},{"startTime":3066.63,"endTime":3068.73,"body":"I think I said nuclear"},{"startTime":3066.63,"endTime":3068.73,"body":"power plants earlier."},{"startTime":3068.73,"endTime":3070.38,"body":"This isn't quite that."},{"startTime":3070.38,"endTime":3072.58,"body":"It's nuclear reactor still, but,"},{"startTime":3072.58,"endTime":3075.28,"body":"so the pitch you see is of the"},{"startTime":3072.58,"endTime":3075.28,"body":"Oak Ridge National Laboratory"},{"startTime":3075.28,"endTime":3077.38,"body":"in Tennessee, in America."},{"startTime":3077.38,"endTime":3081.4,"body":"And there they have the"},{"startTime":3077.38,"endTime":3081.4,"body":"high flux isotope reactor,"},{"startTime":3081.4,"endTime":3085.31,"body":"and they're not interested"},{"startTime":3081.4,"endTime":3085.31,"body":"in making energy."},{"startTime":3085.31,"endTime":3089.93,"body":"What they're interested in doing"},{"startTime":3085.31,"endTime":3089.93,"body":"is producing heavy isotopes"},{"startTime":3089.93,"endTime":3093.24,"body":"of elements for various"},{"startTime":3089.93,"endTime":3093.24,"body":"scientific applications."},{"startTime":3093.24,"endTime":3097.65,"body":"And what you do, there's a nice"},{"startTime":3093.24,"endTime":3097.65,"body":"video about this on YouTube,"},{"startTime":3097.65,"endTime":3099.58,"body":"but I'll put a link in the transcript,"},{"startTime":3099.58,"endTime":3102.01,"body":"but I've just got a couple"},{"startTime":3099.58,"endTime":3102.01,"body":"of pictures from it."},{"startTime":3102.01,"endTime":3102.877,"body":"What you do in in a nuclear reaction"},{"startTime":3102.877,"endTime":3106.26,"body":"and what they're trying to"},{"startTime":3102.877,"endTime":3106.26,"body":"do is to make heavy isotopes."},{"startTime":3106.26,"endTime":3110.56,"body":"So they want to bombard"},{"startTime":3106.26,"endTime":3110.56,"body":"lighter isotopes with neutrons,"},{"startTime":3110.56,"endTime":3113.03,"body":"and then they become heavier, right?"},{"startTime":3113.03,"endTime":3115.47,"body":"Simple guide to nuclear actions."},{"startTime":3115.47,"endTime":3117.65,"body":"So what you do is you have your fuel,"},{"startTime":3117.65,"endTime":3119.72,"body":"which is uranium oxide,"},{"startTime":3119.72,"endTime":3121.85,"body":"and you don't want to have"},{"startTime":3119.72,"endTime":3121.85,"body":"too much of it together"},{"startTime":3121.85,"endTime":3123.92,"body":"in one place 'cause that's not good."},{"startTime":3123.92,"endTime":3125.73,"body":"But inside the reactor core,"},{"startTime":3125.73,"endTime":3129.14,"body":"inside these cylindrical containers,"},{"startTime":3129.14,"endTime":3131.95,"body":"you have fuel strips"},{"startTime":3129.14,"endTime":3131.95,"body":"and they are the fuel,"},{"startTime":3131.95,"endTime":3136.26,"body":"the uranium oxide, sort of"},{"startTime":3131.95,"endTime":3136.26,"body":"sandwiched into aluminium strips."},{"startTime":3136.26,"endTime":3137.811,"body":"And then you have a whole bunch of those,"},{"startTime":3137.811,"endTime":3142.19,"body":"and you want to put them"},{"startTime":3137.811,"endTime":3142.19,"body":"into your cylindrical core"},{"startTime":3142.19,"endTime":3145.02,"body":"in such a way that we"},{"startTime":3142.19,"endTime":3145.02,"body":"don't want them touching"},{"startTime":3145.02,"endTime":3146.21,"body":"because that's not good."},{"startTime":3146.21,"endTime":3148.56,"body":"And we need to leave a little"},{"startTime":3146.21,"endTime":3148.56,"body":"bit of space between them,"},{"startTime":3148.56,"endTime":3151.648,"body":"and the same amount of space,"},{"startTime":3148.56,"endTime":3151.648,"body":"constant amount of space,"},{"startTime":3151.648,"endTime":3154.16,"body":"because you really need to"},{"startTime":3151.648,"endTime":3154.16,"body":"control very much the amount"},{"startTime":3154.16,"endTime":3154.993,"body":"of heat."},{"startTime":3154.993,"endTime":3156.22,"body":"And between them is the coolant."},{"startTime":3156.22,"endTime":3157.89,"body":"The water is going between them"},{"startTime":3157.89,"endTime":3159.83,"body":"constantly cooling them down"},{"startTime":3159.83,"endTime":3161.76,"body":"because there's a lot of heat generated."},{"startTime":3161.76,"endTime":3165.33,"body":"There are a million billion"},{"startTime":3161.76,"endTime":3165.33,"body":"neutrons per square centimeter"},{"startTime":3165.33,"endTime":3166.163,"body":"being produced, right?"},{"startTime":3166.163,"endTime":3168.17,"body":"But hat makes for a lot of heat,"},{"startTime":3168.17,"endTime":3169.86,"body":"so the coolant needs to"},{"startTime":3168.17,"endTime":3169.86,"body":"be able to pass equally"},{"startTime":3169.86,"endTime":3171.23,"body":"between all of these things."},{"startTime":3171.23,"endTime":3173.96,"body":"So you've got these rectangles"},{"startTime":3171.23,"endTime":3173.96,"body":"and you've got to fit them"},{"startTime":3173.96,"endTime":3177.576,"body":"into a cylinder in such a"},{"startTime":3173.96,"endTime":3177.576,"body":"way around a circular center"},{"startTime":3177.576,"endTime":3180.58,"body":"in such a way that they have"},{"startTime":3177.576,"endTime":3180.58,"body":"constant distance apart."},{"startTime":3180.58,"endTime":3183.46,"body":"So this is the kind of thing"},{"startTime":3180.58,"endTime":3183.46,"body":"you you are trying to do."},{"startTime":3183.46,"endTime":3184.68,"body":"So what do you need for that?"},{"startTime":3184.68,"endTime":3186.81,"body":"Well, it turns out that circle involutes"},{"startTime":3186.81,"endTime":3188.0,"body":"have this brilliant property."},{"startTime":3188.0,"endTime":3188.833,"body":"Here's some."},{"startTime":3189.94,"endTime":3192.92,"body":"If you put them equally"},{"startTime":3189.94,"endTime":3192.92,"body":"spaced around a circle,"},{"startTime":3192.92,"endTime":3197.92,"body":"then those curves stay"},{"startTime":3192.92,"endTime":3197.92,"body":"equally spaced forever, right?"},{"startTime":3198.15,"endTime":3199.62,"body":"As long as you want to extend them for,"},{"startTime":3199.62,"endTime":3201.43,"body":"they remain equally spaced."},{"startTime":3201.43,"endTime":3203.58,"body":"So this is it, I quite like this picture."},{"startTime":3203.58,"endTime":3204.54,"body":"They remain equally spaced."},{"startTime":3204.54,"endTime":3206.32,"body":"This would not work for any other curve."},{"startTime":3206.32,"endTime":3207.97,"body":"We tried archs of circles,"},{"startTime":3207.97,"endTime":3210.21,"body":"and they would end up meeting,"},{"startTime":3207.97,"endTime":3210.21,"body":"getting closer together."},{"startTime":3210.21,"endTime":3213.52,"body":"So involutes of circles"},{"startTime":3210.21,"endTime":3213.52,"body":"precisely solve this problem."},{"startTime":3213.52,"endTime":3218.04,"body":"And this is a mock up"},{"startTime":3213.52,"endTime":3218.04,"body":"of one of these insides"},{"startTime":3218.04,"endTime":3219.077,"body":"of the nuclear reactor."},{"startTime":3219.077,"endTime":3221.45,"body":"And you can perhaps see all"},{"startTime":3219.077,"endTime":3221.45,"body":"these beautiful involutes"},{"startTime":3221.45,"endTime":3223.73,"body":"just radiating out from"},{"startTime":3221.45,"endTime":3223.73,"body":"the central circle."},{"startTime":3223.73,"endTime":3228.73,"body":"So that's obviously very real"},{"startTime":3223.73,"endTime":3228.73,"body":"application of involutes."},{"startTime":3230.02,"endTime":3232.26,"body":"And we've really only"},{"startTime":3230.02,"endTime":3232.26,"body":"scratched the surface"},{"startTime":3232.26,"endTime":3234.813,"body":"of what we could say about"},{"startTime":3232.26,"endTime":3234.813,"body":"roulette curves today."},{"startTime":3234.813,"endTime":3237.51,"body":"I'll have to tell you"},{"startTime":3234.813,"endTime":3237.51,"body":"another time what happens"},{"startTime":3237.51,"endTime":3239.58,"body":"if you roll in ellipse along a sin curve,"},{"startTime":3239.58,"endTime":3244.41,"body":"for example, but segue, if you"},{"startTime":3239.58,"endTime":3244.41,"body":"are interested in ellipses,"},{"startTime":3244.41,"endTime":3246.1,"body":"then you should come to my next lecture,"},{"startTime":3246.1,"endTime":3248.72,"body":"which is on The Surprising"},{"startTime":3246.1,"endTime":3248.72,"body":"Uses of Conic Sections."},{"startTime":3248.72,"endTime":3251.42,"body":"The ellipse is one example"},{"startTime":3248.72,"endTime":3251.42,"body":"of the curve you can get"},{"startTime":3251.42,"endTime":3254.09,"body":"when you slice through a"},{"startTime":3251.42,"endTime":3254.09,"body":"cone at different angles."},{"startTime":3254.09,"endTime":3255.42,"body":"Another example is a parabola,"},{"startTime":3255.42,"endTime":3257.633,"body":"which is shown in this waterfall here."},{"startTime":3258.99,"endTime":3261.14,"body":"So I hope you've enjoyed today's talk."},{"startTime":3261.14,"endTime":3262.51,"body":"Next time you have a coffee break,"},{"startTime":3262.51,"endTime":3265.77,"body":"look out for the cardioids,"},{"startTime":3262.51,"endTime":3265.77,"body":"and we will stop there."},{"startTime":3265.77,"endTime":3266.79,"body":"Thank you."},{"startTime":3266.79,"endTime":3269.79,"body":"(audience clapping)"},{"startTime":3271.986,"endTime":3274.3,"body":"- There was a question"},{"startTime":3271.986,"endTime":3274.3,"body":"about the competitions"},{"startTime":3274.3,"endTime":3276.28,"body":"that you were talking"},{"startTime":3274.3,"endTime":3276.28,"body":"about at the beginning."},{"startTime":3276.28,"endTime":3278.56,"body":"Are there any thoughts on"},{"startTime":3276.28,"endTime":3278.56,"body":"what mathematicians were doing"},{"startTime":3278.56,"endTime":3282.61,"body":"at the time or prior outside"},{"startTime":3278.56,"endTime":3282.61,"body":"of Europe on these problems?"},{"startTime":3282.61,"endTime":3284.0,"body":"It seems a bit European, the emphasis."},{"startTime":3284.0,"endTime":3286.003,"body":"- Oh, yeah. You're right."},{"startTime":3290.21,"endTime":3293.21,"body":"I would have to say, yet, I'm not sure."},{"startTime":3293.21,"endTime":3296.16,"body":"I think I would want to look"},{"startTime":3293.21,"endTime":3296.16,"body":"at certainly what was happening"},{"startTime":3296.16,"endTime":3299.023,"body":"in China at this time because I know that"},{"startTime":3299.023,"endTime":3300.99,"body":"the Chinese mathematicians"},{"startTime":3299.023,"endTime":3300.99,"body":"certainly developed something"},{"startTime":3300.99,"endTime":3303.0,"body":"very like Cavalieri's principle."},{"startTime":3303.0,"endTime":3304.64,"body":"I mean, it's actually the same thing,"},{"startTime":3304.64,"endTime":3309.073,"body":"and they use it to come up"},{"startTime":3304.64,"endTime":3309.073,"body":"with a really fantastic way"},{"startTime":3310.08,"endTime":3311.761,"body":"of deriving the volume of a sphere"},{"startTime":3311.761,"endTime":3314.48,"body":"by slicing these shapes."},{"startTime":3314.48,"endTime":3316.24,"body":"You started with a third of a cube,"},{"startTime":3316.24,"endTime":3318.13,"body":"and then you can find the area"},{"startTime":3318.13,"endTime":3320.56,"body":"that the two cylindrical"},{"startTime":3318.13,"endTime":3320.56,"body":"tunnels intersect."},{"startTime":3320.56,"endTime":3322.14,"body":"You can find the volume of that thing,"},{"startTime":3322.14,"endTime":3324.61,"body":"and then you use that in a"},{"startTime":3322.14,"endTime":3324.61,"body":"way, if I had 15 minutes,"},{"startTime":3324.61,"endTime":3326.22,"body":"I could tell you to find"},{"startTime":3324.61,"endTime":3326.22,"body":"the volume of a sphere."},{"startTime":3326.22,"endTime":3328.44,"body":"And it's this idea of slicing,"},{"startTime":3326.22,"endTime":3328.44,"body":"but in three dimensions."},{"startTime":3328.44,"endTime":3330.77,"body":"So they certainly had principles like that"},{"startTime":3332.92,"endTime":3335.24,"body":"perhaps before Cavalieri's"},{"startTime":3332.92,"endTime":3335.24,"body":"principle, I'll have to check."},{"startTime":3335.24,"endTime":3338.45,"body":"And so, yeah, that would be my first place"},{"startTime":3338.45,"endTime":3339.283,"body":"that I would go and look,"},{"startTime":3339.283,"endTime":3341.56,"body":"what was Chinese mathematics"},{"startTime":3339.283,"endTime":3341.56,"body":"doing at this time?"},{"startTime":3341.56,"endTime":3342.849,"body":"Yeah."},{"startTime":3342.849,"endTime":3346.96,"body":"- Are the arches produced by cycloids used"},{"startTime":3346.96,"endTime":3348.89,"body":"in bridge construction"},{"startTime":3346.96,"endTime":3348.89,"body":"and things like that?"},{"startTime":3348.89,"endTime":3352.12,"body":"And I'm thinking of some of"},{"startTime":3348.89,"endTime":3352.12,"body":"the rather essetary arches"},{"startTime":3352.12,"endTime":3356.327,"body":"that in Isambard kingdom,"},{"startTime":3352.12,"endTime":3356.327,"body":"Brunel used to build bridges."},{"startTime":3356.327,"endTime":3360.25,"body":"- So without having"},{"startTime":3356.327,"endTime":3360.25,"body":"knowledge of every single one"},{"startTime":3360.25,"endTime":3363.01,"body":"of those bridges, normally,"},{"startTime":3360.25,"endTime":3363.01,"body":"what you find in bridges"},{"startTime":3363.01,"endTime":3366.267,"body":"is catenary, which is a slightly"},{"startTime":3363.01,"endTime":3366.267,"body":"different kind of curve."},{"startTime":3366.267,"endTime":3371.267,"body":"And that's the curve formed"},{"startTime":3366.267,"endTime":3371.267,"body":"if you take a chain of metal"},{"startTime":3371.95,"endTime":3374.61,"body":"and just let it hang"},{"startTime":3371.95,"endTime":3374.61,"body":"naturally under gravity."},{"startTime":3374.61,"endTime":3376.28,"body":"It forms this curve called a catenary,"},{"startTime":3376.28,"endTime":3380.18,"body":"and quite often in bridges,"},{"startTime":3376.28,"endTime":3380.18,"body":"that's the curve you get."},{"startTime":3380.18,"endTime":3382.09,"body":"So I would say probably catenaries,"},{"startTime":3382.09,"endTime":3386.37,"body":"but I don't absolutely, be sure."},{"startTime":3386.37,"endTime":3387.64,"body":"Yeah."},{"startTime":3387.64,"endTime":3391.18,"body":"- Do we know the equation"},{"startTime":3387.64,"endTime":3391.18,"body":"of different cycloids"},{"startTime":3391.18,"endTime":3392.98,"body":"if we were to go after or something?"},{"startTime":3393.96,"endTime":3396.64,"body":"- Yeah, so yes,"},{"startTime":3396.64,"endTime":3400.03,"body":"we know the equation of cycloid"},{"startTime":3396.64,"endTime":3400.03,"body":"and I didn't write it down,"},{"startTime":3400.03,"endTime":3401.71,"body":"but it's very easy to find online"},{"startTime":3401.71,"endTime":3402.86,"body":"or I could tell you afterwards."},{"startTime":3402.86,"endTime":3405.91,"body":"So it involves sins and coses basically."},{"startTime":3405.91,"endTime":3410.39,"body":"And you can give a, what's"},{"startTime":3405.91,"endTime":3410.39,"body":"called a parametric expression."},{"startTime":3410.39,"endTime":3414.86,"body":"So you can give the X coordinate"},{"startTime":3410.39,"endTime":3414.86,"body":"in terms of cos theater"},{"startTime":3414.86,"endTime":3417.1,"body":"and the Y coordinate"},{"startTime":3414.86,"endTime":3417.1,"body":"in terms of sin theater"},{"startTime":3417.1,"endTime":3418.89,"body":"in some way that I"},{"startTime":3417.1,"endTime":3418.89,"body":"didn't quite write down,"},{"startTime":3418.89,"endTime":3420.93,"body":"but yeah, it's easy to find online."},{"startTime":3420.93,"endTime":3421.95,"body":"Yeah, Galileo, though,"},{"startTime":3421.95,"endTime":3423.9,"body":"those people didn't have that information"},{"startTime":3423.9,"endTime":3427.32,"body":"because it was even before"},{"startTime":3423.9,"endTime":3427.32,"body":"we really had the concept"},{"startTime":3427.32,"endTime":3428.72,"body":"of the equation of a curve."},{"startTime":3428.72,"endTime":3432.9,"body":"So, he actually was doing"},{"startTime":3428.72,"endTime":3432.9,"body":"something fairly sensible,"},{"startTime":3432.9,"endTime":3434.63,"body":"just doing a physical experiment like that"},{"startTime":3434.63,"endTime":3436.57,"body":"because there just wasn't"},{"startTime":3434.63,"endTime":3436.57,"body":"a mathematical machinery"},{"startTime":3436.57,"endTime":3439.8,"body":"at that stage to be able"},{"startTime":3436.57,"endTime":3439.8,"body":"to write down the equations"},{"startTime":3439.8,"endTime":3442.286,"body":"that we now can use to do all this stuff"},{"startTime":3442.286,"endTime":3445.2,"body":"with the techniques available to us now."},{"startTime":3445.2,"endTime":3448.05,"body":"- I'm going to bring it to a close there."},{"startTime":3448.05,"endTime":3450.63,"body":"Thank you to our audience"},{"startTime":3448.05,"endTime":3450.63,"body":"for joining us today,"},{"startTime":3450.63,"endTime":3452.32,"body":"and for your attention online."},{"startTime":3452.32,"endTime":3453.66,"body":"And please join me"},{"startTime":3453.66,"endTime":3456.564,"body":"in thanking Professor Hart one more time."},{"startTime":3456.564,"endTime":3459.647,"body":"(audience clapping)"}]}