{"version":"1.0.0","segments":[{"startTime":0.121,"endTime":2.788,"body":"(air whooshing)"},{"startTime":4.98,"endTime":7.95,"body":"- Welcome, everyone, today,"},{"startTime":4.98,"endTime":7.95,"body":"I'm going to tell you"},{"startTime":7.95,"endTime":9.753,"body":"about Christopher Wren."},{"startTime":9.753,"endTime":12.15,"body":"When we think of Sir Christopher Wren,"},{"startTime":12.15,"endTime":14.58,"body":"we remember him as a great architect,"},{"startTime":14.58,"endTime":17.04,"body":"but he's also a mathematician."},{"startTime":17.04,"endTime":19.56,"body":"And that's what I want"},{"startTime":17.04,"endTime":19.56,"body":"to tell you about today."},{"startTime":19.56,"endTime":20.43,"body":"On the way,"},{"startTime":20.43,"endTime":23.79,"body":"I'm going to share with"},{"startTime":20.43,"endTime":23.79,"body":"you three mathematical gems"},{"startTime":23.79,"endTime":27.87,"body":"that Christopher Wren played"},{"startTime":23.79,"endTime":27.87,"body":"with and investigated."},{"startTime":27.87,"endTime":30.57,"body":"So we'll find out the"},{"startTime":27.87,"endTime":30.57,"body":"mathematical discovery he made"},{"startTime":30.57,"endTime":31.87,"body":"while he was out shopping,"},{"startTime":32.91,"endTime":35.01,"body":"we'll find out why he had a battle of wits"},{"startTime":35.01,"endTime":37.53,"body":"with a man who didn't exist,"},{"startTime":37.53,"endTime":41.103,"body":"and we'll find out what he"},{"startTime":37.53,"endTime":41.103,"body":"thought about seashells."},{"startTime":42.0,"endTime":45.66,"body":"We'll finish today by"},{"startTime":42.0,"endTime":45.66,"body":"talking about the mathematics"},{"startTime":45.66,"endTime":48.36,"body":"behind his greatest"},{"startTime":45.66,"endTime":48.36,"body":"architectural achievement,"},{"startTime":48.36,"endTime":50.043,"body":"the dome of St. Paul's Cathedral."},{"startTime":52.5,"endTime":55.44,"body":"So Christopher Wren, as I've said,"},{"startTime":55.44,"endTime":59.55,"body":"of course, our first"},{"startTime":55.44,"endTime":59.55,"body":"thought is the architecture,"},{"startTime":59.55,"endTime":62.37,"body":"but actually, this was"},{"startTime":59.55,"endTime":62.37,"body":"something he did not do"},{"startTime":62.37,"endTime":63.203,"body":"to begin with."},{"startTime":63.203,"endTime":65.25,"body":"He moved into architecture later,"},{"startTime":65.25,"endTime":69.48,"body":"but he began as a"},{"startTime":65.25,"endTime":69.48,"body":"mathematician, as a scientist."},{"startTime":69.48,"endTime":73.47,"body":"He died 300 years ago this year."},{"startTime":73.47,"endTime":76.95,"body":"2023 is this year, 1723 he died."},{"startTime":76.95,"endTime":79.26,"body":"And among other things,"},{"startTime":79.26,"endTime":81.87,"body":"he was the Gresham professor of astronomy."},{"startTime":81.87,"endTime":84.3,"body":"So obviously good at astronomy,"},{"startTime":84.3,"endTime":88.59,"body":"but he was also really a polymath."},{"startTime":88.59,"endTime":92.97,"body":"He investigated everything"},{"startTime":88.59,"endTime":92.97,"body":"from meteorology to anatomy."},{"startTime":92.97,"endTime":96.45,"body":"He made progress in mathematics."},{"startTime":96.45,"endTime":98.49,"body":"He was into everything."},{"startTime":98.49,"endTime":101.19,"body":"And it was at Gresham College"},{"startTime":101.19,"endTime":104.67,"body":"that he was part of the movement,"},{"startTime":104.67,"endTime":106.89,"body":"one of the founder members of a society"},{"startTime":106.89,"endTime":111.89,"body":"for the promotion of"},{"startTime":106.89,"endTime":111.89,"body":"physico-mathematical learning."},{"startTime":112.26,"endTime":114.18,"body":"And that name, not super catchy,"},{"startTime":114.18,"endTime":116.58,"body":"but it would become known"},{"startTime":114.18,"endTime":116.58,"body":"as the Royal Society."},{"startTime":116.58,"endTime":117.413,"body":"So this is, you know,"},{"startTime":117.413,"endTime":121.95,"body":"one of the most important"},{"startTime":117.413,"endTime":121.95,"body":"scientific societies that we have."},{"startTime":121.95,"endTime":123.95,"body":"And that was started at Gresham College."},{"startTime":125.79,"endTime":130.79,"body":"Now, we've said he started out"},{"startTime":125.79,"endTime":130.79,"body":"in mathematics and science,"},{"startTime":130.8,"endTime":132.48,"body":"then he moved into architecture."},{"startTime":132.48,"endTime":134.73,"body":"I mean, that's a slight"},{"startTime":132.48,"endTime":134.73,"body":"simplification of what happened."},{"startTime":134.73,"endTime":139.71,"body":"That step is not such a leap"},{"startTime":134.73,"endTime":139.71,"body":"as it might now appear to be,"},{"startTime":139.71,"endTime":142.26,"body":"because for Wren, and at the time,"},{"startTime":142.26,"endTime":145.83,"body":"architecture was one of"},{"startTime":142.26,"endTime":145.83,"body":"the mathematical arts"},{"startTime":145.83,"endTime":148.2,"body":"with navigation, with"},{"startTime":145.83,"endTime":148.2,"body":"astronomy, with surveying,"},{"startTime":148.2,"endTime":149.61,"body":"other practical things."},{"startTime":149.61,"endTime":152.25,"body":"It was part of mathematics."},{"startTime":152.25,"endTime":154.35,"body":"And Christopher Wren was"},{"startTime":152.25,"endTime":154.35,"body":"a good mathematician."},{"startTime":154.35,"endTime":158.49,"body":"So it wasn't such a leap, as"},{"startTime":154.35,"endTime":158.49,"body":"I say, as it might now appear."},{"startTime":158.49,"endTime":161.55,"body":"And actually, when"},{"startTime":158.49,"endTime":161.55,"body":"Christopher Wren was writing"},{"startTime":161.55,"endTime":164.4,"body":"about architecture, he put mathematics,"},{"startTime":164.4,"endTime":165.99,"body":"in particular, geometry,"},{"startTime":165.99,"endTime":169.83,"body":"at the heart of what"},{"startTime":165.99,"endTime":169.83,"body":"architectural beauty is all about."},{"startTime":169.83,"endTime":173.82,"body":"So I've got this quote from Wren here"},{"startTime":173.82,"endTime":177.87,"body":"where he's talking about"},{"startTime":173.82,"endTime":177.87,"body":"what beauty is defined by,"},{"startTime":177.87,"endTime":178.95,"body":"the causes of beauty."},{"startTime":178.95,"endTime":180.57,"body":"Says there are two causes."},{"startTime":180.57,"endTime":183.93,"body":"So natural beauty, that"},{"startTime":180.57,"endTime":183.93,"body":"comes from geometry."},{"startTime":183.93,"endTime":188.34,"body":"And that's uniformity, symmetry,"},{"startTime":183.93,"endTime":188.34,"body":"proportion, those things."},{"startTime":188.34,"endTime":189.687,"body":"That's from geometry,"},{"startTime":189.687,"endTime":191.82,"body":"and that's the cause of natural beauty."},{"startTime":191.82,"endTime":193.8,"body":"Then there's customary beauty, you know,"},{"startTime":193.8,"endTime":195.36,"body":"which is in the eye of the beholder."},{"startTime":195.36,"endTime":196.5,"body":"So a familiar thing,"},{"startTime":196.5,"endTime":199.17,"body":"we may come to love how it looks just"},{"startTime":199.17,"endTime":203.1,"body":"because it's lovely to us because"},{"startTime":199.17,"endTime":203.1,"body":"we are familiar with them,"},{"startTime":203.1,"endTime":206.79,"body":"even though the thing"},{"startTime":203.1,"endTime":206.79,"body":"may not be itself lovely."},{"startTime":206.79,"endTime":210.66,"body":"That ragged, old toy that"},{"startTime":206.79,"endTime":210.66,"body":"we had and loved as a child"},{"startTime":210.66,"endTime":213.753,"body":"is not beautiful, but we love"},{"startTime":210.66,"endTime":213.753,"body":"it because it's our thing."},{"startTime":214.59,"endTime":216.72,"body":"But he says, you know,"},{"startTime":214.59,"endTime":216.72,"body":"if you are ever in doubt"},{"startTime":216.72,"endTime":220.383,"body":"about which of these"},{"startTime":216.72,"endTime":220.383,"body":"should take precedence,"},{"startTime":221.49,"endTime":226.11,"body":"the error is made if you"},{"startTime":221.49,"endTime":226.11,"body":"don't have natural beauty"},{"startTime":226.11,"endTime":227.58,"body":"in the forefront."},{"startTime":227.58,"endTime":228.413,"body":"So he said,"},{"startTime":228.413,"endTime":230.37,"body":"\"This is where the"},{"startTime":228.413,"endTime":230.37,"body":"architect's judgment is tried,"},{"startTime":230.37,"endTime":233.07,"body":"the true test is natural"},{"startTime":230.37,"endTime":233.07,"body":"or geometric beauty.\""},{"startTime":233.07,"endTime":235.89,"body":"So he's got geometry"},{"startTime":233.07,"endTime":235.89,"body":"right at the top there."},{"startTime":235.89,"endTime":237.15,"body":"And he goes further than this."},{"startTime":237.15,"endTime":239.73,"body":"He places mathematics itself kind of"},{"startTime":239.73,"endTime":241.95,"body":"at the top of a hierarchy of truth."},{"startTime":241.95,"endTime":245.49,"body":"So he says mathematical"},{"startTime":241.95,"endTime":245.49,"body":"demonstrations, proofs,"},{"startTime":245.49,"endTime":248.7,"body":"are built on impregnable"},{"startTime":245.49,"endTime":248.7,"body":"foundations of geometry"},{"startTime":248.7,"endTime":250.02,"body":"and of arithmetic,"},{"startTime":250.02,"endTime":252.57,"body":"and they're the only truths that can sink"},{"startTime":252.57,"endTime":255.45,"body":"into the mind of man void of uncertainty."},{"startTime":255.45,"endTime":256.86,"body":"And he goes even further."},{"startTime":256.86,"endTime":261.57,"body":"Everything else is participating"},{"startTime":256.86,"endTime":261.57,"body":"more or less of truth"},{"startTime":261.57,"endTime":263.25,"body":"according as their subjects"},{"startTime":263.25,"endTime":265.65,"body":"are more or less capable of"},{"startTime":263.25,"endTime":265.65,"body":"mathematical demonstrations."},{"startTime":265.65,"endTime":267.48,"body":"So that's a really strong statement"},{"startTime":267.48,"endTime":269.85,"body":"about kind of the"},{"startTime":267.48,"endTime":269.85,"body":"importance of mathematics"},{"startTime":269.85,"endTime":272.34,"body":"and mathematical truth"},{"startTime":269.85,"endTime":272.34,"body":"and geometrical truth"},{"startTime":272.34,"endTime":274.533,"body":"being absolutely at the pinnacle."},{"startTime":277.59,"endTime":279.69,"body":"So we can see, yes,"},{"startTime":279.69,"endTime":282.51,"body":"he places mathematics"},{"startTime":279.69,"endTime":282.51,"body":"very highly in geometry."},{"startTime":282.51,"endTime":283.76,"body":"And actually there's a"},{"startTime":282.51,"endTime":283.76,"body":"clue in this picture,"},{"startTime":283.76,"endTime":286.35,"body":"in this portrait of Wren,"},{"startTime":286.35,"endTime":290.13,"body":"he's of course got the plans"},{"startTime":286.35,"endTime":290.13,"body":"of St. Paul's Cathedral there,"},{"startTime":290.13,"endTime":293.1,"body":"but he also has mathematical"},{"startTime":290.13,"endTime":293.1,"body":"instrument dividers,"},{"startTime":293.1,"endTime":295.05,"body":"and there's a book that's on top of it,"},{"startTime":295.05,"endTime":298.32,"body":"holding it all there, and that"},{"startTime":295.05,"endTime":298.32,"body":"book is \"Euclid's Elements.\""},{"startTime":298.32,"endTime":299.64,"body":"So that's a geometry book."},{"startTime":299.64,"endTime":301.89,"body":"And so there's the clue"},{"startTime":299.64,"endTime":301.89,"body":"there in the picture"},{"startTime":301.89,"endTime":305.583,"body":"to this importance that"},{"startTime":301.89,"endTime":305.583,"body":"Wren placed on mathematics."},{"startTime":306.69,"endTime":310.8,"body":"Now I've said he was a Gresham"},{"startTime":306.69,"endTime":310.8,"body":"professor of astronomy."},{"startTime":310.8,"endTime":312.21,"body":"This is Gresham College,"},{"startTime":312.21,"endTime":315.99,"body":"as it roughly as it"},{"startTime":312.21,"endTime":315.99,"body":"would've been in his time."},{"startTime":315.99,"endTime":320.4,"body":"There was a really exciting"},{"startTime":315.99,"endTime":320.4,"body":"and vibrant community"},{"startTime":320.4,"endTime":323.58,"body":"of mathematicians and"},{"startTime":320.4,"endTime":323.58,"body":"scientists in London,"},{"startTime":323.58,"endTime":326.673,"body":"and in Gresham College at that time."},{"startTime":327.66,"endTime":328.8,"body":"It was a great place to be"},{"startTime":328.8,"endTime":331.62,"body":"and to study science and mathematics."},{"startTime":331.62,"endTime":335.1,"body":"He, in fact, this is"},{"startTime":331.62,"endTime":335.1,"body":"another quotation of Wren,"},{"startTime":335.1,"endTime":336.69,"body":"he was very impressed with London"},{"startTime":336.69,"endTime":338.1,"body":"and we are here in London now."},{"startTime":338.1,"endTime":341.34,"body":"So I extend to you those congratulations."},{"startTime":341.34,"endTime":343.237,"body":"He says, \"I must congratulate this city,"},{"startTime":343.237,"endTime":346.44,"body":"\"that I find in it's so general"},{"startTime":343.237,"endTime":346.44,"body":"a relish of mathematics.\""},{"startTime":346.44,"endTime":348.057,"body":"And you know, we've got a full room today"},{"startTime":348.057,"endTime":350.46,"body":"and I'm pleased that that continues."},{"startTime":350.46,"endTime":353.777,"body":"So well done, London, for"},{"startTime":350.46,"endTime":353.777,"body":"relishing mathematics so,"},{"startTime":353.777,"endTime":355.053,"body":"it's a good thing."},{"startTime":357.15,"endTime":360.63,"body":"So he was educated in Oxford,"},{"startTime":360.63,"endTime":362.82,"body":"became Gresham professor of geometry."},{"startTime":362.82,"endTime":365.1,"body":"He did later get a chair,"},{"startTime":362.82,"endTime":365.1,"body":"a very prestigious chair,"},{"startTime":365.1,"endTime":369.12,"body":"the Civilian Professorship"},{"startTime":365.1,"endTime":369.12,"body":"of Astronomy back in Oxford."},{"startTime":369.12,"endTime":371.88,"body":"But really in this community"},{"startTime":369.12,"endTime":371.88,"body":"around Gresham College,"},{"startTime":371.88,"endTime":374.85,"body":"and also what later"},{"startTime":371.88,"endTime":374.85,"body":"became the Royal Society,"},{"startTime":374.85,"endTime":376.38,"body":"he was really at the heart of it."},{"startTime":376.38,"endTime":378.36,"body":"He was a founder member"},{"startTime":376.38,"endTime":378.36,"body":"of the Royal Society,"},{"startTime":378.36,"endTime":379.8,"body":"he contributed to the meetings,"},{"startTime":379.8,"endTime":382.5,"body":"he had papers in the transactions"},{"startTime":379.8,"endTime":382.5,"body":"of the Royal Society,"},{"startTime":382.5,"endTime":387.3,"body":"and you can also see kind of"},{"startTime":382.5,"endTime":387.3,"body":"the esteem of his colleagues."},{"startTime":387.3,"endTime":389.7,"body":"His contributions were really valuable."},{"startTime":389.7,"endTime":391.59,"body":"So just sort of mention a few names,"},{"startTime":391.59,"endTime":393.36,"body":"there's Christopher"},{"startTime":391.59,"endTime":393.36,"body":"Wren on the left, again,"},{"startTime":393.36,"endTime":394.98,"body":"you've got Robert Hooke,"},{"startTime":394.98,"endTime":399.3,"body":"who was similar time Gresham"},{"startTime":394.98,"endTime":399.3,"body":"professor of geometry,"},{"startTime":399.3,"endTime":400.44,"body":"famous for Hooke's law."},{"startTime":400.44,"endTime":402.633,"body":"He's got a spring there showing you that."},{"startTime":403.5,"endTime":406.59,"body":"Isaac Newton next, and it"},{"startTime":403.5,"endTime":406.59,"body":"wasn't just in England,"},{"startTime":406.59,"endTime":409.68,"body":"he corresponded with"},{"startTime":406.59,"endTime":409.68,"body":"mathematicians on the continent."},{"startTime":409.68,"endTime":411.57,"body":"And there's Pascal on"},{"startTime":409.68,"endTime":411.57,"body":"the right hand side there"},{"startTime":411.57,"endTime":415.62,"body":"with whom he corresponded, among others."},{"startTime":415.62,"endTime":418.11,"body":"And so Newton, I just"},{"startTime":415.62,"endTime":418.11,"body":"want to mention this,"},{"startTime":418.11,"endTime":423.11,"body":"this here is a little extract"},{"startTime":418.11,"endTime":423.11,"body":"from \"Principia Mathematica.\""},{"startTime":423.33,"endTime":425.64,"body":"Now Newton is not someone"},{"startTime":423.33,"endTime":425.64,"body":"we necessarily think of"},{"startTime":425.64,"endTime":427.89,"body":"as being the first person to credit others"},{"startTime":427.89,"endTime":430.32,"body":"with discoveries and be great."},{"startTime":430.32,"endTime":433.35,"body":"But, so he mentions"},{"startTime":430.32,"endTime":433.35,"body":"Wren in the \"Principia,\""},{"startTime":433.35,"endTime":435.51,"body":"and this bit that I've quoted there,"},{"startTime":435.51,"endTime":440.31,"body":"he's talking about inverse"},{"startTime":435.51,"endTime":440.31,"body":"square laws appearing in nature."},{"startTime":440.31,"endTime":443.28,"body":"And he mentions as an example,"},{"startTime":440.31,"endTime":443.28,"body":"the motion of the planets."},{"startTime":443.28,"endTime":446.88,"body":"And he says, as Wren, there's Wrennus,"},{"startTime":446.88,"endTime":448.38,"body":"it's in, you know, the Latin version,"},{"startTime":448.38,"endTime":452.97,"body":"Wren, Hooke and Halley"},{"startTime":448.38,"endTime":452.97,"body":"have severally observed,"},{"startTime":452.97,"endTime":455.25,"body":"and Wren is first in that list."},{"startTime":455.25,"endTime":457.863,"body":"And so he's really given him clear credit"},{"startTime":457.863,"endTime":461.07,"body":"for observations about this."},{"startTime":461.07,"endTime":463.83,"body":"Wren's name is mentioned seven"},{"startTime":461.07,"endTime":463.83,"body":"times in the \"Principia.\""},{"startTime":463.83,"endTime":468.63,"body":"So, you know, he's clearly"},{"startTime":463.83,"endTime":468.63,"body":"up there speaking and working"},{"startTime":468.63,"endTime":471.363,"body":"with the other great"},{"startTime":468.63,"endTime":471.363,"body":"mathematicians at the time."},{"startTime":473.07,"endTime":476.4,"body":"So we've got here someone"},{"startTime":476.4,"endTime":479.313,"body":"who is clearly involved"},{"startTime":476.4,"endTime":479.313,"body":"with these developments."},{"startTime":481.02,"endTime":484.23,"body":"I'm not going to talk about"},{"startTime":481.02,"endTime":484.23,"body":"his astronomy really today"},{"startTime":484.23,"endTime":487.65,"body":"because that's been addressed"},{"startTime":484.23,"endTime":487.65,"body":"in a lecture by my colleague,"},{"startTime":487.65,"endTime":489.99,"body":"Gresham professor of"},{"startTime":487.65,"endTime":489.99,"body":"astronomy, Katherine Blundell,"},{"startTime":489.99,"endTime":491.61,"body":"which you can go and watch online"},{"startTime":491.61,"endTime":493.59,"body":"and find online at your leisure."},{"startTime":493.59,"endTime":496.56,"body":"I'm going to talk about the"},{"startTime":493.59,"endTime":496.56,"body":"mathematics and I'm going to give you,"},{"startTime":496.56,"endTime":500.85,"body":"as I say, three little examples"},{"startTime":500.85,"endTime":504.093,"body":"of mathematics that"},{"startTime":500.85,"endTime":504.093,"body":"Christopher Wren has done."},{"startTime":505.11,"endTime":506.433,"body":"So let's begin."},{"startTime":508.17,"endTime":512.13,"body":"One of the challenges at this time,"},{"startTime":512.13,"endTime":516.24,"body":"at the time of Christopher"},{"startTime":512.13,"endTime":516.24,"body":"Wren was making lenses"},{"startTime":516.24,"endTime":518.85,"body":"for the exciting new"},{"startTime":516.24,"endTime":518.85,"body":"scientific instruments,"},{"startTime":518.85,"endTime":522.81,"body":"telescopes, microscopes, and so on."},{"startTime":522.81,"endTime":525.173,"body":"And when you think, what is a lens doing?"},{"startTime":525.173,"endTime":529.35,"body":"I mean, the basic idea is"},{"startTime":525.173,"endTime":529.35,"body":"you've got a piece of curved,"},{"startTime":529.35,"endTime":534.12,"body":"usually glass, a lens, and the"},{"startTime":529.35,"endTime":534.12,"body":"ray of light are coming in,"},{"startTime":534.12,"endTime":538.86,"body":"and then we would like to"},{"startTime":534.12,"endTime":538.86,"body":"focus them to a single point."},{"startTime":538.86,"endTime":541.11,"body":"And how is that focusing happening?"},{"startTime":541.11,"endTime":543.42,"body":"Well, it's the rays are coming in"},{"startTime":543.42,"endTime":546.15,"body":"and they're passing through the glass,"},{"startTime":546.15,"endTime":548.34,"body":"and then when they hit the curved surface,"},{"startTime":548.34,"endTime":553.14,"body":"they are refracted and we"},{"startTime":548.34,"endTime":553.14,"body":"know we can work out the angle"},{"startTime":553.14,"endTime":555.27,"body":"by which their direction"},{"startTime":555.27,"endTime":557.79,"body":"is changed using something"},{"startTime":555.27,"endTime":557.79,"body":"called Snell's laws,"},{"startTime":557.79,"endTime":560.04,"body":"which tell you, depending"},{"startTime":560.04,"endTime":561.6,"body":"on what mediums you're passing through"},{"startTime":561.6,"endTime":565.05,"body":"and what the angle is, that"},{"startTime":561.6,"endTime":565.05,"body":"you are hitting the glass at,"},{"startTime":565.05,"endTime":567.243,"body":"what the new angle of the ray will be."},{"startTime":568.56,"endTime":572.13,"body":"Now the easiest kind of"},{"startTime":568.56,"endTime":572.13,"body":"lens to make, to grind"},{"startTime":572.13,"endTime":573.81,"body":"and at this time, probably the only one"},{"startTime":573.81,"endTime":577.02,"body":"that could easily be done"},{"startTime":573.81,"endTime":577.02,"body":"was a spherical lens,"},{"startTime":577.02,"endTime":579.75,"body":"so the spherical kind of curvature."},{"startTime":579.75,"endTime":580.583,"body":"But the problem with that"},{"startTime":580.583,"endTime":584.19,"body":"is it doesn't actually give"},{"startTime":580.583,"endTime":584.19,"body":"you this beautiful focusing"},{"startTime":584.19,"endTime":585.93,"body":"to a single point."},{"startTime":585.93,"endTime":588.15,"body":"What you get is something more like this."},{"startTime":588.15,"endTime":590.31,"body":"It's gets a little bit"},{"startTime":588.15,"endTime":590.31,"body":"blurry at the edges."},{"startTime":590.31,"endTime":591.84,"body":"It's called spherical aberration."},{"startTime":591.84,"endTime":594.273,"body":"It's not the perfect shape."},{"startTime":595.17,"endTime":597.36,"body":"And this, of course is a problem."},{"startTime":597.36,"endTime":601.77,"body":"You want your instruments to"},{"startTime":597.36,"endTime":601.77,"body":"give you really precise images."},{"startTime":601.77,"endTime":603.42,"body":"So you would like to know what"},{"startTime":603.42,"endTime":606.51,"body":"is actually the right curve to use."},{"startTime":606.51,"endTime":610.29,"body":"Now, just to say, remember"},{"startTime":606.51,"endTime":610.29,"body":"this is refraction."},{"startTime":610.29,"endTime":612.6,"body":"If it were reflection, if we"},{"startTime":610.29,"endTime":612.6,"body":"were reflecting in a mirror,"},{"startTime":612.6,"endTime":615.09,"body":"like if we are thinking"},{"startTime":612.6,"endTime":615.09,"body":"of a telescope where you,"},{"startTime":615.09,"endTime":616.05,"body":"the rays are coming in"},{"startTime":616.05,"endTime":618.84,"body":"and they're reflecting off a reflector"},{"startTime":618.84,"endTime":621.09,"body":"and focusing in on a point,"},{"startTime":618.84,"endTime":621.09,"body":"we know the answer to that,"},{"startTime":621.09,"endTime":622.92,"body":"we would use the parabola for that."},{"startTime":622.92,"endTime":624.63,"body":"But this isn't reflection,"},{"startTime":622.92,"endTime":624.63,"body":"it's refraction."},{"startTime":624.63,"endTime":626.46,"body":"So it's a different thing going on."},{"startTime":626.46,"endTime":629.58,"body":"However, conic sections"},{"startTime":626.46,"endTime":629.58,"body":"are still relevant."},{"startTime":629.58,"endTime":632.04,"body":"So I'll just remind you"},{"startTime":629.58,"endTime":632.04,"body":"what that phrase is."},{"startTime":632.04,"endTime":635.22,"body":"So I mentioned a parabola"},{"startTime":632.04,"endTime":635.22,"body":"already, that's one."},{"startTime":635.22,"endTime":638.19,"body":"Conic sections are the curves you get"},{"startTime":638.19,"endTime":641.91,"body":"when you slice through a"},{"startTime":638.19,"endTime":641.91,"body":"cone at different angles."},{"startTime":641.91,"endTime":645.9,"body":"So if the plane that"},{"startTime":641.91,"endTime":645.9,"body":"you are slicing with is"},{"startTime":645.9,"endTime":650.37,"body":"has a shallow or is horizontal"},{"startTime":645.9,"endTime":650.37,"body":"or has a very shallow angle,"},{"startTime":650.37,"endTime":651.69,"body":"you get ellipses."},{"startTime":651.69,"endTime":653.94,"body":"And then if you keep increasing the angle,"},{"startTime":653.94,"endTime":656.04,"body":"going the other way, if you"},{"startTime":653.94,"endTime":656.04,"body":"keep increasing the angle,"},{"startTime":656.04,"endTime":658.65,"body":"at some point you'll hit"},{"startTime":656.04,"endTime":658.65,"body":"precisely the same slope"},{"startTime":658.65,"endTime":661.41,"body":"as that of the cone, and"},{"startTime":658.65,"endTime":661.41,"body":"that's when you get a parabola."},{"startTime":661.41,"endTime":663.51,"body":"And then if you go on past that point,"},{"startTime":663.51,"endTime":666.45,"body":"you start to hit both halves of the cone,"},{"startTime":666.45,"endTime":668.55,"body":"and that's when you get a hyperbola."},{"startTime":668.55,"endTime":669.887,"body":"So I've just taken that hyperbola shape,"},{"startTime":669.887,"endTime":672.75,"body":"it's a bit hard to see,"},{"startTime":669.887,"endTime":672.75,"body":"I've turned it on its side."},{"startTime":672.75,"endTime":675.843,"body":"So that's the shape of"},{"startTime":672.75,"endTime":675.843,"body":"this hyperbola curve."},{"startTime":676.89,"endTime":681.51,"body":"And it turns out that this is the right"},{"startTime":681.51,"endTime":685.5,"body":"or a better at least curved shape"},{"startTime":685.5,"endTime":688.89,"body":"for your lens in cross section."},{"startTime":688.89,"endTime":691.53,"body":"So that's the kind of curve"},{"startTime":688.89,"endTime":691.53,"body":"we're mountain to aim for"},{"startTime":691.53,"endTime":693.27,"body":"with our lens."},{"startTime":693.27,"endTime":696.0,"body":"But course lenses are"},{"startTime":693.27,"endTime":696.0,"body":"three-dimensional objects."},{"startTime":696.0,"endTime":699.39,"body":"So just having a cross"},{"startTime":696.0,"endTime":699.39,"body":"section is one thing."},{"startTime":699.39,"endTime":701.58,"body":"If you take that hyperbola shape"},{"startTime":701.58,"endTime":704.79,"body":"and then you've just rotated"},{"startTime":701.58,"endTime":704.79,"body":"about it the vertical axis,"},{"startTime":704.79,"endTime":707.76,"body":"you get a three dimensional"},{"startTime":704.79,"endTime":707.76,"body":"shape called a hyperboloid,"},{"startTime":707.76,"endTime":711.72,"body":"which actually nowadays we"},{"startTime":707.76,"endTime":711.72,"body":"most often encounter this shape"},{"startTime":711.72,"endTime":713.64,"body":"in the form of a cooling tower."},{"startTime":713.64,"endTime":715.41,"body":"And there's mathematical reasons"},{"startTime":715.41,"endTime":717.0,"body":"why this is a good shape"},{"startTime":715.41,"endTime":717.0,"body":"for a cooling tower,"},{"startTime":717.0,"endTime":718.89,"body":"which I will not go into now,"},{"startTime":718.89,"endTime":723.48,"body":"but that's the kind of shape"},{"startTime":718.89,"endTime":723.48,"body":"that I'm talking about."},{"startTime":723.48,"endTime":727.83,"body":"And it turns out that yes,"},{"startTime":723.48,"endTime":727.83,"body":"this hyperboloid shape"},{"startTime":727.83,"endTime":732.0,"body":"is a better solution than"},{"startTime":727.83,"endTime":732.0,"body":"spherical, one, full lenses."},{"startTime":732.0,"endTime":734.67,"body":"But the problem is how do"},{"startTime":732.0,"endTime":734.67,"body":"you actually make that,"},{"startTime":734.67,"endTime":739.47,"body":"how do you make a lens with"},{"startTime":734.67,"endTime":739.47,"body":"that particular curvature to it?"},{"startTime":739.47,"endTime":743.957,"body":"And so here is where the mind"},{"startTime":739.47,"endTime":743.957,"body":"of Christopher Wren comes in,"},{"startTime":743.957,"endTime":747.453,"body":"and in particular he's"},{"startTime":743.957,"endTime":747.453,"body":"out shopping one day"},{"startTime":747.453,"endTime":751.05,"body":"and he has a mathematical inspiration."},{"startTime":751.05,"endTime":753.45,"body":"And so we should all do more shopping."},{"startTime":753.45,"endTime":755.07,"body":"This is what happens, he sees a basket,"},{"startTime":755.07,"endTime":756.99,"body":"okay, history doesn't recall"},{"startTime":755.07,"endTime":756.99,"body":"whether he bought the basket."},{"startTime":756.99,"endTime":761.37,"body":"But anyway, I've drawn this"},{"startTime":756.99,"endTime":761.37,"body":"basket here with my imagination."},{"startTime":761.37,"endTime":764.85,"body":"He sees a basket, a wicker"},{"startTime":761.37,"endTime":764.85,"body":"basket, and he notices,"},{"startTime":764.85,"endTime":767.19,"body":"so it's got this nice curved shape,"},{"startTime":767.19,"endTime":772.19,"body":"but the way it's made is with"},{"startTime":767.19,"endTime":772.19,"body":"canes of willow, straight,"},{"startTime":773.94,"endTime":776.37,"body":"straight rods or sticks of willow."},{"startTime":776.37,"endTime":777.57,"body":"But the way they're arranged,"},{"startTime":777.57,"endTime":781.65,"body":"it gives this curved"},{"startTime":777.57,"endTime":781.65,"body":"shape, this curved basket."},{"startTime":781.65,"endTime":784.353,"body":"And he realizes, hang on a minute,"},{"startTime":785.88,"endTime":789.36,"body":"I think that means that"},{"startTime":785.88,"endTime":789.36,"body":"actually a hyperboloid"},{"startTime":789.36,"endTime":792.81,"body":"can be made with straight lights."},{"startTime":792.81,"endTime":796.23,"body":"And I'll show you this sort of happening,"},{"startTime":796.23,"endTime":801.23,"body":"take two circles and put sticks or rods"},{"startTime":802.44,"endTime":804.87,"body":"or even bits of string"},{"startTime":802.44,"endTime":804.87,"body":"or something between them"},{"startTime":804.87,"endTime":806.94,"body":"all the way around like"},{"startTime":804.87,"endTime":806.94,"body":"this, like I've done."},{"startTime":806.94,"endTime":809.52,"body":"And then what we're going to do"},{"startTime":806.94,"endTime":809.52,"body":"is we're going to twist them bit."},{"startTime":809.52,"endTime":811.53,"body":"Now they'll have to get"},{"startTime":809.52,"endTime":811.53,"body":"a bit closer together"},{"startTime":811.53,"endTime":813.42,"body":"because the sticks are still straight."},{"startTime":813.42,"endTime":815.85,"body":"They're not going to suddenly"},{"startTime":813.42,"endTime":815.85,"body":"become curved or stretched."},{"startTime":815.85,"endTime":816.99,"body":"So they'll get a bit closer together"},{"startTime":816.99,"endTime":818.52,"body":"when we do this twisting."},{"startTime":818.52,"endTime":822.0,"body":"But see what happens, twist"},{"startTime":818.52,"endTime":822.0,"body":"a bit, that's what you get."},{"startTime":822.0,"endTime":825.24,"body":"Twist a bit more and a"},{"startTime":822.0,"endTime":825.24,"body":"bit more and a bit more."},{"startTime":825.24,"endTime":828.24,"body":"And look, you get exactly"},{"startTime":825.24,"endTime":828.24,"body":"this cooling towel,"},{"startTime":828.24,"endTime":830.55,"body":"this hyperboloid shape developing."},{"startTime":830.55,"endTime":831.99,"body":"And if you had infinitely many lines,"},{"startTime":831.99,"endTime":833.54,"body":"you'd get the full thing there."},{"startTime":834.6,"endTime":837.75,"body":"So it turns out what Wren realized"},{"startTime":837.75,"endTime":840.93,"body":"and then showed properly was"},{"startTime":837.75,"endTime":840.93,"body":"that you can make a hyperboloid"},{"startTime":840.93,"endTime":844.17,"body":"with straight lines."},{"startTime":844.17,"endTime":847.11,"body":"Now I've got my circles the same size,"},{"startTime":847.11,"endTime":849.27,"body":"but if you change the sizes of the circles"},{"startTime":849.27,"endTime":850.56,"body":"making different sizes,"},{"startTime":850.56,"endTime":852.78,"body":"what you get is kind of the hyperboloid"},{"startTime":852.78,"endTime":855.39,"body":"but sliced off at different"},{"startTime":852.78,"endTime":855.39,"body":"points, top and bottom."},{"startTime":855.39,"endTime":857.79,"body":"So for instance, that,"},{"startTime":857.79,"endTime":860.85,"body":"and then they see it's just"},{"startTime":857.79,"endTime":860.85,"body":"a short step to your basket."},{"startTime":860.85,"endTime":864.93,"body":"So just not so the other"},{"startTime":860.85,"endTime":864.93,"body":"conic sections don't feel sad,"},{"startTime":864.93,"endTime":866.613,"body":"I've made the handle a parabola."},{"startTime":868.32,"endTime":870.36,"body":"I dunno whether it was in the shop"},{"startTime":870.36,"endTime":871.56,"body":"that Christopher Wren saw it."},{"startTime":871.56,"endTime":873.87,"body":"But there we are, so that's the basket."},{"startTime":873.87,"endTime":876.87,"body":"And this is not just a"},{"startTime":873.87,"endTime":876.87,"body":"theoretical observation,"},{"startTime":876.87,"endTime":879.75,"body":"and this is one of the hallmarks"},{"startTime":876.87,"endTime":879.75,"body":"of Christopher Wren's work"},{"startTime":879.75,"endTime":883.11,"body":"and his attitude to mathematics."},{"startTime":883.11,"endTime":885.6,"body":"It's a practical thing."},{"startTime":885.6,"endTime":887.46,"body":"So of course we use it in architecture,"},{"startTime":887.46,"endTime":889.77,"body":"but we, you know, we have"},{"startTime":887.46,"endTime":889.77,"body":"practical applications of it."},{"startTime":889.77,"endTime":892.56,"body":"And he doesn't just say, I've"},{"startTime":889.77,"endTime":892.56,"body":"had a nice theoretical idea."},{"startTime":892.56,"endTime":894.63,"body":"He does think about"},{"startTime":892.56,"endTime":894.63,"body":"how it could be useful."},{"startTime":894.63,"endTime":898.71,"body":"So he realized that this straight line way"},{"startTime":898.71,"endTime":903.21,"body":"of making hyperboloids could be then used,"},{"startTime":903.21,"endTime":904.56,"body":"this idea could be used"},{"startTime":904.56,"endTime":907.77,"body":"because you could then"},{"startTime":904.56,"endTime":907.77,"body":"make hyperboloid with a,"},{"startTime":907.77,"endTime":909.48,"body":"you start with a cylinder on a lathe,"},{"startTime":909.48,"endTime":914.48,"body":"and then you just need to"},{"startTime":909.48,"endTime":914.48,"body":"have a straight cutting device"},{"startTime":915.51,"endTime":918.3,"body":"at an angle, at a fixed angle,"},{"startTime":915.51,"endTime":918.3,"body":"then you rotate the cylinder"},{"startTime":918.3,"endTime":920.25,"body":"and it will just produce,"},{"startTime":920.25,"endTime":922.5,"body":"then eventually produce a hyperboloid,"},{"startTime":922.5,"endTime":925.83,"body":"just with a rotating lathe"},{"startTime":922.5,"endTime":925.83,"body":"and one straight edge."},{"startTime":925.83,"endTime":930.24,"body":"So that already is a good"},{"startTime":925.83,"endTime":930.24,"body":"thing to have noticed."},{"startTime":930.24,"endTime":932.04,"body":"But he also, then he went further,"},{"startTime":932.04,"endTime":934.14,"body":"he came up with a design for a machine"},{"startTime":934.14,"endTime":936.15,"body":"to grind hyperbolic lenses."},{"startTime":936.15,"endTime":938.01,"body":"And this is, so this is his paper"},{"startTime":938.01,"endTime":941.4,"body":"from his paper in the"},{"startTime":938.01,"endTime":941.4,"body":"transactions of the Royal Society"},{"startTime":941.4,"endTime":943.74,"body":"and what you've got here,"},{"startTime":943.74,"endTime":946.71,"body":"so you'll start with two"},{"startTime":943.74,"endTime":946.71,"body":"cylinders at an angle,"},{"startTime":946.71,"endTime":949.41,"body":"and then at the back you've"},{"startTime":946.71,"endTime":949.41,"body":"got this piece of glass"},{"startTime":949.41,"endTime":951.87,"body":"and they're all kind of"},{"startTime":949.41,"endTime":951.87,"body":"working against each other"},{"startTime":951.87,"endTime":954.15,"body":"to gradually grind the right shape."},{"startTime":954.15,"endTime":956.31,"body":"So the cylinders will"},{"startTime":954.15,"endTime":956.31,"body":"work against each other"},{"startTime":956.31,"endTime":958.92,"body":"to produce two hyperboloids."},{"startTime":958.92,"endTime":961.95,"body":"And then the cylinder at the"},{"startTime":958.92,"endTime":961.95,"body":"back will also simultaneously"},{"startTime":961.95,"endTime":964.02,"body":"be working against the piece of glass"},{"startTime":964.02,"endTime":966.09,"body":"to grind a hyperbolic lens."},{"startTime":966.09,"endTime":969.45,"body":"So he designed a machine"},{"startTime":966.09,"endTime":969.45,"body":"and I do like this diagram."},{"startTime":969.45,"endTime":971.34,"body":"So that's an example of a time"},{"startTime":971.34,"endTime":973.59,"body":"where he made a theoretical observation,"},{"startTime":973.59,"endTime":976.92,"body":"he did the mathematics,"},{"startTime":973.59,"endTime":976.92,"body":"but he also then described"},{"startTime":976.92,"endTime":980.913,"body":"and made a machine to show"},{"startTime":976.92,"endTime":980.913,"body":"a practical application."},{"startTime":983.22,"endTime":984.813,"body":"So that's gem number one."},{"startTime":986.01,"endTime":991.01,"body":"Next up, a story that"},{"startTime":986.01,"endTime":991.01,"body":"began in February, 1658"},{"startTime":994.14,"endTime":996.96,"body":"when several mathematicians"},{"startTime":996.96,"endTime":1000.41,"body":"in England received a mysterious challenge"},{"startTime":1000.41,"endTime":1003.563,"body":"from France, from one Jean de Montfort,"},{"startTime":1004.46,"endTime":1007.91,"body":"who greatly desired that"},{"startTime":1004.46,"endTime":1007.91,"body":"those distinguished gentlemen,"},{"startTime":1007.91,"endTime":1011.42,"body":"the professors of mathematics"},{"startTime":1007.91,"endTime":1011.42,"body":"and others in England renowned"},{"startTime":1011.42,"endTime":1014.18,"body":"for mathematical skill may condescend"},{"startTime":1014.18,"endTime":1015.38,"body":"to resolve this problem."},{"startTime":1015.38,"endTime":1017.93,"body":"I'll tell you more about"},{"startTime":1015.38,"endTime":1017.93,"body":"the problem in a minute."},{"startTime":1017.93,"endTime":1021.59,"body":"But they were issued this"},{"startTime":1017.93,"endTime":1021.59,"body":"challenge by Jean de Montfort."},{"startTime":1021.59,"endTime":1023.33,"body":"There is no Jean de Montfort,"},{"startTime":1023.33,"endTime":1025.73,"body":"it's a mirage and it's an illusion."},{"startTime":1025.73,"endTime":1028.55,"body":"There's no one called Jean de"},{"startTime":1025.73,"endTime":1028.55,"body":"Montfort or no mathematician."},{"startTime":1028.55,"endTime":1032.09,"body":"This was a pseudonym"},{"startTime":1028.55,"endTime":1032.09,"body":"for we don't know who."},{"startTime":1032.09,"endTime":1035.27,"body":"Now there's speculation,"},{"startTime":1032.09,"endTime":1035.27,"body":"it might have been Pascal,"},{"startTime":1035.27,"endTime":1037.04,"body":"we don't know, we'll never know."},{"startTime":1037.04,"endTime":1040.22,"body":"But anyway, the challenge was laid down."},{"startTime":1040.22,"endTime":1042.62,"body":"Can the distinguished gentlemen,"},{"startTime":1042.62,"endTime":1047.62,"body":"professors of mathematics"},{"startTime":1042.62,"endTime":1047.62,"body":"condescend to resolve the problem?"},{"startTime":1048.05,"endTime":1049.82,"body":"So I won't give you the full detail of it,"},{"startTime":1049.82,"endTime":1053.48,"body":"but the basic problem was"},{"startTime":1049.82,"endTime":1053.48,"body":"you are given any ellipse"},{"startTime":1053.48,"endTime":1054.67,"body":"that you know the dimensions of,"},{"startTime":1054.67,"endTime":1057.44,"body":"so you know the length of"},{"startTime":1054.67,"endTime":1057.44,"body":"the like the major axis,"},{"startTime":1057.44,"endTime":1058.88,"body":"you know the length of the minor axis,"},{"startTime":1058.88,"endTime":1060.89,"body":"so you know the ellipse"},{"startTime":1060.89,"endTime":1064.19,"body":"and you are told there's a"},{"startTime":1060.89,"endTime":1064.19,"body":"cord through the ellipse,"},{"startTime":1064.19,"endTime":1067.37,"body":"you are told where it"},{"startTime":1064.19,"endTime":1067.37,"body":"crosses the major axis"},{"startTime":1067.37,"endTime":1069.11,"body":"and at what angle."},{"startTime":1069.11,"endTime":1071.51,"body":"And the challenge is"},{"startTime":1069.11,"endTime":1071.51,"body":"given that information,"},{"startTime":1071.51,"endTime":1072.95,"body":"can you determine the lengths"},{"startTime":1072.95,"endTime":1077.9,"body":"of those two segments into"},{"startTime":1072.95,"endTime":1077.9,"body":"which the cord is divided, okay?"},{"startTime":1077.9,"endTime":1080.27,"body":"So that's the challenge."},{"startTime":1080.27,"endTime":1083.868,"body":"And Christopher Wren did"},{"startTime":1080.27,"endTime":1083.868,"body":"condescend to resolve it."},{"startTime":1083.868,"endTime":1086.09,"body":"So he managed to solve this problem."},{"startTime":1086.09,"endTime":1088.37,"body":"And so he wrote up his solution,"},{"startTime":1088.37,"endTime":1091.64,"body":"and then what he did was to"},{"startTime":1088.37,"endTime":1091.64,"body":"say, so I've done your problem."},{"startTime":1091.64,"endTime":1093.74,"body":"Here's a problem for you guys to solve."},{"startTime":1093.74,"endTime":1096.41,"body":"And he issued a challenge back in return."},{"startTime":1096.41,"endTime":1098.81,"body":"So let me tell you what"},{"startTime":1096.41,"endTime":1098.81,"body":"this return challenge was,"},{"startTime":1098.81,"endTime":1101.063,"body":"and it has to do with Kepler's laws."},{"startTime":1102.05,"endTime":1106.28,"body":"So here's Kepler and this is an ellipse,"},{"startTime":1106.28,"endTime":1109.34,"body":"it's illustrating one of Kepler's laws."},{"startTime":1109.34,"endTime":1112.31,"body":"So these were describing planetary motion"},{"startTime":1112.31,"endTime":1117.31,"body":"and planets move around the"},{"startTime":1112.31,"endTime":1117.31,"body":"sun in elliptical orbits"},{"startTime":1117.35,"endTime":1118.76,"body":"with the sun at one focus,"},{"startTime":1118.76,"endTime":1121.73,"body":"which I've sort of got on"},{"startTime":1118.76,"endTime":1121.73,"body":"the left hand side there."},{"startTime":1121.73,"endTime":1126.73,"body":"They don't move at a constant"},{"startTime":1121.73,"endTime":1126.73,"body":"speed, their speed changes,"},{"startTime":1127.37,"endTime":1130.55,"body":"but there is something"},{"startTime":1127.37,"endTime":1130.55,"body":"that does remain constant."},{"startTime":1130.55,"endTime":1133.34,"body":"And this is what one of"},{"startTime":1130.55,"endTime":1133.34,"body":"Kepler's laws tells us."},{"startTime":1133.34,"endTime":1137.72,"body":"It's the area swept out by the planet"},{"startTime":1137.72,"endTime":1139.55,"body":"in a given amount of time,"},{"startTime":1139.55,"endTime":1141.9,"body":"is constant no matter"},{"startTime":1139.55,"endTime":1141.9,"body":"where it is in its orbit."},{"startTime":1142.91,"endTime":1145.1,"body":"So the area is constant,"},{"startTime":1142.91,"endTime":1145.1,"body":"but then obviously"},{"startTime":1145.1,"endTime":1146.66,"body":"at different times in the orbit,"},{"startTime":1146.66,"endTime":1149.21,"body":"that will mean the distance it's traveled"},{"startTime":1149.21,"endTime":1150.59,"body":"around the ellipse may not be constant,"},{"startTime":1150.59,"endTime":1151.97,"body":"so the speed will vary."},{"startTime":1151.97,"endTime":1154.46,"body":"But this thing about"},{"startTime":1151.97,"endTime":1154.46,"body":"the area is a constant."},{"startTime":1154.46,"endTime":1158.03,"body":"So Kepler observed this, and in order"},{"startTime":1158.03,"endTime":1161.993,"body":"to kind of do the mathematics"},{"startTime":1158.03,"endTime":1161.993,"body":"that lies behind that,"},{"startTime":1163.19,"endTime":1166.22,"body":"there's a problem that"},{"startTime":1163.19,"endTime":1166.22,"body":"he raised, which is okay,"},{"startTime":1166.22,"endTime":1170.03,"body":"if you've got an ellipse, and"},{"startTime":1166.22,"endTime":1170.03,"body":"I'm going to divide it in two,"},{"startTime":1170.03,"endTime":1172.19,"body":"so make it a semi ellipse."},{"startTime":1172.19,"endTime":1176.51,"body":"The challenge is can you draw a line"},{"startTime":1176.51,"endTime":1179.96,"body":"from your favorite focus to the ellipse"},{"startTime":1179.96,"endTime":1184.04,"body":"that divides that semi"},{"startTime":1179.96,"endTime":1184.04,"body":"ellipse in a specified ratio?"},{"startTime":1184.04,"endTime":1186.41,"body":"So you come to me and"},{"startTime":1184.04,"endTime":1186.41,"body":"say, I want the ratio,"},{"startTime":1186.41,"endTime":1188.27,"body":"I want it to be 50 50 split,"},{"startTime":1188.27,"endTime":1190.4,"body":"and then I've got to"},{"startTime":1188.27,"endTime":1190.4,"body":"be able to draw a line,"},{"startTime":1190.4,"endTime":1192.47,"body":"the relevant line from the focus"},{"startTime":1192.47,"endTime":1195.77,"body":"to the ellipse that"},{"startTime":1192.47,"endTime":1195.77,"body":"divides it in that ratio."},{"startTime":1195.77,"endTime":1197.927,"body":"I tried for 50 50, I'm not"},{"startTime":1195.77,"endTime":1197.927,"body":"sure I quite managed it,"},{"startTime":1197.927,"endTime":1200.51,"body":"but the yellow might be a bit bigger."},{"startTime":1200.51,"endTime":1202.91,"body":"But anyway, Kepler had this problem"},{"startTime":1202.91,"endTime":1207.02,"body":"and he realized that you could turn it"},{"startTime":1207.02,"endTime":1209.9,"body":"into a related problem about semi-circles."},{"startTime":1209.9,"endTime":1211.52,"body":"So it be an equivalent problem,"},{"startTime":1211.52,"endTime":1213.68,"body":"if you could do the semi-circle problem,"},{"startTime":1213.68,"endTime":1214.52,"body":"which I'll show you,"},{"startTime":1214.52,"endTime":1216.38,"body":"then you'd be able to"},{"startTime":1214.52,"endTime":1216.38,"body":"do the ellipse problem."},{"startTime":1216.38,"endTime":1219.62,"body":"So the semi-circle problem was this time,"},{"startTime":1219.62,"endTime":1221.3,"body":"you've got a semi-circle"},{"startTime":1221.3,"endTime":1225.56,"body":"and you fix a point"},{"startTime":1221.3,"endTime":1225.56,"body":"somewhere on the diameter"},{"startTime":1225.56,"endTime":1228.32,"body":"of that semi-circle, and then"},{"startTime":1225.56,"endTime":1228.32,"body":"you have to do the same thing."},{"startTime":1228.32,"endTime":1232.76,"body":"Can you find the line from that point"},{"startTime":1232.76,"endTime":1235.28,"body":"on the diameter to the circumference"},{"startTime":1235.28,"endTime":1239.603,"body":"that divides the semicircle in"},{"startTime":1235.28,"endTime":1239.603,"body":"the specified desired ratio?"},{"startTime":1241.43,"endTime":1244.52,"body":"Okay, so that became"},{"startTime":1241.43,"endTime":1244.52,"body":"known as Kepler's Problem"},{"startTime":1244.52,"endTime":1249.383,"body":"and Wren challenged the French"},{"startTime":1244.52,"endTime":1249.383,"body":"mathematicians to solve it."},{"startTime":1250.73,"endTime":1254.3,"body":"Now Wren had actually solved"},{"startTime":1250.73,"endTime":1254.3,"body":"this problem himself already,"},{"startTime":1254.3,"endTime":1256.76,"body":"and it's quite good to already know"},{"startTime":1256.76,"endTime":1257.87,"body":"if you're going to issue a challenge,"},{"startTime":1257.87,"endTime":1259.85,"body":"it's good to know that"},{"startTime":1257.87,"endTime":1259.85,"body":"you could do it yourself."},{"startTime":1259.85,"endTime":1260.72,"body":"So he'd solved this problem"},{"startTime":1260.72,"endTime":1263.06,"body":"and he did it using a"},{"startTime":1260.72,"endTime":1263.06,"body":"lovely mathematical curve"},{"startTime":1263.06,"endTime":1264.32,"body":"called a cycloid."},{"startTime":1264.32,"endTime":1267.14,"body":"So I'm showing you a"},{"startTime":1264.32,"endTime":1267.14,"body":"cycloid now being made."},{"startTime":1267.14,"endTime":1271.13,"body":"It's the curve that is"},{"startTime":1267.14,"endTime":1271.13,"body":"produced by following the path"},{"startTime":1271.13,"endTime":1274.7,"body":"of a point on the rim of a circle"},{"startTime":1274.7,"endTime":1276.473,"body":"as it rolls along a straight line."},{"startTime":1277.76,"endTime":1280.04,"body":"So that's it just being produced there."},{"startTime":1280.04,"endTime":1281.33,"body":"And you can see you get these kind"},{"startTime":1281.33,"endTime":1284.84,"body":"of arches repeating shapes, arches."},{"startTime":1284.84,"endTime":1287.69,"body":"And that's what a cycloid is."},{"startTime":1287.69,"endTime":1292.1,"body":"Now cycloids are lovely, the"},{"startTime":1287.69,"endTime":1292.1,"body":"cycloid is a lovely thing."},{"startTime":1292.1,"endTime":1294.26,"body":"It was studied, it was all the rage really"},{"startTime":1294.26,"endTime":1296.24,"body":"in the 17th century."},{"startTime":1296.24,"endTime":1299.48,"body":"Galileo started studying cycloids,"},{"startTime":1299.48,"endTime":1300.89,"body":"one of the first to think about them."},{"startTime":1300.89,"endTime":1303.8,"body":"He gave this curve the name cycloid,"},{"startTime":1303.8,"endTime":1306.05,"body":"he wrote that he'd"},{"startTime":1303.8,"endTime":1306.05,"body":"studied them for 50 years."},{"startTime":1307.19,"endTime":1308.45,"body":"I can't tell you everything"},{"startTime":1308.45,"endTime":1310.4,"body":"that's marvelous about them right away,"},{"startTime":1310.4,"endTime":1311.93,"body":"but I did give a lecture"},{"startTime":1310.4,"endTime":1311.93,"body":"on them last year."},{"startTime":1311.93,"endTime":1313.07,"body":"So you can go and find that"},{"startTime":1313.07,"endTime":1314.75,"body":"and watch it if you want to know more."},{"startTime":1314.75,"endTime":1319.75,"body":"But like everyone was studying"},{"startTime":1314.75,"endTime":1319.75,"body":"cycloids in 17th century."},{"startTime":1319.97,"endTime":1323.03,"body":"It started with, sort"},{"startTime":1319.97,"endTime":1323.03,"body":"of started with Galileo"},{"startTime":1323.03,"endTime":1325.82,"body":"and then Mersenne of Mersenne Primes fame,"},{"startTime":1325.82,"endTime":1328.22,"body":"Pascal of Pascal's Triangles fame,"},{"startTime":1328.22,"endTime":1333.22,"body":"Newton of being Newton"},{"startTime":1328.22,"endTime":1333.22,"body":"fame, Descartes, Fermat,"},{"startTime":1333.323,"endTime":1335.537,"body":"they were all looking at the cycloid"},{"startTime":1335.537,"endTime":1337.4,"body":"and they all studied it."},{"startTime":1337.4,"endTime":1341.39,"body":"And the two main questions that"},{"startTime":1337.4,"endTime":1341.39,"body":"were being asked about this"},{"startTime":1341.39,"endTime":1343.19,"body":"were questions that"},{"startTime":1341.39,"endTime":1343.19,"body":"actually tended to be asked"},{"startTime":1343.19,"endTime":1345.83,"body":"about any curve that was being studied."},{"startTime":1345.83,"endTime":1347.15,"body":"And they were known as the problems"},{"startTime":1347.15,"endTime":1349.67,"body":"of quadrature and rectification."},{"startTime":1349.67,"endTime":1352.58,"body":"So to our modern"},{"startTime":1349.67,"endTime":1352.58,"body":"language, quadrature means"},{"startTime":1352.58,"endTime":1354.443,"body":"what's the area under the curve?"},{"startTime":1355.37,"endTime":1357.02,"body":"So in our case, obviously the cycloid"},{"startTime":1357.02,"endTime":1358.82,"body":"could continue forever, but we just think,"},{"startTime":1358.82,"endTime":1361.31,"body":"we take one arch of it"},{"startTime":1358.82,"endTime":1361.31,"body":"and care about that."},{"startTime":1361.31,"endTime":1363.68,"body":"And we're interested in what's"},{"startTime":1361.31,"endTime":1363.68,"body":"the area between that arch"},{"startTime":1363.68,"endTime":1364.97,"body":"and the straight line."},{"startTime":1364.97,"endTime":1366.14,"body":"So what's the area, you know,"},{"startTime":1366.14,"endTime":1369.05,"body":"in terms of the original"},{"startTime":1366.14,"endTime":1369.05,"body":"circle that generated it?"},{"startTime":1369.05,"endTime":1372.59,"body":"That's quadrature and then rectification,"},{"startTime":1372.59,"endTime":1375.97,"body":"that is what is the length of that curve,"},{"startTime":1375.97,"endTime":1377.453,"body":"of the length of the arch?"},{"startTime":1378.95,"endTime":1381.62,"body":"And those two questions"},{"startTime":1378.95,"endTime":1381.62,"body":"were asked about the cycloid"},{"startTime":1381.62,"endTime":1383.9,"body":"as they were about many other curves."},{"startTime":1383.9,"endTime":1385.91,"body":"And these are difficult things to do."},{"startTime":1385.91,"endTime":1386.75,"body":"I mean, you've got to remember"},{"startTime":1386.75,"endTime":1388.64,"body":"at the start of the 17th century,"},{"startTime":1388.64,"endTime":1390.47,"body":"we didn't really even have what you"},{"startTime":1390.47,"endTime":1392.75,"body":"might call modern algebraic notation."},{"startTime":1392.75,"endTime":1396.17,"body":"The equations of things"},{"startTime":1392.75,"endTime":1396.17,"body":"were not, we, you know,"},{"startTime":1396.17,"endTime":1398.72,"body":"wouldn't have written it"},{"startTime":1396.17,"endTime":1398.72,"body":"down in quite the same way."},{"startTime":1398.72,"endTime":1399.89,"body":"Calculus hadn't come along."},{"startTime":1399.89,"endTime":1402.32,"body":"So these are difficult things to do."},{"startTime":1402.32,"endTime":1404.6,"body":"Galileo couldn't resolve them."},{"startTime":1404.6,"endTime":1406.94,"body":"To try and find out the"},{"startTime":1404.6,"endTime":1406.94,"body":"area under the cycloid,"},{"startTime":1406.94,"endTime":1409.88,"body":"he actually drew on a"},{"startTime":1406.94,"endTime":1409.88,"body":"piece of sheet metal,"},{"startTime":1409.88,"endTime":1413.96,"body":"cut it out and weighed it to"},{"startTime":1409.88,"endTime":1413.96,"body":"try, and no, and this is fine,"},{"startTime":1413.96,"endTime":1416.53,"body":"it's a valid thing to do, but you know,"},{"startTime":1416.53,"endTime":1418.49,"body":"he wasn't quite able,"},{"startTime":1416.53,"endTime":1418.49,"body":"he got a rough answer"},{"startTime":1418.49,"endTime":1420.29,"body":"but he wasn't quite able to determine it."},{"startTime":1420.29,"endTime":1423.77,"body":"It took a chap called Gilles de Roberval"},{"startTime":1423.77,"endTime":1425.33,"body":"to work out the quadratic."},{"startTime":1425.33,"endTime":1427.31,"body":"He was a mathematician"},{"startTime":1425.33,"endTime":1427.31,"body":"who worked this out."},{"startTime":1427.31,"endTime":1428.45,"body":"He didn't tell anyone for a while."},{"startTime":1428.45,"endTime":1433.34,"body":"And then he kept his secrets"},{"startTime":1428.45,"endTime":1433.34,"body":"close to his chest, Roberval."},{"startTime":1433.34,"endTime":1437.6,"body":"But he did work out that"},{"startTime":1433.34,"endTime":1437.6,"body":"the area under the cycloid,"},{"startTime":1437.6,"endTime":1440.24,"body":"if you've got a circle"},{"startTime":1437.6,"endTime":1440.24,"body":"of area pi R squared,"},{"startTime":1440.24,"endTime":1442.01,"body":"making it r is the radius,"},{"startTime":1442.01,"endTime":1443.93,"body":"and it's a beautiful simple formula,"},{"startTime":1443.93,"endTime":1446.72,"body":"the area under each arch"},{"startTime":1443.93,"endTime":1446.72,"body":"is three pi R squared."},{"startTime":1446.72,"endTime":1447.62,"body":"So it's three times,"},{"startTime":1447.62,"endTime":1450.5,"body":"exactly three times"},{"startTime":1447.62,"endTime":1450.5,"body":"the area of the circle."},{"startTime":1450.5,"endTime":1454.61,"body":"Lovely, but rectification"},{"startTime":1450.5,"endTime":1454.61,"body":"hadn't been done yet."},{"startTime":1454.61,"endTime":1457.28,"body":"What's the length of the cycloid?"},{"startTime":1457.28,"endTime":1458.113,"body":"And Christopher Wren solved it."},{"startTime":1458.113,"endTime":1460.013,"body":"He was the first person to solve it."},{"startTime":1461.42,"endTime":1464.03,"body":"And here is the expression"},{"startTime":1461.42,"endTime":1464.03,"body":"again, it's really simple."},{"startTime":1464.03,"endTime":1467.27,"body":"If your circle has a diameter d,"},{"startTime":1467.27,"endTime":1469.22,"body":"then it's circumference is pi d, right?"},{"startTime":1469.22,"endTime":1472.1,"body":"But the length of the"},{"startTime":1469.22,"endTime":1472.1,"body":"arch is four times d."},{"startTime":1472.1,"endTime":1475.55,"body":"So another lovely whole number,"},{"startTime":1472.1,"endTime":1475.55,"body":"really simple expression."},{"startTime":1475.55,"endTime":1479.6,"body":"And Christopher Wren was"},{"startTime":1475.55,"endTime":1479.6,"body":"the one who proved that."},{"startTime":1479.6,"endTime":1481.04,"body":"So that was great."},{"startTime":1481.04,"endTime":1484.82,"body":"And he used his knowledge of the cycloid,"},{"startTime":1484.82,"endTime":1487.04,"body":"the rectification of the cycloid actually"},{"startTime":1487.04,"endTime":1490.55,"body":"to solve this problem of Kepler."},{"startTime":1490.55,"endTime":1493.01,"body":"And it's a bit complicated."},{"startTime":1493.01,"endTime":1495.68,"body":"There is a diagram that sort"},{"startTime":1493.01,"endTime":1495.68,"body":"of tells some of the story."},{"startTime":1495.68,"endTime":1497.21,"body":"We're not going to go into the details,"},{"startTime":1497.21,"endTime":1498.62,"body":"but he managed to solve it."},{"startTime":1498.62,"endTime":1500.42,"body":"Essentially what he did,"},{"startTime":1500.42,"endTime":1503.78,"body":"and this is written up"},{"startTime":1500.42,"endTime":1503.78,"body":"in a book by John Wallis,"},{"startTime":1503.78,"endTime":1507.656,"body":"an English mathematician, he"},{"startTime":1503.78,"endTime":1507.656,"body":"wrote a treaties on cycloids"},{"startTime":1507.656,"endTime":1511.67,"body":"in 1659, so just one year"},{"startTime":1507.656,"endTime":1511.67,"body":"after that original challenge."},{"startTime":1511.67,"endTime":1515.24,"body":"And you can see, I mean, we don't need"},{"startTime":1515.24,"endTime":1518.12,"body":"to be able to understand"},{"startTime":1515.24,"endTime":1518.12,"body":"Latin to see that it looks"},{"startTime":1518.12,"endTime":1521.42,"body":"like it's saying Kepler's"},{"startTime":1518.12,"endTime":1521.42,"body":"problems solved by cycloids."},{"startTime":1521.42,"endTime":1523.28,"body":"And that's exactly what happens."},{"startTime":1523.28,"endTime":1525.95,"body":"So Wallis writes up in this"},{"startTime":1523.28,"endTime":1525.95,"body":"book everything that's known"},{"startTime":1525.95,"endTime":1527.36,"body":"about the cycloid, in particular,"},{"startTime":1527.36,"endTime":1531.56,"body":"he writes up Christopher Wren's"},{"startTime":1527.36,"endTime":1531.56,"body":"proof of the rectification,"},{"startTime":1531.56,"endTime":1534.74,"body":"the length of the cycloid and"},{"startTime":1531.56,"endTime":1534.74,"body":"Christopher Wren's solution"},{"startTime":1534.74,"endTime":1537.02,"body":"of the Kepler problem using a cycloid."},{"startTime":1537.02,"endTime":1539.57,"body":"And essentially what he does,"},{"startTime":1539.57,"endTime":1541.43,"body":"I mean, you can see there's"},{"startTime":1539.57,"endTime":1541.43,"body":"a semi-circle in there,"},{"startTime":1541.43,"endTime":1544.52,"body":"which is divided up by a line"},{"startTime":1544.52,"endTime":1547.19,"body":"from the diameter to the circumference."},{"startTime":1547.19,"endTime":1551.09,"body":"He relates the areas in question to arcs"},{"startTime":1551.09,"endTime":1552.67,"body":"of a particular circle,"},{"startTime":1552.67,"endTime":1555.38,"body":"of the circumference"},{"startTime":1552.67,"endTime":1555.38,"body":"of a particular circle,"},{"startTime":1555.38,"endTime":1560.38,"body":"and then in turn, those to parts"},{"startTime":1555.38,"endTime":1560.38,"body":"of the length of a cycloid."},{"startTime":1561.32,"endTime":1564.29,"body":"Or actually it's a sort"},{"startTime":1561.32,"endTime":1564.29,"body":"of stretched cycloid,"},{"startTime":1564.29,"endTime":1567.02,"body":"but it's still within that family."},{"startTime":1567.02,"endTime":1569.9,"body":"And of course, Wren himself"},{"startTime":1567.02,"endTime":1569.9,"body":"knew he'd found the formula"},{"startTime":1569.9,"endTime":1571.28,"body":"for the length of cycloid."},{"startTime":1571.28,"endTime":1576.173,"body":"So he was able to then use"},{"startTime":1571.28,"endTime":1576.173,"body":"that to solve Kepler's problem."},{"startTime":1577.49,"endTime":1579.35,"body":"John Wallis, we'll meet"},{"startTime":1577.49,"endTime":1579.35,"body":"him again in a moment,"},{"startTime":1579.35,"endTime":1583.37,"body":"but he couldn't resist having little jab"},{"startTime":1583.37,"endTime":1585.53,"body":"at the French mathematicians"},{"startTime":1583.37,"endTime":1585.53,"body":"a few years later,"},{"startTime":1585.53,"endTime":1587.273,"body":"in a letter to a friend,"},{"startTime":1588.44,"endTime":1591.29,"body":"10 years after the original"},{"startTime":1588.44,"endTime":1591.29,"body":"challenge from Jean de Montfort,"},{"startTime":1591.29,"endTime":1594.62,"body":"whoever he was, he couldn't"},{"startTime":1591.29,"endTime":1594.62,"body":"quite resist saying"},{"startTime":1594.62,"endTime":1598.5,"body":"to his friend, no Frenchman"},{"startTime":1594.62,"endTime":1598.5,"body":"has returned any solution"},{"startTime":1599.81,"endTime":1601.49,"body":"to Christopher Wren's problem."},{"startTime":1601.49,"endTime":1603.74,"body":"Now I don't quite know if that's the case,"},{"startTime":1603.74,"endTime":1605.69,"body":"but anyway, he was, you know,"},{"startTime":1605.69,"endTime":1608.29,"body":"take that, Jean de Montfort,"},{"startTime":1605.69,"endTime":1608.29,"body":"was what he was saying."},{"startTime":1609.77,"endTime":1610.91,"body":"But of course we're all together"},{"startTime":1610.91,"endTime":1614.66,"body":"in the community of"},{"startTime":1610.91,"endTime":1614.66,"body":"mathematics, of course."},{"startTime":1614.66,"endTime":1619.66,"body":"So that is Wren's brilliant solution"},{"startTime":1620.09,"endTime":1623.693,"body":"to Kepler's problem and his"},{"startTime":1620.09,"endTime":1623.693,"body":"rectification of the cycloid."},{"startTime":1628.4,"endTime":1632.93,"body":"Here's a piece of Christopher"},{"startTime":1628.4,"endTime":1632.93,"body":"Wren's mathematics next"},{"startTime":1632.93,"endTime":1636.203,"body":"that is related to spirals."},{"startTime":1637.25,"endTime":1642.25,"body":"And this spiral shape"},{"startTime":1637.25,"endTime":1642.25,"body":"represents or is reminiscent"},{"startTime":1643.55,"endTime":1647.363,"body":"of many of the shapes"},{"startTime":1643.55,"endTime":1647.363,"body":"that we see in nature."},{"startTime":1649.13,"endTime":1652.64,"body":"You can think of shells, some"},{"startTime":1649.13,"endTime":1652.64,"body":"of the ways some plants grow"},{"startTime":1652.64,"endTime":1654.59,"body":"in a kind of spiral formation."},{"startTime":1654.59,"endTime":1658.73,"body":"This is a shape that we do see in nature."},{"startTime":1658.73,"endTime":1662.09,"body":"And this particular kind of spiral"},{"startTime":1662.09,"endTime":1663.92,"body":"was one that was known to Wren."},{"startTime":1663.92,"endTime":1665.873,"body":"It's called a logarithmic spiral."},{"startTime":1667.04,"endTime":1670.05,"body":"I believe the first mention"},{"startTime":1667.04,"endTime":1670.05,"body":"of this kind of spiral"},{"startTime":1671.12,"endTime":1675.83,"body":"in writing is the German artist Durer."},{"startTime":1675.83,"endTime":1678.98,"body":"Later on, the mathematician"},{"startTime":1675.83,"endTime":1678.98,"body":"Jacob Bernoulli"},{"startTime":1678.98,"endTime":1680.42,"body":"would study it at length"},{"startTime":1680.42,"endTime":1683.06,"body":"and he gave it the"},{"startTime":1680.42,"endTime":1683.06,"body":"nickname spira mirabilis,"},{"startTime":1683.06,"endTime":1686.033,"body":"the miraculous spiral because"},{"startTime":1683.06,"endTime":1686.033,"body":"of its lovely properties."},{"startTime":1688.13,"endTime":1690.11,"body":"The reason it's called"},{"startTime":1688.13,"endTime":1690.11,"body":"the logarithmic spiral,"},{"startTime":1690.11,"endTime":1691.58,"body":"so what is a spiral?"},{"startTime":1691.58,"endTime":1694.7,"body":"Well, you start at a central"},{"startTime":1691.58,"endTime":1694.7,"body":"point and then you kind of,"},{"startTime":1694.7,"endTime":1696.77,"body":"well, you spiral outwards literally."},{"startTime":1696.77,"endTime":1699.44,"body":"And in a logarithmic spiral,"},{"startTime":1699.44,"endTime":1702.41,"body":"it's describing how"},{"startTime":1699.44,"endTime":1702.41,"body":"quickly the your distance"},{"startTime":1702.41,"endTime":1704.0,"body":"from the center increases."},{"startTime":1704.0,"endTime":1706.01,"body":"So that equation at the bottom there,"},{"startTime":1706.01,"endTime":1708.41,"body":"which I've written r"},{"startTime":1706.01,"endTime":1708.41,"body":"equals key to the theta,"},{"startTime":1708.41,"endTime":1710.6,"body":"unhelpfully not telling you"},{"startTime":1708.41,"endTime":1710.6,"body":"what any of those letters are."},{"startTime":1710.6,"endTime":1713.0,"body":"Now I will tell you, so r is the distance"},{"startTime":1713.0,"endTime":1715.19,"body":"from the center where you are,"},{"startTime":1715.19,"endTime":1718.1,"body":"theta is the angle that"},{"startTime":1715.19,"endTime":1718.1,"body":"you've traveled through,"},{"startTime":1718.1,"endTime":1721.73,"body":"and in this kind of spiral,"},{"startTime":1718.1,"endTime":1721.73,"body":"that's the formula for it,"},{"startTime":1721.73,"endTime":1724.553,"body":"k is just some constant,"},{"startTime":1721.73,"endTime":1724.553,"body":"it's just some fixed number."},{"startTime":1725.87,"endTime":1728.05,"body":"So you are raising your fixed"},{"startTime":1725.87,"endTime":1728.05,"body":"number to the power of theta"},{"startTime":1728.05,"endTime":1730.91,"body":"at the angle, and that tells"},{"startTime":1728.05,"endTime":1730.91,"body":"you how far out you are."},{"startTime":1730.91,"endTime":1732.71,"body":"So you're spiraling out."},{"startTime":1732.71,"endTime":1735.08,"body":"And for this kind of spiral,"},{"startTime":1735.08,"endTime":1737.36,"body":"there's an example of it"},{"startTime":1735.08,"endTime":1737.36,"body":"that I've shown you there,"},{"startTime":1737.36,"endTime":1739.79,"body":"which is involving number two."},{"startTime":1739.79,"endTime":1744.5,"body":"So here K would be two,"},{"startTime":1739.79,"endTime":1744.5,"body":"and if you can see,"},{"startTime":1744.5,"endTime":1746.03,"body":"but even if you can't, I will tell you"},{"startTime":1746.03,"endTime":1748.49,"body":"that there are points"},{"startTime":1746.03,"endTime":1748.49,"body":"when it hits the axis."},{"startTime":1748.49,"endTime":1750.68,"body":"So you can see at one point it's at two"},{"startTime":1750.68,"endTime":1752.39,"body":"and then you go full revolution round,"},{"startTime":1752.39,"endTime":1755.18,"body":"now it's distance four, and"},{"startTime":1752.39,"endTime":1755.18,"body":"another full revolution round,"},{"startTime":1755.18,"endTime":1757.33,"body":"it gets to eight, and then"},{"startTime":1755.18,"endTime":1757.33,"body":"it'll be 16, and 32 and so on."},{"startTime":1757.33,"endTime":1761.99,"body":"So it's powers of two every"},{"startTime":1757.33,"endTime":1761.99,"body":"time you go a full revolution."},{"startTime":1761.99,"endTime":1763.73,"body":"And the reason it's called logarithmic,"},{"startTime":1763.73,"endTime":1765.35,"body":"so if you know about logarithms,"},{"startTime":1765.35,"endTime":1768.29,"body":"you will notice that if you"},{"startTime":1765.35,"endTime":1768.29,"body":"take logs on both sides,"},{"startTime":1768.29,"endTime":1771.26,"body":"you get that the logarithm"},{"startTime":1768.29,"endTime":1771.26,"body":"of r of the distance"},{"startTime":1771.26,"endTime":1775.94,"body":"is going to be a multiple"},{"startTime":1771.26,"endTime":1775.94,"body":"of the angle theta."},{"startTime":1775.94,"endTime":1778.283,"body":"So that's why it's called logarithmic now."},{"startTime":1779.51,"endTime":1784.51,"body":"So these spirals, as I say,"},{"startTime":1779.51,"endTime":1784.51,"body":"you do see things in nature"},{"startTime":1784.52,"endTime":1785.74,"body":"that look quite like it."},{"startTime":1785.74,"endTime":1788.39,"body":"So just imprint that on"},{"startTime":1785.74,"endTime":1788.39,"body":"your mind for a second."},{"startTime":1788.39,"endTime":1790.22,"body":"And I will go to the next slide."},{"startTime":1790.22,"endTime":1792.68,"body":"And you can see this is a cross section"},{"startTime":1792.68,"endTime":1794.51,"body":"through a nautilus shell."},{"startTime":1794.51,"endTime":1796.16,"body":"It's really, really similar."},{"startTime":1796.16,"endTime":1798.263,"body":"It looks very like that logarithmic spiral"},{"startTime":1798.263,"endTime":1799.94,"body":"that I've just shown you."},{"startTime":1799.94,"endTime":1804.94,"body":"And the reason or a reason we"},{"startTime":1799.94,"endTime":1804.94,"body":"think this might be the case"},{"startTime":1805.22,"endTime":1808.01,"body":"is that actually it's the perfect solution"},{"startTime":1808.01,"endTime":1811.97,"body":"to how nature could arrange something"},{"startTime":1811.97,"endTime":1815.06,"body":"that grows constantly through its lifetime"},{"startTime":1815.06,"endTime":1817.58,"body":"and it's growing from a single point."},{"startTime":1817.58,"endTime":1820.1,"body":"So of course you need to"},{"startTime":1817.58,"endTime":1820.1,"body":"make room for the new growth."},{"startTime":1820.1,"endTime":1822.71,"body":"So what happens is you"},{"startTime":1820.1,"endTime":1822.71,"body":"have this, you know,"},{"startTime":1822.71,"endTime":1824.69,"body":"it's growing kind of at an angle"},{"startTime":1824.69,"endTime":1828.17,"body":"and the older bits are"},{"startTime":1824.69,"endTime":1828.17,"body":"forced to move outwards,"},{"startTime":1828.17,"endTime":1829.94,"body":"but they're still growing as well."},{"startTime":1829.94,"endTime":1832.76,"body":"So it's getting bigger"},{"startTime":1829.94,"endTime":1832.76,"body":"as it grows outwards"},{"startTime":1832.76,"endTime":1834.83,"body":"but the new material"},{"startTime":1832.76,"endTime":1834.83,"body":"is still being produced"},{"startTime":1834.83,"endTime":1836.33,"body":"in this spiral formation."},{"startTime":1836.33,"endTime":1837.92,"body":"And the great advantage of this"},{"startTime":1837.92,"endTime":1841.76,"body":"is that you keep the same"},{"startTime":1837.92,"endTime":1841.76,"body":"basic shape the whole time."},{"startTime":1841.76,"endTime":1843.65,"body":"It's getting bigger, but"},{"startTime":1841.76,"endTime":1843.65,"body":"it's the same basic shape."},{"startTime":1843.65,"endTime":1845.6,"body":"So these logarithmic spirals"},{"startTime":1845.6,"endTime":1847.52,"body":"have this self similarity property"},{"startTime":1847.52,"endTime":1851.09,"body":"that when you zoom in or"},{"startTime":1847.52,"endTime":1851.09,"body":"zoom out, it looks the same"},{"startTime":1851.09,"endTime":1852.62,"body":"and that means the animal can, you know,"},{"startTime":1852.62,"endTime":1854.03,"body":"it can be bigger or smaller,"},{"startTime":1854.03,"endTime":1857.09,"body":"but it's still got the"},{"startTime":1854.03,"endTime":1857.09,"body":"same shape body to live in"},{"startTime":1857.09,"endTime":1859.19,"body":"or to, you know, to deal with."},{"startTime":1859.19,"endTime":1861.98,"body":"It doesn't have to learn new"},{"startTime":1859.19,"endTime":1861.98,"body":"rules of being, whatever."},{"startTime":1861.98,"endTime":1864.98,"body":"It's just growing but it's"},{"startTime":1861.98,"endTime":1864.98,"body":"keeping the same shape."},{"startTime":1864.98,"endTime":1868.13,"body":"Now one thing is, okay, nice spiral."},{"startTime":1868.13,"endTime":1870.11,"body":"This is a cross section though,"},{"startTime":1870.11,"endTime":1872.3,"body":"shells and other things"},{"startTime":1870.11,"endTime":1872.3,"body":"are three-dimensional."},{"startTime":1872.3,"endTime":1875.18,"body":"So we haven't quite, we need"},{"startTime":1872.3,"endTime":1875.18,"body":"to do something else here."},{"startTime":1875.18,"endTime":1876.98,"body":"And there's something else again,"},{"startTime":1876.98,"endTime":1878.543,"body":"Christopher Wren thought of it."},{"startTime":1879.56,"endTime":1882.65,"body":"So brief timeline, John"},{"startTime":1879.56,"endTime":1882.65,"body":"Wallis, we just met him,"},{"startTime":1882.65,"endTime":1887.42,"body":"he was able to perform the rectification,"},{"startTime":1887.42,"endTime":1890.57,"body":"i.e. finding the length"},{"startTime":1887.42,"endTime":1890.57,"body":"of a logarithmic spiral"},{"startTime":1890.57,"endTime":1894.77,"body":"or actually more properly, any"},{"startTime":1890.57,"endTime":1894.77,"body":"portion of it that you desire"},{"startTime":1894.77,"endTime":1898.67,"body":"by a process which he called convolution,"},{"startTime":1898.67,"endTime":1901.64,"body":"which is basically unrolling the spiral"},{"startTime":1901.64,"endTime":1902.96,"body":"until it's a straight line."},{"startTime":1902.96,"endTime":1903.92,"body":"But of course you have to do that"},{"startTime":1903.92,"endTime":1905.78,"body":"in a very, very careful way"},{"startTime":1905.78,"endTime":1907.91,"body":"to make sure you haven't"},{"startTime":1905.78,"endTime":1907.91,"body":"changed the length"},{"startTime":1907.91,"endTime":1910.25,"body":"by doing whatever"},{"startTime":1907.91,"endTime":1910.25,"body":"mathematical transformation."},{"startTime":1910.25,"endTime":1911.57,"body":"But he worked out how to do it,"},{"startTime":1911.57,"endTime":1913.82,"body":"sort of unraveling the spiral"},{"startTime":1911.57,"endTime":1913.82,"body":"to make a straight line."},{"startTime":1913.82,"endTime":1916.64,"body":"And then you can see the length."},{"startTime":1916.64,"endTime":1920.72,"body":"Now Christopher Wren, who"},{"startTime":1916.64,"endTime":1920.72,"body":"was thinking about seashells"},{"startTime":1920.72,"endTime":1923.87,"body":"and snails shells and things, he realized"},{"startTime":1923.87,"endTime":1927.11,"body":"that you'd need to have a"},{"startTime":1923.87,"endTime":1927.11,"body":"kind of 3D version of this."},{"startTime":1927.11,"endTime":1931.34,"body":"And so what he managed to do"},{"startTime":1927.11,"endTime":1931.34,"body":"was kind of go up a dimension"},{"startTime":1931.34,"endTime":1934.34,"body":"and to do this convoluting"},{"startTime":1931.34,"endTime":1934.34,"body":"thing, but in reverse."},{"startTime":1934.34,"endTime":1937.73,"body":"So he found a way to"},{"startTime":1934.34,"endTime":1937.73,"body":"basically roll up a cone"},{"startTime":1937.73,"endTime":1940.43,"body":"to make a three-dimensional version"},{"startTime":1940.43,"endTime":1942.56,"body":"of the logarithmic spiral."},{"startTime":1942.56,"endTime":1945.26,"body":"So he had the idea of"},{"startTime":1942.56,"endTime":1945.26,"body":"how you could do this"},{"startTime":1945.26,"endTime":1948.02,"body":"to make a sort of three-dimensional"},{"startTime":1945.26,"endTime":1948.02,"body":"logarithmic spiral"},{"startTime":1948.02,"endTime":1951.8,"body":"that would then, he suggested,"},{"startTime":1951.8,"endTime":1954.203,"body":"give you the shapes of shells."},{"startTime":1955.49,"endTime":1958.58,"body":"And more recently, these"},{"startTime":1955.49,"endTime":1958.58,"body":"ideas have been generalized"},{"startTime":1958.58,"endTime":1960.44,"body":"to something called the"},{"startTime":1958.58,"endTime":1960.44,"body":"power cone construction,"},{"startTime":1960.44,"endTime":1962.87,"body":"which I'll talk more on in a second."},{"startTime":1962.87,"endTime":1965.09,"body":"But I want to show, I'm so"},{"startTime":1962.87,"endTime":1965.09,"body":"pleased with this slide,"},{"startTime":1965.09,"endTime":1966.83,"body":"it might be the favorite"},{"startTime":1965.09,"endTime":1966.83,"body":"slide I've ever prepared"},{"startTime":1966.83,"endTime":1971.83,"body":"for Gresham because, so"},{"startTime":1966.83,"endTime":1971.83,"body":"what I did was to take"},{"startTime":1971.9,"endTime":1975.38,"body":"that equation for a solid spiral"},{"startTime":1975.38,"endTime":1977.66,"body":"and it has some parameters in it"},{"startTime":1977.66,"endTime":1980.51,"body":"so you can change like how"},{"startTime":1977.66,"endTime":1980.51,"body":"twisty you want it to be"},{"startTime":1980.51,"endTime":1982.79,"body":"just by changing some of the"},{"startTime":1980.51,"endTime":1982.79,"body":"parameters in the equation,"},{"startTime":1982.79,"endTime":1984.35,"body":"in the general equation."},{"startTime":1984.35,"endTime":1987.11,"body":"And by doing that you can play around"},{"startTime":1987.11,"endTime":1989.69,"body":"and you can try and make"},{"startTime":1987.11,"endTime":1989.69,"body":"things that look like shells."},{"startTime":1989.69,"endTime":1993.51,"body":"And here look, these things"},{"startTime":1989.69,"endTime":1993.51,"body":"really look like shells"},{"startTime":1994.37,"endTime":1996.26,"body":"and I just sort of, you know,"},{"startTime":1994.37,"endTime":1996.26,"body":"did them on the computer."},{"startTime":1996.26,"endTime":1997.94,"body":"But you can by changing the parameters,"},{"startTime":1997.94,"endTime":2001.06,"body":"this is coming exactly from Wren's idea"},{"startTime":2001.06,"endTime":2003.7,"body":"about this three-dimensional"},{"startTime":2001.06,"endTime":2003.7,"body":"algorithmic spiral."},{"startTime":2003.7,"endTime":2006.49,"body":"You can create things that"},{"startTime":2003.7,"endTime":2006.49,"body":"really do look very much"},{"startTime":2006.49,"endTime":2009.79,"body":"like a wide array of different shells."},{"startTime":2009.79,"endTime":2011.92,"body":"And so this is a, you know, a great idea"},{"startTime":2011.92,"endTime":2013.87,"body":"from Christopher Wren."},{"startTime":2013.87,"endTime":2017.47,"body":"And as I say, it's really"},{"startTime":2013.87,"endTime":2017.47,"body":"still very relevant."},{"startTime":2017.47,"endTime":2020.83,"body":"And now, I was looking"},{"startTime":2017.47,"endTime":2020.83,"body":"at a paper from 2021"},{"startTime":2020.83,"endTime":2023.65,"body":"where they'd made a slightly"},{"startTime":2020.83,"endTime":2023.65,"body":"more general version"},{"startTime":2023.65,"endTime":2024.64,"body":"of the equation,"},{"startTime":2024.64,"endTime":2027.16,"body":"which they call the"},{"startTime":2024.64,"endTime":2027.16,"body":"power cone construction,"},{"startTime":2027.16,"endTime":2029.23,"body":"which can model not just shells"},{"startTime":2029.23,"endTime":2033.37,"body":"but also things like"},{"startTime":2029.23,"endTime":2033.37,"body":"antlers, horns, tusks, teeth,"},{"startTime":2033.37,"endTime":2036.46,"body":"any kind of sharp pointy"},{"startTime":2033.37,"endTime":2036.46,"body":"bits animals may have,"},{"startTime":2036.46,"endTime":2038.98,"body":"even things like rose"},{"startTime":2036.46,"endTime":2038.98,"body":"hips as well in plants."},{"startTime":2038.98,"endTime":2042.1,"body":"And so I just want to show"},{"startTime":2038.98,"endTime":2042.1,"body":"you two or three things"},{"startTime":2042.1,"endTime":2043.453,"body":"actually from nature."},{"startTime":2044.65,"endTime":2047.74,"body":"Look right, look at the spirals on those."},{"startTime":2047.74,"endTime":2050.23,"body":"You've got giant eland on the left."},{"startTime":2050.23,"endTime":2051.85,"body":"I mean, you can see in the middle"},{"startTime":2051.85,"endTime":2055.99,"body":"you've got a fantastic curly horned ram."},{"startTime":2055.99,"endTime":2058.99,"body":"And the one on the right, I"},{"startTime":2055.99,"endTime":2058.99,"body":"think this is my favorite one."},{"startTime":2058.99,"endTime":2060.16,"body":"I mean, it's fabulous."},{"startTime":2060.16,"endTime":2062.8,"body":"So this is a markhor, it's a kind of goat."},{"startTime":2062.8,"endTime":2064.69,"body":"Another name for it is,"},{"startTime":2062.8,"endTime":2064.69,"body":"for obvious reasons,"},{"startTime":2064.69,"endTime":2066.4,"body":"the screw horned goat."},{"startTime":2066.4,"endTime":2068.71,"body":"And look at those corkscrew horns."},{"startTime":2068.71,"endTime":2070.09,"body":"I mean, they're just fantastic."},{"startTime":2070.09,"endTime":2074.71,"body":"And you can get those, you"},{"startTime":2070.09,"endTime":2074.71,"body":"can mimic those pretty exactly"},{"startTime":2074.71,"endTime":2077.29,"body":"using these kinds of equations."},{"startTime":2077.29,"endTime":2081.04,"body":"So this is an example"},{"startTime":2077.29,"endTime":2081.04,"body":"of how Christopher Wren,"},{"startTime":2081.04,"endTime":2083.29,"body":"he was, you know, he was"},{"startTime":2081.04,"endTime":2083.29,"body":"interested in everything"},{"startTime":2083.29,"endTime":2084.49,"body":"as all the best people are."},{"startTime":2084.49,"endTime":2088.06,"body":"And he had, you know, great"},{"startTime":2084.49,"endTime":2088.06,"body":"ideas and mathematical ideas"},{"startTime":2088.06,"endTime":2090.55,"body":"for how you could model"},{"startTime":2088.06,"endTime":2090.55,"body":"lots of different things,"},{"startTime":2090.55,"endTime":2094.513,"body":"and seashells and snail"},{"startTime":2090.55,"endTime":2094.513,"body":"shells are just one of those."},{"startTime":2098.2,"endTime":2102.91,"body":"So that is some three little gems"},{"startTime":2102.91,"endTime":2104.74,"body":"of Christopher Wren's mathematics."},{"startTime":2104.74,"endTime":2105.573,"body":"I want to spend the rest"},{"startTime":2105.573,"endTime":2108.043,"body":"of the time talking about architecture."},{"startTime":2110.5,"endTime":2113.5,"body":"Now there is lots of"},{"startTime":2110.5,"endTime":2113.5,"body":"mathematics in architecture."},{"startTime":2113.5,"endTime":2115.45,"body":"I want to spend a little while talking"},{"startTime":2115.45,"endTime":2119.77,"body":"about the mathematics that"},{"startTime":2115.45,"endTime":2119.77,"body":"Christopher Wren used,"},{"startTime":2119.77,"endTime":2122.71,"body":"but not for all of the"},{"startTime":2119.77,"endTime":2122.71,"body":"architecture he ever did."},{"startTime":2122.71,"endTime":2124.96,"body":"You know, we could have"},{"startTime":2122.71,"endTime":2124.96,"body":"a whole lecture series"},{"startTime":2124.96,"endTime":2127.75,"body":"about the links between"},{"startTime":2124.96,"endTime":2127.75,"body":"mathematics and architecture"},{"startTime":2127.75,"endTime":2129.19,"body":"because it's everywhere, right?"},{"startTime":2129.19,"endTime":2132.43,"body":"You need it for the kind"},{"startTime":2129.19,"endTime":2132.43,"body":"of the engineering aspect,"},{"startTime":2132.43,"endTime":2134.59,"body":"load bearing beams and you know,"},{"startTime":2134.59,"endTime":2136.06,"body":"forces that are acting on the building"},{"startTime":2136.06,"endTime":2138.37,"body":"and being able to construct"},{"startTime":2136.06,"endTime":2138.37,"body":"it so it doesn't fall down,"},{"startTime":2138.37,"endTime":2140.617,"body":"lots of mathematics there."},{"startTime":2140.617,"endTime":2144.34,"body":"There's also mathematics"},{"startTime":2140.617,"endTime":2144.34,"body":"around the aesthetic side."},{"startTime":2144.34,"endTime":2146.65,"body":"We already heard how Wren says, you know,"},{"startTime":2146.65,"endTime":2149.68,"body":"geometry is where it's at,"},{"startTime":2146.65,"endTime":2149.68,"body":"that's what beauty is about."},{"startTime":2149.68,"endTime":2153.34,"body":"It's got to have beautiful"},{"startTime":2149.68,"endTime":2153.34,"body":"proportions and symmetry"},{"startTime":2153.34,"endTime":2157.18,"body":"and you know, these wonderful things"},{"startTime":2157.18,"endTime":2159.4,"body":"that make our buildings look beautiful."},{"startTime":2159.4,"endTime":2161.92,"body":"That's geometry, that's mathematics."},{"startTime":2161.92,"endTime":2163.9,"body":"And then of course,"},{"startTime":2161.92,"endTime":2163.9,"body":"while you are doing that,"},{"startTime":2163.9,"endTime":2165.67,"body":"you have to address the practical,"},{"startTime":2165.67,"endTime":2168.16,"body":"what is this building actually for?"},{"startTime":2168.16,"endTime":2171.4,"body":"And can I design it in"},{"startTime":2168.16,"endTime":2171.4,"body":"such a way that it will be,"},{"startTime":2171.4,"endTime":2175.0,"body":"you know, the best feeling"},{"startTime":2171.4,"endTime":2175.0,"body":"as you use this building,"},{"startTime":2175.0,"endTime":2177.16,"body":"you know, this machine for living in."},{"startTime":2177.16,"endTime":2179.56,"body":"So all of these things"},{"startTime":2177.16,"endTime":2179.56,"body":"have mathematics in them."},{"startTime":2180.52,"endTime":2182.56,"body":"What I want to talk about specifically"},{"startTime":2182.56,"endTime":2185.38,"body":"is Christopher Wren's"},{"startTime":2182.56,"endTime":2185.38,"body":"most famous building,"},{"startTime":2185.38,"endTime":2188.8,"body":"St. Paul's Cathedral, and"},{"startTime":2185.38,"endTime":2188.8,"body":"even more specifically,"},{"startTime":2188.8,"endTime":2190.78,"body":"the dome of St. Paul's Cathedral."},{"startTime":2190.78,"endTime":2193.51,"body":"And the mathematics"},{"startTime":2190.78,"endTime":2193.51,"body":"behind that is absolutely,"},{"startTime":2193.51,"endTime":2196.42,"body":"I hope you will soon agree"},{"startTime":2193.51,"endTime":2196.42,"body":"with me, fascinating."},{"startTime":2196.42,"endTime":2199.75,"body":"So let's just look at the"},{"startTime":2196.42,"endTime":2199.75,"body":"outside of it, the dome."},{"startTime":2199.75,"endTime":2204.75,"body":"And you may know that"},{"startTime":2199.75,"endTime":2204.75,"body":"the design for St Paul's"},{"startTime":2205.39,"endTime":2208.51,"body":"went through many iterations."},{"startTime":2208.51,"endTime":2210.79,"body":"And you know, Christopher"},{"startTime":2208.51,"endTime":2210.79,"body":"Wren's getting more"},{"startTime":2210.79,"endTime":2213.04,"body":"and more frustrated 'cause"},{"startTime":2210.79,"endTime":2213.04,"body":"people kept saying, oh,"},{"startTime":2213.04,"endTime":2215.14,"body":"so first of all, the first design,"},{"startTime":2215.14,"endTime":2216.67,"body":"I think I've got the"},{"startTime":2215.14,"endTime":2216.67,"body":"order right, you know,"},{"startTime":2216.67,"endTime":2221.05,"body":"one design was far too"},{"startTime":2216.67,"endTime":2221.05,"body":"ambitious and it was too much"},{"startTime":2221.05,"endTime":2223.42,"body":"and we could never afford"},{"startTime":2221.05,"endTime":2223.42,"body":"it, no, tone it down a bit."},{"startTime":2223.42,"endTime":2225.28,"body":"So he toned it down a"},{"startTime":2223.42,"endTime":2225.28,"body":"bit and then he was told,"},{"startTime":2225.28,"endTime":2227.14,"body":"well, that's not nearly splendid enough"},{"startTime":2227.14,"endTime":2229.48,"body":"for, you know, the great"},{"startTime":2227.14,"endTime":2229.48,"body":"cathedral of London."},{"startTime":2229.48,"endTime":2231.28,"body":"You know, you need to"},{"startTime":2229.48,"endTime":2231.28,"body":"make it more splendid"},{"startTime":2231.28,"endTime":2233.71,"body":"but similar, you know, but"},{"startTime":2231.28,"endTime":2233.71,"body":"obviously not too expensive"},{"startTime":2233.71,"endTime":2234.79,"body":"but really splendid."},{"startTime":2234.79,"endTime":2236.77,"body":"And then the church"},{"startTime":2234.79,"endTime":2236.77,"body":"authorities weighed in saying,"},{"startTime":2236.77,"endTime":2239.68,"body":"well, it's not quite,"},{"startTime":2236.77,"endTime":2239.68,"body":"the shape isn't right"},{"startTime":2239.68,"endTime":2242.05,"body":"for what, you know, the"},{"startTime":2239.68,"endTime":2242.05,"body":"purpose of the building"},{"startTime":2242.05,"endTime":2244.3,"body":"and is it too Catholic"},{"startTime":2242.05,"endTime":2244.3,"body":"or not Catholic enough?"},{"startTime":2244.3,"endTime":2246.194,"body":"All of that, there was all that going on"},{"startTime":2246.194,"endTime":2247.27,"body":"and there were several designs"},{"startTime":2247.27,"endTime":2249.7,"body":"and I think at some point he"},{"startTime":2247.27,"endTime":2249.7,"body":"sort of went, you know what?"},{"startTime":2249.7,"endTime":2252.25,"body":"I'm just going to start"},{"startTime":2249.7,"endTime":2252.25,"body":"building a cathedral"},{"startTime":2252.25,"endTime":2256.72,"body":"and let's see how it goes,"},{"startTime":2252.25,"endTime":2256.72,"body":"I'm paraphrasing little bit."},{"startTime":2256.72,"endTime":2260.59,"body":"So the thing that we all"},{"startTime":2256.72,"endTime":2260.59,"body":"known recognize instantly"},{"startTime":2260.59,"endTime":2263.96,"body":"about the London skyline"},{"startTime":2260.59,"endTime":2263.96,"body":"is this fabulous dome"},{"startTime":2265.72,"endTime":2269.713,"body":"and the dome that we see is a hemisphere."},{"startTime":2270.61,"endTime":2275.61,"body":"And really that's for both"},{"startTime":2270.61,"endTime":2275.61,"body":"aesthetic and symbolic reasons."},{"startTime":2276.43,"endTime":2278.71,"body":"So when you think of the celestial sphere,"},{"startTime":2278.71,"endTime":2281.65,"body":"this hemisphere is kind of representing"},{"startTime":2281.65,"endTime":2285.07,"body":"that universe but also the hemisphere."},{"startTime":2285.07,"endTime":2287.74,"body":"I mean, it's made of circles and circles"},{"startTime":2287.74,"endTime":2290.14,"body":"because you can go round"},{"startTime":2287.74,"endTime":2290.14,"body":"and round them forever,"},{"startTime":2290.14,"endTime":2292.27,"body":"they symbolize the eternal."},{"startTime":2292.27,"endTime":2294.79,"body":"So this is a great shape to associate,"},{"startTime":2294.79,"endTime":2297.64,"body":"not just because it's beautiful,"},{"startTime":2294.79,"endTime":2297.64,"body":"it's highly symmetrical,"},{"startTime":2297.64,"endTime":2299.08,"body":"but it's a great shape to associate"},{"startTime":2299.08,"endTime":2300.52,"body":"with, you know, a cathedral"},{"startTime":2300.52,"endTime":2302.923,"body":"where you're worshiping"},{"startTime":2300.52,"endTime":2302.923,"body":"the divine, the eternal."},{"startTime":2303.91,"endTime":2307.12,"body":"So that was the decision for the dome"},{"startTime":2307.12,"endTime":2309.463,"body":"that we all see, a hemisphere."},{"startTime":2310.3,"endTime":2312.13,"body":"But there's a slight problem,"},{"startTime":2312.13,"endTime":2313.3,"body":"there's a slight problem"},{"startTime":2312.13,"endTime":2313.3,"body":"with the hemisphere"},{"startTime":2313.3,"endTime":2318.3,"body":"because it's not actually"},{"startTime":2313.3,"endTime":2318.3,"body":"a particularly strong shape"},{"startTime":2319.66,"endTime":2322.24,"body":"and a hemisphere of this size"},{"startTime":2322.24,"endTime":2325.24,"body":"is really going to"},{"startTime":2322.24,"endTime":2325.24,"body":"struggle to support itself"},{"startTime":2325.24,"endTime":2327.31,"body":"even just as it is."},{"startTime":2327.31,"endTime":2331.18,"body":"But we've got this gigantic"},{"startTime":2327.31,"endTime":2331.18,"body":"lantern on the top,"},{"startTime":2331.18,"endTime":2336.04,"body":"which looks great, but it"},{"startTime":2331.18,"endTime":2336.04,"body":"weighs something like 850 tons."},{"startTime":2336.04,"endTime":2339.31,"body":"And that added to the"},{"startTime":2336.04,"endTime":2339.31,"body":"weight of the dome itself"},{"startTime":2339.31,"endTime":2342.31,"body":"would absolutely mean this"},{"startTime":2339.31,"endTime":2342.31,"body":"could not support itself"},{"startTime":2342.31,"endTime":2343.69,"body":"as a structure."},{"startTime":2343.69,"endTime":2345.4,"body":"So we need to have something"},{"startTime":2345.4,"endTime":2347.11,"body":"that is supporting the structure."},{"startTime":2347.11,"endTime":2348.61,"body":"But you don't want to ruin the look"},{"startTime":2348.61,"endTime":2349.96,"body":"with lots of, you know, buttresses"},{"startTime":2349.96,"endTime":2351.76,"body":"and things on the outside."},{"startTime":2351.76,"endTime":2354.19,"body":"So you want to put them inside,"},{"startTime":2354.19,"endTime":2356.08,"body":"but then that's going to"},{"startTime":2354.19,"endTime":2356.08,"body":"ruin the look of the inside."},{"startTime":2356.08,"endTime":2359.29,"body":"You don't be looking up and"},{"startTime":2356.08,"endTime":2359.29,"body":"just seeing load of horrible,"},{"startTime":2359.29,"endTime":2361.87,"body":"you know, things that don't look good."},{"startTime":2361.87,"endTime":2364.27,"body":"So it's a real problem,"},{"startTime":2361.87,"endTime":2364.27,"body":"what are we going to do?"},{"startTime":2364.27,"endTime":2366.88,"body":"Where are we going to put"},{"startTime":2364.27,"endTime":2366.88,"body":"the supporting structure"},{"startTime":2366.88,"endTime":2369.46,"body":"that allows this hemispherical dome"},{"startTime":2369.46,"endTime":2371.86,"body":"to not, you know, fall"},{"startTime":2369.46,"endTime":2371.86,"body":"over and crush everybody?"},{"startTime":2372.94,"endTime":2376.21,"body":"So the solution was that, okay,"},{"startTime":2376.21,"endTime":2377.89,"body":"we talk about the dome of St. Paul's,"},{"startTime":2377.89,"endTime":2380.74,"body":"I've got news for you,"},{"startTime":2377.89,"endTime":2380.74,"body":"the domes of St. Paul's."},{"startTime":2380.74,"endTime":2384.28,"body":"The solution was to have an inner dome"},{"startTime":2384.28,"endTime":2387.88,"body":"that would kind of be in"},{"startTime":2384.28,"endTime":2387.88,"body":"between the outer and the inner."},{"startTime":2387.88,"endTime":2391.27,"body":"Then you could put all the mess"},{"startTime":2387.88,"endTime":2391.27,"body":"in between those two things."},{"startTime":2391.27,"endTime":2392.264,"body":"So no one would see,"},{"startTime":2392.264,"endTime":2393.28,"body":"so from the inside, it's"},{"startTime":2392.264,"endTime":2393.28,"body":"going to look great,"},{"startTime":2393.28,"endTime":2394.81,"body":"from the outside, it's"},{"startTime":2393.28,"endTime":2394.81,"body":"going to look great."},{"startTime":2394.81,"endTime":2397.42,"body":"Okay, then what shape"},{"startTime":2394.81,"endTime":2397.42,"body":"should that inner dome be"},{"startTime":2397.42,"endTime":2398.385,"body":"is the next question."},{"startTime":2398.385,"endTime":2399.76,"body":"So you've got a hemisphere on the outside."},{"startTime":2399.76,"endTime":2401.47,"body":"What's on the inside?"},{"startTime":2401.47,"endTime":2406.15,"body":"So actually at the time,"},{"startTime":2401.47,"endTime":2406.15,"body":"there was active discussion"},{"startTime":2406.15,"endTime":2409.69,"body":"about the best shape for various things"},{"startTime":2409.69,"endTime":2411.16,"body":"that you could make of masonry."},{"startTime":2411.16,"endTime":2412.843,"body":"So one of these is an arch."},{"startTime":2415.522,"endTime":2416.355,"body":"This is such a lovely picture."},{"startTime":2416.355,"endTime":2420.79,"body":"These are arches in the"},{"startTime":2416.355,"endTime":2420.79,"body":"loft of a building by Gaudi"},{"startTime":2421.734,"endTime":2425.95,"body":"in Spain and they are a beautiful shape"},{"startTime":2425.95,"endTime":2428.503,"body":"and they illustrate the"},{"startTime":2425.95,"endTime":2428.503,"body":"best shape for an arch."},{"startTime":2429.52,"endTime":2432.31,"body":"But this is being addressed"},{"startTime":2429.52,"endTime":2432.31,"body":"at the Royal Society,"},{"startTime":2432.31,"endTime":2433.39,"body":"what's the best shape?"},{"startTime":2433.39,"endTime":2436.36,"body":"And there's record of meeting in 1671."},{"startTime":2436.36,"endTime":2441.36,"body":"So Dr. Wren tells a meeting of"},{"startTime":2436.36,"endTime":2441.36,"body":"the society what line it is,"},{"startTime":2443.89,"endTime":2446.26,"body":"which is difficult, sentence"},{"startTime":2443.89,"endTime":2446.26,"body":"three, what line it is"},{"startTime":2446.26,"endTime":2450.64,"body":"which an arch fit to sustain"},{"startTime":2446.26,"endTime":2450.64,"body":"any assigned weight makes?"},{"startTime":2450.64,"endTime":2452.92,"body":"In other words, what's the"},{"startTime":2450.64,"endTime":2452.92,"body":"best shape for an arch?"},{"startTime":2452.92,"endTime":2457.18,"body":"And then they asked Mr."},{"startTime":2452.92,"endTime":2457.18,"body":"Hooke as well, Robert Hooke,"},{"startTime":2457.18,"endTime":2460.9,"body":"Wren's collaborator, to"},{"startTime":2457.18,"endTime":2460.9,"body":"also weigh in on this."},{"startTime":2460.9,"endTime":2462.07,"body":"And they promised to write it up"},{"startTime":2462.07,"endTime":2464.86,"body":"and send it to the society."},{"startTime":2464.86,"endTime":2467.29,"body":"So they're both thinking about"},{"startTime":2464.86,"endTime":2467.29,"body":"the best shape for an arch."},{"startTime":2467.29,"endTime":2469.12,"body":"They think they've got it worked out."},{"startTime":2469.12,"endTime":2472.06,"body":"Hooke, well, they did have"},{"startTime":2469.12,"endTime":2472.06,"body":"it worked out and Hooke,"},{"startTime":2472.06,"endTime":2476.29,"body":"he had this habit of writing"},{"startTime":2472.06,"endTime":2476.29,"body":"his answers to things in Latin"},{"startTime":2476.29,"endTime":2479.17,"body":"as an anagram so that"},{"startTime":2476.29,"endTime":2479.17,"body":"he could later on say,"},{"startTime":2479.17,"endTime":2481.72,"body":"I knew that first, look,"},{"startTime":2479.17,"endTime":2481.72,"body":"here's the anagram, right?"},{"startTime":2481.72,"endTime":2484.15,"body":"He did it with Hooke's law"},{"startTime":2481.72,"endTime":2484.15,"body":"and he did it with this."},{"startTime":2484.15,"endTime":2487.27,"body":"And what he had worked out was that"},{"startTime":2487.27,"endTime":2490.63,"body":"if you're trying to work out"},{"startTime":2487.27,"endTime":2490.63,"body":"the best shape for an arch,"},{"startTime":2490.63,"endTime":2492.73,"body":"there's various forces acting."},{"startTime":2492.73,"endTime":2495.97,"body":"But he knew something about a"},{"startTime":2492.73,"endTime":2495.97,"body":"seemingly different question."},{"startTime":2495.97,"endTime":2500.29,"body":"And this picture, this is a"},{"startTime":2495.97,"endTime":2500.29,"body":"not a contemporaneous picture,"},{"startTime":2500.29,"endTime":2503.08,"body":"this was done in 2009 by Rita Greer."},{"startTime":2503.08,"endTime":2504.77,"body":"So he's holding there a chain"},{"startTime":2505.99,"endTime":2508.063,"body":"and it's just sort of hanging there,"},{"startTime":2508.99,"endTime":2511.84,"body":"and the shape that a chain"},{"startTime":2508.99,"endTime":2511.84,"body":"makes when it's just hanging"},{"startTime":2511.84,"endTime":2514.54,"body":"between two points and"},{"startTime":2511.84,"endTime":2514.54,"body":"you know, under gravity,"},{"startTime":2514.54,"endTime":2517.48,"body":"it's called a catenary"},{"startTime":2514.54,"endTime":2517.48,"body":"from the Latin for chain,"},{"startTime":2517.48,"endTime":2521.44,"body":"and Hooke, who's studying catenaries,"},{"startTime":2521.44,"endTime":2523.48,"body":"he realized that the equation"},{"startTime":2523.48,"endTime":2526.183,"body":"that describes the force"},{"startTime":2523.48,"endTime":2526.183,"body":"acting on a catenary,"},{"startTime":2526.183,"endTime":2530.62,"body":"on a chain, sorry, is"},{"startTime":2526.183,"endTime":2530.62,"body":"exactly the same equation"},{"startTime":2530.62,"endTime":2533.2,"body":"that describes the"},{"startTime":2530.62,"endTime":2533.2,"body":"force acting on an arch."},{"startTime":2533.2,"endTime":2536.71,"body":"Now the forces are different,"},{"startTime":2533.2,"endTime":2536.71,"body":"with the chain it's gravity"},{"startTime":2536.71,"endTime":2539.02,"body":"and the tension in the chain,"},{"startTime":2539.02,"endTime":2541.63,"body":"with an arch it's gravity and compression."},{"startTime":2541.63,"endTime":2546.58,"body":"But you get the same"},{"startTime":2541.63,"endTime":2546.58,"body":"equation of each situation."},{"startTime":2546.58,"endTime":2549.31,"body":"And so here I've got some catenaries"},{"startTime":2549.31,"endTime":2550.78,"body":"and it's the same curve essentially,"},{"startTime":2550.78,"endTime":2552.16,"body":"but you're just sort of stretching it"},{"startTime":2552.16,"endTime":2555.28,"body":"depending on how far apart"},{"startTime":2552.16,"endTime":2555.28,"body":"you're holding the ends"},{"startTime":2555.28,"endTime":2556.15,"body":"of the chain."},{"startTime":2556.15,"endTime":2558.703,"body":"So these are catenaries,"},{"startTime":2556.15,"endTime":2558.703,"body":"some shallow, some deep."},{"startTime":2559.75,"endTime":2562.33,"body":"You can just turn those upside down"},{"startTime":2562.33,"endTime":2565.06,"body":"and get the right shape for arches"},{"startTime":2565.06,"endTime":2566.62,"body":"because it's the same equation,"},{"startTime":2566.62,"endTime":2568.36,"body":"and therefore, has the same solution."},{"startTime":2568.36,"endTime":2572.71,"body":"So Hooke had worked this"},{"startTime":2568.36,"endTime":2572.71,"body":"out and Wren also knew this,"},{"startTime":2572.71,"endTime":2574.51,"body":"they were working together,"},{"startTime":2574.51,"endTime":2578.68,"body":"that the best shape for"},{"startTime":2574.51,"endTime":2578.68,"body":"an arch is this catenary."},{"startTime":2578.68,"endTime":2581.17,"body":"And so an arch doesn't"},{"startTime":2578.68,"endTime":2581.17,"body":"have to be that shape,"},{"startTime":2581.17,"endTime":2582.97,"body":"but that's the line of thrust."},{"startTime":2582.97,"endTime":2584.92,"body":"So any arch that's not going to fall down"},{"startTime":2584.92,"endTime":2586.587,"body":"will have that catenary line in it."},{"startTime":2586.587,"endTime":2589.45,"body":"And so of course that means"},{"startTime":2586.587,"endTime":2589.45,"body":"the most efficient way to do it"},{"startTime":2589.45,"endTime":2590.86,"body":"is just to have a catenary."},{"startTime":2590.86,"endTime":2594.52,"body":"So the strongest arches"},{"startTime":2590.86,"endTime":2594.52,"body":"are catenary shapes."},{"startTime":2594.52,"endTime":2597.94,"body":"And so then in the design of St Paul's,"},{"startTime":2597.94,"endTime":2601.39,"body":"the inner dome that you can"},{"startTime":2597.94,"endTime":2601.39,"body":"see there, that's a catenary,"},{"startTime":2601.39,"endTime":2603.31,"body":"has a catenary cross section."},{"startTime":2603.31,"endTime":2605.35,"body":"And that means like"},{"startTime":2603.31,"endTime":2605.35,"body":"visually it fits in with,"},{"startTime":2605.35,"endTime":2609.34,"body":"it chimes in with the other"},{"startTime":2605.35,"endTime":2609.34,"body":"elements of the internal design."},{"startTime":2609.34,"endTime":2610.93,"body":"It all looks very nice."},{"startTime":2610.93,"endTime":2612.73,"body":"So when you are inside the cathedral,"},{"startTime":2612.73,"endTime":2614.93,"body":"it all feels like a"},{"startTime":2612.73,"endTime":2614.93,"body":"harmonious lovely thing."},{"startTime":2616.63,"endTime":2619.78,"body":"So we've got an inner"},{"startTime":2616.63,"endTime":2619.78,"body":"dome, got an outer dome."},{"startTime":2619.78,"endTime":2622.9,"body":"The question is what is"},{"startTime":2619.78,"endTime":2622.9,"body":"going on between them?"},{"startTime":2622.9,"endTime":2624.58,"body":"Because we still haven't"},{"startTime":2622.9,"endTime":2624.58,"body":"solved the problem"},{"startTime":2624.58,"endTime":2627.58,"body":"of supporting that gigantic weight"},{"startTime":2627.58,"endTime":2630.883,"body":"of the hemispherical outer"},{"startTime":2627.58,"endTime":2630.883,"body":"dome and the lantern."},{"startTime":2632.2,"endTime":2636.64,"body":"And Wren and Hooked believed"},{"startTime":2636.64,"endTime":2641.64,"body":"that the strongest shape you"},{"startTime":2636.64,"endTime":2641.64,"body":"could actually have for a dome,"},{"startTime":2642.31,"endTime":2643.75,"body":"which isn't guaranteed to be the same"},{"startTime":2643.75,"endTime":2646.18,"body":"as the answer for an arch"},{"startTime":2643.75,"endTime":2646.18,"body":"because of course an arch is a,"},{"startTime":2646.18,"endTime":2647.62,"body":"you know, essentially"},{"startTime":2646.18,"endTime":2647.62,"body":"two-dimensional thing"},{"startTime":2647.62,"endTime":2648.88,"body":"and a dome is a three-dimensional thing."},{"startTime":2648.88,"endTime":2650.95,"body":"So we may have different things going on."},{"startTime":2650.95,"endTime":2654.9,"body":"They thought the best"},{"startTime":2650.95,"endTime":2654.9,"body":"curve to go with is this."},{"startTime":2654.9,"endTime":2658.81,"body":"So this is the curve Y equals"},{"startTime":2654.9,"endTime":2658.81,"body":"X cubed, not the whole thing."},{"startTime":2658.81,"endTime":2662.02,"body":"But if you just take half of"},{"startTime":2658.81,"endTime":2662.02,"body":"it, say that bottom bit there"},{"startTime":2662.02,"endTime":2663.94,"body":"and maybe spin it around on the axis,"},{"startTime":2663.94,"endTime":2667.75,"body":"you'll get a kind of long thin dome shape."},{"startTime":2667.75,"endTime":2671.77,"body":"And they thought this would"},{"startTime":2667.75,"endTime":2671.77,"body":"be the strongest shape."},{"startTime":2671.77,"endTime":2674.41,"body":"And I want to know why they thought that."},{"startTime":2674.41,"endTime":2676.24,"body":"Why would you think"},{"startTime":2674.41,"endTime":2676.24,"body":"suddenly out of nowhere,"},{"startTime":2676.24,"endTime":2678.95,"body":"why is a cubic equation come into things?"},{"startTime":2678.95,"endTime":2682.57,"body":"And I to show you that, I"},{"startTime":2678.95,"endTime":2682.57,"body":"want to give you an extremely,"},{"startTime":2682.57,"endTime":2684.67,"body":"like my hands will be"},{"startTime":2682.57,"endTime":2684.67,"body":"waving like anything,"},{"startTime":2684.67,"endTime":2688.66,"body":"approximate rough and ready guide"},{"startTime":2688.66,"endTime":2692.86,"body":"to actually the answer for a catenary"},{"startTime":2692.86,"endTime":2694.0,"body":"and the equation of a catenary."},{"startTime":2694.0,"endTime":2695.754,"body":"So this what will come out of this,"},{"startTime":2695.754,"endTime":2697.45,"body":"this is not the equation of a catenary,"},{"startTime":2697.45,"endTime":2700.03,"body":"it's an approximation to"},{"startTime":2697.45,"endTime":2700.03,"body":"the equation of a catenary"},{"startTime":2700.03,"endTime":2703.87,"body":"that I hope it'll be clear"},{"startTime":2700.03,"endTime":2703.87,"body":"to you why I've done this."},{"startTime":2703.87,"endTime":2708.04,"body":"So there's a few symbols in"},{"startTime":2703.87,"endTime":2708.04,"body":"the next couple of minutes."},{"startTime":2708.04,"endTime":2710.5,"body":"If you don't like them, just"},{"startTime":2708.04,"endTime":2710.5,"body":"let your eyes glaze over"},{"startTime":2710.5,"endTime":2713.11,"body":"for a couple minutes so we"},{"startTime":2710.5,"endTime":2713.11,"body":"can come back after that."},{"startTime":2713.11,"endTime":2715.42,"body":"So you've been warned, what"},{"startTime":2713.11,"endTime":2715.42,"body":"do they say in the advert?"},{"startTime":2715.42,"endTime":2718.06,"body":"Here comes the science bit,"},{"startTime":2715.42,"endTime":2718.06,"body":"here comes the science bit."},{"startTime":2718.06,"endTime":2719.68,"body":"So what have we got here?"},{"startTime":2719.68,"endTime":2723.11,"body":"The yellow bit here is your"},{"startTime":2719.68,"endTime":2723.11,"body":"chain that's hanging down"},{"startTime":2724.15,"endTime":2726.46,"body":"and there are various forces acting on it."},{"startTime":2726.46,"endTime":2728.74,"body":"I am interested in the force acting"},{"startTime":2728.74,"endTime":2731.65,"body":"on the bit that I've"},{"startTime":2728.74,"endTime":2731.65,"body":"got in that rectangle."},{"startTime":2731.65,"endTime":2733.84,"body":"So just that little"},{"startTime":2731.65,"endTime":2733.84,"body":"bit of the chain there."},{"startTime":2733.84,"endTime":2735.67,"body":"What are the forces acting on it?"},{"startTime":2735.67,"endTime":2737.47,"body":"So it's just hanging there."},{"startTime":2737.47,"endTime":2742.47,"body":"So you've got the whole of this"},{"startTime":2737.47,"endTime":2742.47,"body":"left hand half of the chain"},{"startTime":2744.04,"endTime":2746.11,"body":"so that the origin in there I put"},{"startTime":2746.11,"endTime":2746.943,"body":"is the lowest part of the chain,"},{"startTime":2746.943,"endTime":2749.02,"body":"the whole left hand half of it,"},{"startTime":2749.02,"endTime":2753.49,"body":"there's tension pulling and"},{"startTime":2749.02,"endTime":2753.49,"body":"I've called that force F."},{"startTime":2753.49,"endTime":2756.34,"body":"You've got that bit of the"},{"startTime":2753.49,"endTime":2756.34,"body":"chain that we're interested in"},{"startTime":2756.34,"endTime":2759.13,"body":"is being pulled down due to gravity."},{"startTime":2759.13,"endTime":2762.01,"body":"So that's downward, its"},{"startTime":2759.13,"endTime":2762.01,"body":"weight is pulling it down."},{"startTime":2762.01,"endTime":2763.24,"body":"I've called that W."},{"startTime":2763.24,"endTime":2765.4,"body":"And then at the right hand end,"},{"startTime":2765.4,"endTime":2767.5,"body":"you've got the rest of the chain"},{"startTime":2767.5,"endTime":2769.51,"body":"and that will have a"},{"startTime":2767.5,"endTime":2769.51,"body":"tension that's pulling"},{"startTime":2769.51,"endTime":2770.92,"body":"on our little bit of chain"},{"startTime":2770.92,"endTime":2773.17,"body":"and that's pulling not exactly upwards,"},{"startTime":2773.17,"endTime":2777.583,"body":"but it's at an angle towards"},{"startTime":2773.17,"endTime":2777.583,"body":"sort of the northeast."},{"startTime":2778.54,"endTime":2782.26,"body":"So if this chain is just"},{"startTime":2778.54,"endTime":2782.26,"body":"at rest, it's come to rest,"},{"startTime":2782.26,"endTime":2783.46,"body":"then everything should be in equilibrium."},{"startTime":2783.46,"endTime":2785.65,"body":"All the forces should balance out."},{"startTime":2785.65,"endTime":2786.82,"body":"So what have we got here?"},{"startTime":2786.82,"endTime":2789.64,"body":"Well, we've got the"},{"startTime":2786.82,"endTime":2789.64,"body":"vertical forces at play,"},{"startTime":2789.64,"endTime":2792.1,"body":"got gravity kind of downwards"},{"startTime":2792.1,"endTime":2794.59,"body":"or the force due to gravity"},{"startTime":2792.1,"endTime":2794.59,"body":"pulling you downwards."},{"startTime":2794.59,"endTime":2797.62,"body":"That's got to be balanced"},{"startTime":2794.59,"endTime":2797.62,"body":"out by the vertical bit"},{"startTime":2797.62,"endTime":2801.67,"body":"of the tension up there,"},{"startTime":2797.62,"endTime":2801.67,"body":"which will be T sin theta."},{"startTime":2801.67,"endTime":2804.88,"body":"And the tension pulling to the left"},{"startTime":2804.88,"endTime":2806.98,"body":"caused by half of the chain,"},{"startTime":2806.98,"endTime":2809.41,"body":"that has got to be balanced"},{"startTime":2806.98,"endTime":2809.41,"body":"out by the horizontal component"},{"startTime":2809.41,"endTime":2811.36,"body":"of the tension in this direction,"},{"startTime":2811.36,"endTime":2812.66,"body":"which will be T cos theta."},{"startTime":2814.24,"endTime":2815.62,"body":"So if you're do a few grade level maths,"},{"startTime":2815.62,"endTime":2817.66,"body":"you'll be familiar with"},{"startTime":2815.62,"endTime":2817.66,"body":"this kind of calculation."},{"startTime":2817.66,"endTime":2819.427,"body":"So we've got these two"},{"startTime":2817.66,"endTime":2819.427,"body":"equations, T sin theta"},{"startTime":2819.427,"endTime":2822.73,"body":"has got to equal W, T cos"},{"startTime":2819.427,"endTime":2822.73,"body":"theta's got to equal F."},{"startTime":2822.73,"endTime":2824.26,"body":"This is great, Sarah, we dunno what any"},{"startTime":2824.26,"endTime":2827.41,"body":"of those letters are and we"},{"startTime":2824.26,"endTime":2827.41,"body":"dunno any of the values here."},{"startTime":2827.41,"endTime":2828.85,"body":"We can get rid of one of them"},{"startTime":2828.85,"endTime":2831.76,"body":"because we can divide the top"},{"startTime":2828.85,"endTime":2831.76,"body":"equation by the bottom one"},{"startTime":2831.76,"endTime":2833.83,"body":"and that sin over cos,"},{"startTime":2831.76,"endTime":2833.83,"body":"that'll give you tan,"},{"startTime":2833.83,"endTime":2835.36,"body":"the t's will cancel out,"},{"startTime":2835.36,"endTime":2837.07,"body":"and we get tan theta is W over F."},{"startTime":2837.07,"endTime":2838.72,"body":"Okay, so far, so good."},{"startTime":2838.72,"endTime":2840.67,"body":"Now here the hand waving starts in earnest"},{"startTime":2840.67,"endTime":2843.37,"body":"because I'm going to"},{"startTime":2840.67,"endTime":2843.37,"body":"make two approximations."},{"startTime":2843.37,"endTime":2847.42,"body":"The first one is the force"},{"startTime":2843.37,"endTime":2847.42,"body":"downwards due to gravity."},{"startTime":2847.42,"endTime":2849.49,"body":"Now force is mass times acceleration."},{"startTime":2849.49,"endTime":2852.79,"body":"So this will be dependent on proportional"},{"startTime":2852.79,"endTime":2855.28,"body":"to the mass of that bit of chain."},{"startTime":2855.28,"endTime":2859.39,"body":"And the mass of that bit of"},{"startTime":2855.28,"endTime":2859.39,"body":"chain will depend on its length."},{"startTime":2859.39,"endTime":2860.38,"body":"We don't know the length"},{"startTime":2860.38,"endTime":2862.39,"body":"'cause we dunno the"},{"startTime":2860.38,"endTime":2862.39,"body":"equation of this curve yet."},{"startTime":2862.39,"endTime":2863.44,"body":"So we're going to approximate it,"},{"startTime":2863.44,"endTime":2865.6,"body":"we're going to pretend it's"},{"startTime":2863.44,"endTime":2865.6,"body":"a straight line this bit."},{"startTime":2865.6,"endTime":2867.94,"body":"It's not quite, it's gently curving."},{"startTime":2867.94,"endTime":2869.71,"body":"We're pretending it's a straight line."},{"startTime":2869.71,"endTime":2871.93,"body":"And so then the mass would"},{"startTime":2869.71,"endTime":2871.93,"body":"be proportional actually"},{"startTime":2871.93,"endTime":2876.16,"body":"to the horizontal distance"},{"startTime":2871.93,"endTime":2876.16,"body":"you've traveled, which is X."},{"startTime":2876.16,"endTime":2879.25,"body":"So I'm going to approximate W,"},{"startTime":2879.25,"endTime":2880.51,"body":"I'm going to say it's a proportional"},{"startTime":2880.51,"endTime":2883.69,"body":"to this X horizontal distance."},{"startTime":2883.69,"endTime":2887.35,"body":"And now F is going to"},{"startTime":2883.69,"endTime":2887.35,"body":"be some constant thing"},{"startTime":2887.35,"endTime":2889.12,"body":"because it's representing"},{"startTime":2887.35,"endTime":2889.12,"body":"half of the chain,"},{"startTime":2889.12,"endTime":2891.49,"body":"which is, you know, we know in theory,"},{"startTime":2891.49,"endTime":2893.56,"body":"we know what that is, it's fixed."},{"startTime":2893.56,"endTime":2897.43,"body":"And so we can say that W over"},{"startTime":2893.56,"endTime":2897.43,"body":"F is also proportional to x"},{"startTime":2897.43,"endTime":2900.73,"body":"and we'll say that it's"},{"startTime":2897.43,"endTime":2900.73,"body":"some constant a times x."},{"startTime":2900.73,"endTime":2904.06,"body":"So that's W over F,"},{"startTime":2900.73,"endTime":2904.06,"body":"some constant a times x,"},{"startTime":2904.06,"endTime":2905.17,"body":"approximation number one."},{"startTime":2905.17,"endTime":2906.79,"body":"Approximation number two,"},{"startTime":2906.79,"endTime":2909.55,"body":"even a bit more dodgy is to say,"},{"startTime":2909.55,"endTime":2911.44,"body":"right, what about this angle theta?"},{"startTime":2911.44,"endTime":2913.12,"body":"Well, I don't know what it is."},{"startTime":2913.12,"endTime":2916.3,"body":"If I sort of extend"},{"startTime":2913.12,"endTime":2916.3,"body":"this green line dottedly"},{"startTime":2916.3,"endTime":2920.35,"body":"towards the origin, I'm"},{"startTime":2916.3,"endTime":2920.35,"body":"going to make a triangle"},{"startTime":2920.35,"endTime":2923.14,"body":"and the triangle will"},{"startTime":2920.35,"endTime":2923.14,"body":"have width x and height y"},{"startTime":2923.14,"endTime":2927.76,"body":"and approximately,"},{"startTime":2923.14,"endTime":2927.76,"body":"approximately angle theta."},{"startTime":2927.76,"endTime":2929.98,"body":"It won't be quite, it"},{"startTime":2927.76,"endTime":2929.98,"body":"won't be quite theta,"},{"startTime":2929.98,"endTime":2933.43,"body":"but it's close enough"},{"startTime":2929.98,"endTime":2933.43,"body":"for our approximation."},{"startTime":2933.43,"endTime":2936.79,"body":"So now I've got the tan theta is y over x."},{"startTime":2936.79,"endTime":2939.37,"body":"And so putting it all together,"},{"startTime":2939.37,"endTime":2942.757,"body":"y over x is roughly a"},{"startTime":2939.37,"endTime":2942.757,"body":"times x, a is any constant."},{"startTime":2942.757,"endTime":2947.59,"body":"And so that gives you y is"},{"startTime":2942.757,"endTime":2947.59,"body":"approximately ax squared."},{"startTime":2947.59,"endTime":2951.1,"body":"And that, if you've ever"},{"startTime":2947.59,"endTime":2951.1,"body":"drawn a curve in class,"},{"startTime":2951.1,"endTime":2953.92,"body":"the first curved line you"},{"startTime":2951.1,"endTime":2953.92,"body":"probably plotted in class"},{"startTime":2953.92,"endTime":2956.89,"body":"was y was x squared, which is a parabola."},{"startTime":2956.89,"endTime":2961.75,"body":"So approximately a catenary"},{"startTime":2956.89,"endTime":2961.75,"body":"is a bit like a parabola."},{"startTime":2961.75,"endTime":2964.39,"body":"Now we do know the actual"},{"startTime":2961.75,"endTime":2964.39,"body":"equation for a catenary."},{"startTime":2964.39,"endTime":2968.38,"body":"It's this, yeah, now we fully understand,"},{"startTime":2968.38,"endTime":2969.88,"body":"it's a bit more complicated"},{"startTime":2969.88,"endTime":2972.22,"body":"and you can change the shallowness"},{"startTime":2972.22,"endTime":2974.26,"body":"by changing this letter B,"},{"startTime":2974.26,"endTime":2976.777,"body":"which is a parameter you can change."},{"startTime":2976.777,"endTime":2979.54,"body":"That's got some powers in it and things."},{"startTime":2979.54,"endTime":2981.644,"body":"There's a way of writing that differently"},{"startTime":2981.644,"endTime":2986.44,"body":"as a kind of a thing that just"},{"startTime":2981.644,"endTime":2986.44,"body":"involves x's, powers of x."},{"startTime":2986.44,"endTime":2988.84,"body":"It's an infinite thing"},{"startTime":2986.44,"endTime":2988.84,"body":"that carries on forever,"},{"startTime":2988.84,"endTime":2990.91,"body":"but it looks like there's"},{"startTime":2988.84,"endTime":2990.91,"body":"the first few terms,"},{"startTime":2990.91,"endTime":2992.77,"body":"something x squared,"},{"startTime":2990.91,"endTime":2992.77,"body":"something x to the four,"},{"startTime":2992.77,"endTime":2994.9,"body":"something x to the six, dot, dot, dot."},{"startTime":2994.9,"endTime":2997.333,"body":"Now if you just noticed"},{"startTime":2994.9,"endTime":2997.333,"body":"the first term of that,"},{"startTime":2998.8,"endTime":3002.97,"body":"that is looking like that"},{"startTime":2998.8,"endTime":3002.97,"body":"parabola approximation I've got."},{"startTime":3002.97,"endTime":3007.62,"body":"And if x is small, then if x"},{"startTime":3002.97,"endTime":3007.62,"body":"is like less than one, right?"},{"startTime":3007.62,"endTime":3010.74,"body":"Then x squared is smaller,"},{"startTime":3007.62,"endTime":3010.74,"body":"x to the four is tiny,"},{"startTime":3010.74,"endTime":3012.39,"body":"x to six is even tinier."},{"startTime":3012.39,"endTime":3014.19,"body":"So actually when X is small,"},{"startTime":3014.19,"endTime":3016.23,"body":"those later terms get very, very tiny"},{"startTime":3016.23,"endTime":3018.21,"body":"and you can almost ignore them."},{"startTime":3018.21,"endTime":3020.31,"body":"So I'm going to superimpose"},{"startTime":3018.21,"endTime":3020.31,"body":"now just to prove this,"},{"startTime":3020.31,"endTime":3024.06,"body":"I'll superimpose for"},{"startTime":3020.31,"endTime":3024.06,"body":"the correct values of B,"},{"startTime":3024.06,"endTime":3026.28,"body":"the relevant parabola onto these catenary."},{"startTime":3026.28,"endTime":3028.26,"body":"And you'll see, okay look,"},{"startTime":3028.26,"endTime":3030.69,"body":"the top one you can't even tell"},{"startTime":3030.69,"endTime":3034.59,"body":"'cause it's got a very"},{"startTime":3030.69,"endTime":3034.59,"body":"small bit of the catenary."},{"startTime":3034.59,"endTime":3036.18,"body":"Next one you start to see it at the edges"},{"startTime":3036.18,"endTime":3038.67,"body":"and the bottom one you can really notice."},{"startTime":3038.67,"endTime":3040.95,"body":"So the parabola is a"},{"startTime":3038.67,"endTime":3040.95,"body":"really good approximation"},{"startTime":3040.95,"endTime":3044.28,"body":"to the catenary for small-ish angles,"},{"startTime":3044.28,"endTime":3045.753,"body":"for shallow catenaries."},{"startTime":3047.19,"endTime":3049.2,"body":"And so what's that got"},{"startTime":3047.19,"endTime":3049.2,"body":"to do with anything?"},{"startTime":3049.2,"endTime":3052.32,"body":"Well, Wren and Hooke knew this,"},{"startTime":3052.32,"endTime":3055.41,"body":"they knew the parabola"},{"startTime":3052.32,"endTime":3055.41,"body":"was a good approximation."},{"startTime":3055.41,"endTime":3059.85,"body":"And you can do a similar"},{"startTime":3055.41,"endTime":3059.85,"body":"but more complicated thing"},{"startTime":3059.85,"endTime":3060.84,"body":"if you go up a dimensions."},{"startTime":3060.84,"endTime":3063.18,"body":"If you're trying to solve"},{"startTime":3060.84,"endTime":3063.18,"body":"the same sort of situation"},{"startTime":3063.18,"endTime":3065.58,"body":"for a dome, then you know, at some point I"},{"startTime":3065.58,"endTime":3069.84,"body":"had my approximation led"},{"startTime":3065.58,"endTime":3069.84,"body":"me to y over X equals ax."},{"startTime":3069.84,"endTime":3073.44,"body":"While in the dome situation,"},{"startTime":3069.84,"endTime":3073.44,"body":"we've gone up a direction,"},{"startTime":3073.44,"endTime":3075.93,"body":"the ideal dome as an approximation,"},{"startTime":3075.93,"endTime":3077.97,"body":"at some point in that we get to something"},{"startTime":3077.97,"endTime":3081.75,"body":"like y over X equals"},{"startTime":3077.97,"endTime":3081.75,"body":"approximately ax squared."},{"startTime":3081.75,"endTime":3083.13,"body":"And so then if you multiply up,"},{"startTime":3083.13,"endTime":3086.01,"body":"y is approximately some"},{"startTime":3083.13,"endTime":3086.01,"body":"constant times x cubed."},{"startTime":3086.01,"endTime":3090.15,"body":"And that's where the cubic idea came from."},{"startTime":3090.15,"endTime":3092.91,"body":"Really it's an approximation."},{"startTime":3092.91,"endTime":3095.91,"body":"And actually they thought,"},{"startTime":3092.91,"endTime":3095.91,"body":"Wren and Hooke thought"},{"startTime":3095.91,"endTime":3097.65,"body":"this was the answer."},{"startTime":3097.65,"endTime":3099.45,"body":"It's not quite the answer."},{"startTime":3099.45,"endTime":3100.92,"body":"The real answer wasn't known for a long,"},{"startTime":3100.92,"endTime":3104.07,"body":"long time after then, it's"},{"startTime":3100.92,"endTime":3104.07,"body":"extremely complicated."},{"startTime":3104.07,"endTime":3105.557,"body":"Here's the first few bits of it."},{"startTime":3105.557,"endTime":3108.51,"body":"It's a very difficult,"},{"startTime":3105.557,"endTime":3108.51,"body":"complicated expression."},{"startTime":3108.51,"endTime":3112.2,"body":"But the first term there"},{"startTime":3108.51,"endTime":3112.2,"body":"is that x cubed term."},{"startTime":3112.2,"endTime":3115.38,"body":"And so the the cube,"},{"startTime":3112.2,"endTime":3115.38,"body":"the cubic, y was x cubic"},{"startTime":3115.38,"endTime":3117.9,"body":"is a really good approximation"},{"startTime":3117.9,"endTime":3119.43,"body":"to the real shape of an ideal,"},{"startTime":3119.43,"endTime":3120.72,"body":"the strongest possible dome"},{"startTime":3120.72,"endTime":3123.09,"body":"which they wanted to"},{"startTime":3120.72,"endTime":3123.09,"body":"support this huge weight."},{"startTime":3123.09,"endTime":3127.08,"body":"So the dome of St. Paul's"},{"startTime":3123.09,"endTime":3127.08,"body":"actually is two domes."},{"startTime":3127.08,"endTime":3130.8,"body":"Well, no, how many domes does"},{"startTime":3127.08,"endTime":3130.8,"body":"it take to build a cathedral?"},{"startTime":3130.8,"endTime":3134.7,"body":"Three domes, there are three"},{"startTime":3130.8,"endTime":3134.7,"body":"domes in St. Paul's Cathedral."},{"startTime":3134.7,"endTime":3135.533,"body":"Here is a picture,"},{"startTime":3135.533,"endTime":3137.04,"body":"this is drawn by Christopher Wren himself."},{"startTime":3137.04,"endTime":3140.1,"body":"It says at the bottom,"},{"startTime":3137.04,"endTime":3140.1,"body":"Christopher Wren's hand,"},{"startTime":3140.1,"endTime":3141.99,"body":"illustrating the three domes."},{"startTime":3141.99,"endTime":3146.99,"body":"So, and they called this the"},{"startTime":3141.99,"endTime":3146.99,"body":"cubicle parabolic conoid,"},{"startTime":3148.262,"endTime":3152.247,"body":"conoid catchy, what it is there,"},{"startTime":3152.247,"endTime":3154.83,"body":"and it's might be hard"},{"startTime":3152.247,"endTime":3154.83,"body":"to see at this distance,"},{"startTime":3154.83,"endTime":3156.03,"body":"but you can see there's some numbers"},{"startTime":3156.03,"endTime":3157.95,"body":"down the side perhaps."},{"startTime":3157.95,"endTime":3161.37,"body":"What those numbers are, he's"},{"startTime":3157.95,"endTime":3161.37,"body":"actually sketching the graph"},{"startTime":3161.37,"endTime":3162.6,"body":"of y equals x cubed."},{"startTime":3162.6,"endTime":3165.57,"body":"So it goes up in eights so"},{"startTime":3162.6,"endTime":3165.57,"body":"I've drawn that like this,"},{"startTime":3165.57,"endTime":3168.75,"body":"and he's marked off to help him with this,"},{"startTime":3168.75,"endTime":3170.1,"body":"look, I've highlighted it in blue,"},{"startTime":3170.1,"endTime":3172.08,"body":"this is that exact curve y was x cubed."},{"startTime":3172.08,"endTime":3172.913,"body":"I've just superimposed it"},{"startTime":3172.913,"endTime":3174.9,"body":"and you can see it's a"},{"startTime":3172.913,"endTime":3174.9,"body":"really, really good match"},{"startTime":3174.9,"endTime":3176.25,"body":"'cause that's what he was drawing."},{"startTime":3176.25,"endTime":3178.68,"body":"And you can actually see"},{"startTime":3176.25,"endTime":3178.68,"body":"where he's marked off,"},{"startTime":3178.68,"endTime":3181.89,"body":"one cubed is one, two cubed"},{"startTime":3178.68,"endTime":3181.89,"body":"is eight, three cubed is 27."},{"startTime":3181.89,"endTime":3185.812,"body":"You can see the dot at"},{"startTime":3181.89,"endTime":3185.812,"body":"27, four cubed is 64."},{"startTime":3185.812,"endTime":3186.645,"body":"And that's the same at the bottom."},{"startTime":3186.645,"endTime":3190.89,"body":"So he's sketched the x"},{"startTime":3186.645,"endTime":3190.89,"body":"cubed graph in his picture"},{"startTime":3190.89,"endTime":3194.01,"body":"of the not one, not two, but"},{"startTime":3190.89,"endTime":3194.01,"body":"three domes of St. Paul's."},{"startTime":3194.01,"endTime":3195.96,"body":"So that's really cool to see."},{"startTime":3195.96,"endTime":3199.11,"body":"And it's what a wonderful"},{"startTime":3195.96,"endTime":3199.11,"body":"mathematical marvel"},{"startTime":3199.11,"endTime":3201.93,"body":"that the domes of St. Paul's are."},{"startTime":3201.93,"endTime":3204.45,"body":"You've got a catenary, you've got a cubic,"},{"startTime":3204.45,"endTime":3205.773,"body":"you've got a hemisphere."},{"startTime":3207.09,"endTime":3210.42,"body":"So Christopher Wren then architect, yes."},{"startTime":3210.42,"endTime":3212.913,"body":"But look how mathematical his work is."},{"startTime":3214.05,"endTime":3217.41,"body":"And I want to finish by just,"},{"startTime":3217.41,"endTime":3220.41,"body":"this is a justification of how it comes"},{"startTime":3220.41,"endTime":3225.12,"body":"that the Gresham professors"},{"startTime":3220.41,"endTime":3225.12,"body":"in their academic year,"},{"startTime":3225.12,"endTime":3228.15,"body":"we're supposed to give six"},{"startTime":3225.12,"endTime":3228.15,"body":"linked lectures on a theme."},{"startTime":3228.15,"endTime":3229.65,"body":"And my first lectures this year,"},{"startTime":3229.65,"endTime":3231.15,"body":"three of them were on Matson Money."},{"startTime":3231.15,"endTime":3233.52,"body":"We did coins and currency,"},{"startTime":3231.15,"endTime":3233.52,"body":"we did game theory,"},{"startTime":3233.52,"endTime":3235.08,"body":"we did how to win the lottery."},{"startTime":3235.08,"endTime":3236.31,"body":"And those people who attended the how"},{"startTime":3236.31,"endTime":3238.47,"body":"to win the lottery lecture"},{"startTime":3236.31,"endTime":3238.47,"body":"are now sitting here"},{"startTime":3238.47,"endTime":3241.563,"body":"in top hats and fur"},{"startTime":3238.47,"endTime":3241.563,"body":"coats, drinking champagne."},{"startTime":3242.586,"endTime":3243.93,"body":"So that was at Matson Money."},{"startTime":3243.93,"endTime":3246.27,"body":"And now that the second half of this year,"},{"startTime":3246.27,"endTime":3248.7,"body":"we have three lectures on"},{"startTime":3246.27,"endTime":3248.7,"body":"unexpected in mathematical lives,"},{"startTime":3248.7,"endTime":3251.76,"body":"Christopher Wren, architect,"},{"startTime":3248.7,"endTime":3251.76,"body":"also mathematician."},{"startTime":3251.76,"endTime":3254.37,"body":"Next up Florence Nightingale, nurse,"},{"startTime":3254.37,"endTime":3256.08,"body":"also a pioneer of statistics."},{"startTime":3256.08,"endTime":3257.91,"body":"In June, Alan Turing."},{"startTime":3257.91,"endTime":3259.53,"body":"Okay, we do know he's a mathematician."},{"startTime":3259.53,"endTime":3263.46,"body":"What we think of his work"},{"startTime":3259.53,"endTime":3263.46,"body":"mainly in cryptography,"},{"startTime":3263.46,"endTime":3265.68,"body":"cracking the Enigma code"},{"startTime":3263.46,"endTime":3265.68,"body":"in the Second World War."},{"startTime":3265.68,"endTime":3268.23,"body":"But he also did brilliant"},{"startTime":3265.68,"endTime":3268.23,"body":"work in mathematical biology."},{"startTime":3268.23,"endTime":3269.37,"body":"So I'll talk about that."},{"startTime":3269.37,"endTime":3271.23,"body":"So how did I get away with this?"},{"startTime":3271.23,"endTime":3274.17,"body":"So here's my justification, my excuse,"},{"startTime":3274.17,"endTime":3275.82,"body":"I wanted to call these lectures,"},{"startTime":3275.82,"endTime":3278.913,"body":"mathematicians of note because"},{"startTime":3275.82,"endTime":3278.913,"body":"have you spotted the link?"},{"startTime":3280.53,"endTime":3284.37,"body":"They're all on bank notes,"},{"startTime":3280.53,"endTime":3284.37,"body":"so that's my excuse."},{"startTime":3284.37,"endTime":3287.01,"body":"So Christopher Wren, he got 50 pound note."},{"startTime":3287.01,"endTime":3288.93,"body":"Florence Nightingale,"},{"startTime":3287.01,"endTime":3288.93,"body":"the old 10 pound note."},{"startTime":3288.93,"endTime":3291.39,"body":"And on the new 50 pound note, Alan Turing."},{"startTime":3291.39,"endTime":3293.25,"body":"So we'll finish there"},{"startTime":3291.39,"endTime":3293.25,"body":"and I hope to see you"},{"startTime":3293.25,"endTime":3296.737,"body":"in May for my lecture on"},{"startTime":3293.25,"endTime":3296.737,"body":"Florence Nightingale, thank you."},{"startTime":3296.737,"endTime":3299.904,"body":"(audience applauding)"},{"startTime":3305.081,"endTime":3308.248,"body":"(audience applauding)"},{"startTime":3309.099,"endTime":3310.759,"body":"- Thank you so much, Professor Hart,"},{"startTime":3310.759,"endTime":3312.074,"body":"for such a fascinating lecture."},{"startTime":3312.074,"endTime":3313.393,"body":"I've got a couple questions from online"},{"startTime":3313.393,"endTime":3315.279,"body":"before going to audience in the room."},{"startTime":3315.279,"endTime":3317.4,"body":"First one is from (mumbles)."},{"startTime":3317.4,"endTime":3319.147,"body":"He says, \"If they, Wren and Hooke,"},{"startTime":3319.147,"endTime":3323.107,"body":"\"knew that the catenary was"},{"startTime":3319.147,"endTime":3323.107,"body":"the best load bearing shape"},{"startTime":3323.107,"endTime":3324.847,"body":"\"and they knew its equation,"},{"startTime":3324.847,"endTime":3328.477,"body":"\"then why approximate to parabola?\""},{"startTime":3328.477,"endTime":3333.36,"body":"- Parabola, yeah, so they"},{"startTime":3328.477,"endTime":3333.36,"body":"weren't approximating"},{"startTime":3333.36,"endTime":3338.22,"body":"to a parabola, but that was"},{"startTime":3333.36,"endTime":3338.22,"body":"kind of my justification"},{"startTime":3338.22,"endTime":3339.63,"body":"for how they got to the cubic essentially."},{"startTime":3339.63,"endTime":3342.42,"body":"So Galileo, so some people say"},{"startTime":3342.42,"endTime":3344.52,"body":"that Galileo thought the"},{"startTime":3342.42,"endTime":3344.52,"body":"catenary was a parabola."},{"startTime":3344.52,"endTime":3345.87,"body":"He didn't think it was a parabola,"},{"startTime":3345.87,"endTime":3347.82,"body":"but he knew that the parabola was close."},{"startTime":3347.82,"endTime":3349.11,"body":"And so this was a well known thing"},{"startTime":3349.11,"endTime":3350.94,"body":"that the catenary was close to a parabola."},{"startTime":3350.94,"endTime":3352.86,"body":"And you can get it, you know,"},{"startTime":3352.86,"endTime":3354.15,"body":"you'll notice what I did there,"},{"startTime":3354.15,"endTime":3356.01,"body":"that little slightly dodgy calculation"},{"startTime":3356.01,"endTime":3359.55,"body":"doesn't involve calculus,"},{"startTime":3356.01,"endTime":3359.55,"body":"but if you are pre-calculus,"},{"startTime":3359.55,"endTime":3361.65,"body":"it's very hard to actually"},{"startTime":3359.55,"endTime":3361.65,"body":"derive these things."},{"startTime":3361.65,"endTime":3364.71,"body":"So a parabola, as we saw is"},{"startTime":3361.65,"endTime":3364.71,"body":"a really good approximation."},{"startTime":3364.71,"endTime":3367.77,"body":"But it was the fact of it"},{"startTime":3364.71,"endTime":3367.77,"body":"being close to a catenary"},{"startTime":3367.77,"endTime":3369.24,"body":"that made them then think"},{"startTime":3369.24,"endTime":3372.39,"body":"of actually maybe the"},{"startTime":3369.24,"endTime":3372.39,"body":"cubic is the right thing"},{"startTime":3372.39,"endTime":3373.713,"body":"for the dome, yeah."},{"startTime":3374.64,"endTime":3376.547,"body":"- There's another question referring"},{"startTime":3376.547,"endTime":3379.631,"body":"to another of the slide,"},{"startTime":3379.631,"endTime":3383.644,"body":"I think the slide just"},{"startTime":3379.631,"endTime":3383.644,"body":"before the ideal dome slide."},{"startTime":3383.644,"endTime":3385.297,"body":"- Yes."},{"startTime":3383.644,"endTime":3385.297,"body":"- And he asks,"},{"startTime":3385.297,"endTime":3388.41,"body":"\"Wouldn't W be closer to the hypothenuse?\""},{"startTime":3388.41,"endTime":3392.748,"body":"- Wouldn't doubly be closer to that?"},{"startTime":3392.748,"endTime":3397.353,"body":"Might have to, oh, there's a W."},{"startTime":3399.36,"endTime":3402.15,"body":"So the hypothenuse of this triangle."},{"startTime":3402.15,"endTime":3404.52,"body":"Well, so I know what you're saying."},{"startTime":3404.52,"endTime":3407.58,"body":"Yes, why wouldn't, I've"},{"startTime":3404.52,"endTime":3407.58,"body":"said it's related to x,"},{"startTime":3407.58,"endTime":3409.38,"body":"well, it's not, it doesn't equal x."},{"startTime":3409.38,"endTime":3411.03,"body":"So that the length, I"},{"startTime":3409.38,"endTime":3411.03,"body":"think what the question"},{"startTime":3411.03,"endTime":3413.55,"body":"is the length of that bit of the catenary,"},{"startTime":3413.55,"endTime":3415.77,"body":"if we're imagining it,"},{"startTime":3413.55,"endTime":3415.77,"body":"it's a straight line,"},{"startTime":3415.77,"endTime":3418.23,"body":"yes, it would be the"},{"startTime":3415.77,"endTime":3418.23,"body":"hypothenuse of that triangle."},{"startTime":3418.23,"endTime":3421.623,"body":"But that would be proportional to x."},{"startTime":3421.623,"endTime":3423.15,"body":"It would be some multiple of x in length."},{"startTime":3423.15,"endTime":3425.97,"body":"So it's not equal to x,"},{"startTime":3423.15,"endTime":3425.97,"body":"it's a multiple of x."},{"startTime":3425.97,"endTime":3427.35,"body":"But that's okay 'cause we're only caring"},{"startTime":3427.35,"endTime":3429.42,"body":"about proportions here."},{"startTime":3429.42,"endTime":3431.01,"body":"Yes, you know, if X were to double,"},{"startTime":3431.01,"endTime":3432.15,"body":"then the hypothenuse would double."},{"startTime":3432.15,"endTime":3436.11,"body":"You can see it's a, yeah, a"},{"startTime":3432.15,"endTime":3436.11,"body":"proportional relationship, yes."},{"startTime":3436.11,"endTime":3437.867,"body":"- [Audience Member] You"},{"startTime":3436.11,"endTime":3437.867,"body":"emphasized at the beginning"},{"startTime":3437.867,"endTime":3442.867,"body":"of the lecture,"},{"startTime":3437.867,"endTime":3442.867,"body":"Christopher Wren's emphasis"},{"startTime":3445.2,"endTime":3448.89,"body":"or interest in the"},{"startTime":3445.2,"endTime":3448.89,"body":"applications of mathematics."},{"startTime":3448.89,"endTime":3451.08,"body":"I just wonder whether you"},{"startTime":3448.89,"endTime":3451.08,"body":"would care to conjecture,"},{"startTime":3451.08,"endTime":3456.08,"body":"if you could imagine Wren"},{"startTime":3451.08,"endTime":3456.08,"body":"meeting, (coughs) excuse me,"},{"startTime":3456.24,"endTime":3460.433,"body":"peers like Birch and Russell,"},{"startTime":3456.24,"endTime":3460.433,"body":"Alfred Whitehead and Godel,"},{"startTime":3461.79,"endTime":3464.427,"body":"would he have any time"},{"startTime":3461.79,"endTime":3464.427,"body":"for what they had to say"},{"startTime":3466.494,"endTime":3470.22,"body":"in the contrast between"},{"startTime":3466.494,"endTime":3470.22,"body":"applications in the purest aspect?"},{"startTime":3470.22,"endTime":3473.76,"body":"- Yeah, so I think, I don't"},{"startTime":3470.22,"endTime":3473.76,"body":"think he'd see it as a contrast."},{"startTime":3473.76,"endTime":3477.48,"body":"I think he would see all of that"},{"startTime":3477.48,"endTime":3481.043,"body":"as being valuable and"},{"startTime":3477.48,"endTime":3481.043,"body":"important parts of what it"},{"startTime":3481.043,"endTime":3483.3,"body":"is that mathematicians are doing"},{"startTime":3483.3,"endTime":3487.41,"body":"in a way that perhaps has"},{"startTime":3483.3,"endTime":3487.41,"body":"been a bit lost more recently."},{"startTime":3487.41,"endTime":3489.93,"body":"So, you know, you get Hardy"},{"startTime":3489.93,"endTime":3492.0,"body":"in his mathematician's"},{"startTime":3489.93,"endTime":3492.0,"body":"apology saying like,"},{"startTime":3492.0,"endTime":3494.22,"body":"I'd rather be at the"},{"startTime":3492.0,"endTime":3494.22,"body":"top of Nelson's column"},{"startTime":3494.22,"endTime":3495.84,"body":"and no one can see me or"},{"startTime":3494.22,"endTime":3495.84,"body":"understand what I'm doing,"},{"startTime":3495.84,"endTime":3498.81,"body":"but as long as my important"},{"startTime":3495.84,"endTime":3498.81,"body":"work is, you know,"},{"startTime":3498.81,"endTime":3501.09,"body":"I don't care and I despise application."},{"startTime":3501.09,"endTime":3502.14,"body":"You know, you get that attitude,"},{"startTime":3502.14,"endTime":3505.5,"body":"which I'm a pure mathematician"},{"startTime":3502.14,"endTime":3505.5,"body":"and I do it 'cause I love it,"},{"startTime":3505.5,"endTime":3507.84,"body":"but I don't despise applications."},{"startTime":3507.84,"endTime":3509.52,"body":"And I think this is what Wren was like."},{"startTime":3509.52,"endTime":3512.16,"body":"He was, yeah, he was"},{"startTime":3509.52,"endTime":3512.16,"body":"rectifying the cycloid"},{"startTime":3512.16,"endTime":3515.07,"body":"like there was no tomorrow"},{"startTime":3512.16,"endTime":3515.07,"body":"and he didn't have in mind"},{"startTime":3515.07,"endTime":3516.78,"body":"when he did that because I might be able"},{"startTime":3516.78,"endTime":3518.91,"body":"to use it for something,"},{"startTime":3516.78,"endTime":3518.91,"body":"he was genuinely interested"},{"startTime":3518.91,"endTime":3523.11,"body":"in just finding things out,"},{"startTime":3518.91,"endTime":3523.11,"body":"he had that curious mind."},{"startTime":3523.11,"endTime":3525.6,"body":"But he didn't then say, oh,"},{"startTime":3525.6,"endTime":3528.3,"body":"I don't care about what"},{"startTime":3525.6,"endTime":3528.3,"body":"it might be used for."},{"startTime":3528.3,"endTime":3529.68,"body":"He was also interested in that."},{"startTime":3529.68,"endTime":3532.56,"body":"So he, you know, it was part"},{"startTime":3529.68,"endTime":3532.56,"body":"of what mathematics was to him."},{"startTime":3532.56,"endTime":3535.71,"body":"The mathematical arts"},{"startTime":3532.56,"endTime":3535.71,"body":"encompass all of these things"},{"startTime":3535.71,"endTime":3539.943,"body":"and I think that's what he"},{"startTime":3535.71,"endTime":3539.943,"body":"what he would've thought, yeah."},{"startTime":3541.14,"endTime":3542.58,"body":"- Thank you so much Professor Hart."},{"startTime":3542.58,"endTime":3545.46,"body":"I'm afraid we are now at times, so that's,"},{"startTime":3545.46,"endTime":3547.65,"body":"I think all the time"},{"startTime":3545.46,"endTime":3547.65,"body":"we have for questions,"},{"startTime":3547.65,"endTime":3550.5,"body":"but can everyone give a very warm round"},{"startTime":3550.5,"endTime":3552.325,"body":"of applause for Professor Sarah Hart?"},{"startTime":3552.325,"endTime":3555.575,"body":"(audience applauding)"}]}