{"version":"1.0.0","segments":[{"startTime":0.24,"endTime":4.04,"body":"- Okay, so conic sections"},{"startTime":0.24,"endTime":4.04,"body":"are the curves made"},{"startTime":4.04,"endTime":7.43,"body":"by slicing through cones"},{"startTime":4.04,"endTime":7.43,"body":"at different angles."},{"startTime":7.43,"endTime":10.86,"body":"There are three types, the ellipse,"},{"startTime":10.86,"endTime":14.49,"body":"which is slicing at a kind of"},{"startTime":10.86,"endTime":14.49,"body":"narrow angle or a low angle."},{"startTime":14.49,"endTime":16.22,"body":"As you steepen the angle of the slice,"},{"startTime":16.22,"endTime":17.87,"body":"you get parabolas"},{"startTime":17.87,"endTime":21.45,"body":"and they occur when the angle of the slice"},{"startTime":21.45,"endTime":24.16,"body":"is exactly the same as"},{"startTime":21.45,"endTime":24.16,"body":"the angle of the cone."},{"startTime":24.16,"endTime":26.1,"body":"And then if you get steeper still,"},{"startTime":26.1,"endTime":28.75,"body":"the slices start to cut"},{"startTime":26.1,"endTime":28.75,"body":"through both halves of the cone"},{"startTime":28.75,"endTime":31.01,"body":"and that's the hyperbola."},{"startTime":31.01,"endTime":34.85,"body":"Now these three types of conic"},{"startTime":31.01,"endTime":34.85,"body":"section as they're called,"},{"startTime":34.85,"endTime":36.8,"body":"the ellipse, parabola and hyperbola"},{"startTime":36.8,"endTime":38.67,"body":"have beautiful and intriguing properties"},{"startTime":38.67,"endTime":40.029,"body":"and we're going to talk about these today,"},{"startTime":40.029,"endTime":42.55,"body":"as well as some applications."},{"startTime":42.55,"endTime":44.83,"body":"We'll see how ellipses"},{"startTime":42.55,"endTime":44.83,"body":"are used in medicine,"},{"startTime":44.83,"endTime":47.51,"body":"how to make a parabola by folding paper"},{"startTime":47.51,"endTime":50.163,"body":"and what hyperbolas have"},{"startTime":47.51,"endTime":50.163,"body":"to do with economics."},{"startTime":51.39,"endTime":53.41,"body":"There are three of these conic sections,"},{"startTime":53.41,"endTime":55.61,"body":"and for each one, I'm"},{"startTime":53.41,"endTime":55.61,"body":"going to show you three"},{"startTime":55.61,"endTime":57.283,"body":"quite varied applications."},{"startTime":58.18,"endTime":61.93,"body":"So let's get started and"},{"startTime":58.18,"endTime":61.93,"body":"talk about the ellipse first."},{"startTime":61.93,"endTime":64.44,"body":"And I should mention there"},{"startTime":61.93,"endTime":64.44,"body":"are three of these things,"},{"startTime":64.44,"endTime":66.13,"body":"but the ellipse will take up"},{"startTime":66.13,"endTime":67.97,"body":"approximately half the time we have,"},{"startTime":67.97,"endTime":70.3,"body":"so don't panic if I'm still"},{"startTime":67.97,"endTime":70.3,"body":"talking about ellipses"},{"startTime":70.3,"endTime":71.133,"body":"halfway through,"},{"startTime":71.133,"endTime":73.41,"body":"and you think you're never"},{"startTime":71.133,"endTime":73.41,"body":"going to, this will never end,"},{"startTime":73.41,"endTime":75.2,"body":"although it's exciting, of course."},{"startTime":75.2,"endTime":76.89,"body":"Right, let's talk about ellipses."},{"startTime":76.89,"endTime":78.43,"body":"This is one way to define an ellipse."},{"startTime":78.43,"endTime":80.15,"body":"If you think about a circle,"},{"startTime":80.15,"endTime":82.33,"body":"which I guess is a"},{"startTime":80.15,"endTime":82.33,"body":"special case of an ellipse"},{"startTime":82.33,"endTime":84.72,"body":"where you slice"},{"startTime":82.33,"endTime":84.72,"body":"horizontally through a cone,"},{"startTime":84.72,"endTime":86.83,"body":"with a circle, you can make a circle"},{"startTime":86.83,"endTime":89.12,"body":"by having a fixed point, the center,"},{"startTime":89.12,"endTime":90.36,"body":"and then the circle is"},{"startTime":89.12,"endTime":90.36,"body":"the collection of points"},{"startTime":90.36,"endTime":93.88,"body":"that are all exactly the same"},{"startTime":90.36,"endTime":93.88,"body":"distance from that center,"},{"startTime":93.88,"endTime":95.24,"body":"a radius R."},{"startTime":95.24,"endTime":98.15,"body":"well, with an ellipse, it's a bit similar,"},{"startTime":98.15,"endTime":100.85,"body":"but this time you have"},{"startTime":98.15,"endTime":100.85,"body":"two important points."},{"startTime":100.85,"endTime":102.47,"body":"Each one is called a focus."},{"startTime":102.47,"endTime":105.35,"body":"So the plural of focus, foci."},{"startTime":105.35,"endTime":108.47,"body":"I've marked them on the"},{"startTime":105.35,"endTime":108.47,"body":"diagram, they're F and F dash."},{"startTime":108.47,"endTime":110.97,"body":"And the definition with this"},{"startTime":110.97,"endTime":114.52,"body":"is that it's the sum of"},{"startTime":110.97,"endTime":114.52,"body":"the distances to the foci"},{"startTime":114.52,"endTime":115.37,"body":"that is a constant."},{"startTime":115.37,"endTime":118.02,"body":"So the points on the"},{"startTime":115.37,"endTime":118.02,"body":"ellipse have this property."},{"startTime":118.02,"endTime":121.46,"body":"Point P has a property"},{"startTime":118.02,"endTime":121.46,"body":"that the distance to F"},{"startTime":121.46,"endTime":123.94,"body":"plus the distance to F dash is a constant,"},{"startTime":123.94,"endTime":127.05,"body":"and I'll call that R and I'll"},{"startTime":123.94,"endTime":127.05,"body":"do that throughout the talk."},{"startTime":127.05,"endTime":129.18,"body":"So this is the definition of an ellipse."},{"startTime":129.18,"endTime":132.6,"body":"That gives us a way to draw ellipses."},{"startTime":132.6,"endTime":134.111,"body":"With a circle, we can just have,"},{"startTime":134.111,"endTime":136.31,"body":"you know, a pair of compasses"},{"startTime":134.111,"endTime":136.31,"body":"and we can draw like this."},{"startTime":136.31,"endTime":138.861,"body":"With an ellipse, you"},{"startTime":136.31,"endTime":138.861,"body":"can get a piece of paper"},{"startTime":138.861,"endTime":140.26,"body":"and you can put two pins in it,"},{"startTime":140.26,"endTime":142.44,"body":"and then you get a piece of string."},{"startTime":142.44,"endTime":144.67,"body":"I'll make this whole piece of string here."},{"startTime":144.67,"endTime":146.38,"body":"You get a piece of"},{"startTime":144.67,"endTime":146.38,"body":"string and loop it round,"},{"startTime":146.38,"endTime":147.77,"body":"not an elastic band or something,"},{"startTime":147.77,"endTime":150.28,"body":"'cause it needs to have a fixed length,"},{"startTime":150.28,"endTime":152.81,"body":"and then you pull it taut"},{"startTime":150.28,"endTime":152.81,"body":"and you get a pen or pencil"},{"startTime":152.81,"endTime":155.63,"body":"and just draw around like that."},{"startTime":155.63,"endTime":158.09,"body":"Now, the reason this"},{"startTime":155.63,"endTime":158.09,"body":"will give you an ellipse"},{"startTime":158.09,"endTime":161.08,"body":"is because the total length"},{"startTime":158.09,"endTime":161.08,"body":"of the string is constant,"},{"startTime":161.08,"endTime":163.2,"body":"and then the distance between the two foci"},{"startTime":163.2,"endTime":164.27,"body":"is always constant,"},{"startTime":164.27,"endTime":165.65,"body":"and then the rest of the string"},{"startTime":165.65,"endTime":167.84,"body":"is giving you the sum of the distances"},{"startTime":167.84,"endTime":172.19,"body":"from the point to the"},{"startTime":167.84,"endTime":172.19,"body":"two foci, F and F dash."},{"startTime":172.19,"endTime":174.645,"body":"So I've done a not very good video"},{"startTime":174.645,"endTime":177.62,"body":"of me drawing an ellipse"},{"startTime":174.645,"endTime":177.62,"body":"using this technique."},{"startTime":177.62,"endTime":179.41,"body":"So I'm about to-"},{"startTime":179.41,"endTime":182.35,"body":"This is not an absolutely"},{"startTime":179.41,"endTime":182.35,"body":"mathematically perfect ellipse."},{"startTime":182.35,"endTime":183.41,"body":"There's a little bit of a bump"},{"startTime":183.41,"endTime":185.14,"body":"where I go over the knot in the string,"},{"startTime":185.14,"endTime":187.77,"body":"but you can see that this"},{"startTime":185.14,"endTime":187.77,"body":"is producing something"},{"startTime":187.77,"endTime":189.74,"body":"that looks sort of elliptical."},{"startTime":189.74,"endTime":191.57,"body":"So that's that."},{"startTime":191.57,"endTime":195.53,"body":"And this gives us, this"},{"startTime":191.57,"endTime":195.53,"body":"definition of an ellipse"},{"startTime":195.53,"endTime":200.453,"body":"gives us the first"},{"startTime":195.53,"endTime":200.453,"body":"application of ellipses."},{"startTime":201.29,"endTime":204.567,"body":"I should say, by the way, it"},{"startTime":201.29,"endTime":204.567,"body":"isn't clear if you're thinking,"},{"startTime":204.567,"endTime":206.253,"body":"\"How does she know that"},{"startTime":204.567,"endTime":206.253,"body":"that's the same curve"},{"startTime":206.253,"endTime":208.33,"body":"that you get by slicing through a cone?\""},{"startTime":208.33,"endTime":210.07,"body":"It's not obvious that that is the case."},{"startTime":210.07,"endTime":211.32,"body":"It is the case,"},{"startTime":211.32,"endTime":213.92,"body":"but I'm not going to give"},{"startTime":211.32,"endTime":213.92,"body":"the detailed proof here now."},{"startTime":213.92,"endTime":215.07,"body":"It takes a little bit of algebra."},{"startTime":215.07,"endTime":217.28,"body":"It's not too hard, but it can be done."},{"startTime":217.28,"endTime":219.58,"body":"But I want to focus on"},{"startTime":217.28,"endTime":219.58,"body":"talking about the properties"},{"startTime":219.58,"endTime":222.08,"body":"and the applications in this talk."},{"startTime":222.08,"endTime":225.78,"body":"So first application of"},{"startTime":222.08,"endTime":225.78,"body":"ellipses is in gear systems."},{"startTime":225.78,"endTime":228.81,"body":"Now, usually when you"},{"startTime":225.78,"endTime":228.81,"body":"have a system of gears,"},{"startTime":228.81,"endTime":230.03,"body":"they are usually circular."},{"startTime":230.03,"endTime":232.03,"body":"So you might have an input gear"},{"startTime":232.03,"endTime":235.26,"body":"that's say, a circle of"},{"startTime":232.03,"endTime":235.26,"body":"some size or some radius,"},{"startTime":235.26,"endTime":238.47,"body":"and then it'll have an output"},{"startTime":235.26,"endTime":238.47,"body":"gear or a series of gears,"},{"startTime":238.47,"endTime":241.53,"body":"and you usually would turn the gears."},{"startTime":241.53,"endTime":244.03,"body":"The input gear would be"},{"startTime":241.53,"endTime":244.03,"body":"turning at a constant rate,"},{"startTime":244.03,"endTime":246.21,"body":"and that would transfer"},{"startTime":244.03,"endTime":246.21,"body":"to the next gear along,"},{"startTime":246.21,"endTime":249.63,"body":"which will then start turning"},{"startTime":246.21,"endTime":249.63,"body":"at a constant rate too."},{"startTime":249.63,"endTime":252.35,"body":"It'll not necessarily be"},{"startTime":249.63,"endTime":252.35,"body":"the same constant rate,"},{"startTime":252.35,"endTime":254.65,"body":"because if the gears are different sizes,"},{"startTime":254.65,"endTime":257.43,"body":"that will affect the speed"},{"startTime":254.65,"endTime":257.43,"body":"at which they rotate."},{"startTime":257.43,"endTime":259.288,"body":"So usually if you-"},{"startTime":259.288,"endTime":262.01,"body":"Well, not usually, always, if"},{"startTime":259.288,"endTime":262.01,"body":"you have a small input gear"},{"startTime":262.01,"endTime":263.61,"body":"turning at a particular speed"},{"startTime":263.61,"endTime":266.06,"body":"and it's next to a large output gear,"},{"startTime":266.06,"endTime":269.35,"body":"then the larger one will"},{"startTime":266.06,"endTime":269.35,"body":"turn at a slower rate."},{"startTime":269.35,"endTime":271.12,"body":"Because of course you"},{"startTime":269.35,"endTime":271.12,"body":"need lots of revolutions"},{"startTime":271.12,"endTime":274.21,"body":"of the small one to get one"},{"startTime":271.12,"endTime":274.21,"body":"revolution of the big one."},{"startTime":274.21,"endTime":275.417,"body":"Whereas if you have a large input gear"},{"startTime":275.417,"endTime":277.14,"body":"and a small output gear,"},{"startTime":277.14,"endTime":279.74,"body":"then that small output gear"},{"startTime":277.14,"endTime":279.74,"body":"will turn at a faster rate"},{"startTime":279.74,"endTime":281.431,"body":"than the input gear."},{"startTime":281.431,"endTime":284.01,"body":"Sometimes in some applications,"},{"startTime":284.01,"endTime":286.45,"body":"you don't want a constant rate of motion."},{"startTime":286.45,"endTime":289.54,"body":"One example is if you"},{"startTime":286.45,"endTime":289.54,"body":"have a production line"},{"startTime":289.54,"endTime":291.31,"body":"where things are moving along a conveyor,"},{"startTime":291.31,"endTime":292.99,"body":"and they're having various"},{"startTime":291.31,"endTime":292.99,"body":"things done to them"},{"startTime":292.99,"endTime":295.94,"body":"at various stages, what you might want"},{"startTime":295.94,"endTime":297.47,"body":"is for it to go very, very slowly"},{"startTime":297.47,"endTime":299.27,"body":"while something is happening,"},{"startTime":299.27,"endTime":301.36,"body":"some bit of the process is being done,"},{"startTime":301.36,"endTime":304.75,"body":"move quickly to the next"},{"startTime":301.36,"endTime":304.75,"body":"place where something happens."},{"startTime":304.75,"endTime":306.73,"body":"So in that case, what"},{"startTime":304.75,"endTime":306.73,"body":"you would like to have"},{"startTime":306.73,"endTime":308.87,"body":"is a variable rate where"},{"startTime":306.73,"endTime":308.87,"body":"it's slow sometimes"},{"startTime":308.87,"endTime":310.63,"body":"and faster other times."},{"startTime":310.63,"endTime":313.56,"body":"Now an elliptical gear"},{"startTime":310.63,"endTime":313.56,"body":"system can do that for you,"},{"startTime":313.56,"endTime":316.22,"body":"and I want to show you why."},{"startTime":316.22,"endTime":317.99,"body":"What we've got here in the picture"},{"startTime":317.99,"endTime":321.545,"body":"is two ellipses that are"},{"startTime":317.99,"endTime":321.545,"body":"the same size and shape."},{"startTime":321.545,"endTime":325.44,"body":"So what that means is we've"},{"startTime":321.545,"endTime":325.44,"body":"got this constant R, right?"},{"startTime":325.44,"endTime":327.0,"body":"For both of them, it's the same,"},{"startTime":327.0,"endTime":329.57,"body":"where points on the"},{"startTime":327.0,"endTime":329.57,"body":"edge of these ellipses,"},{"startTime":329.57,"endTime":331.817,"body":"the sum of the distances to there to foci"},{"startTime":332.94,"endTime":334.28,"body":"is this R, this value R."},{"startTime":334.28,"endTime":336.17,"body":"I've just marked one example of that"},{"startTime":336.17,"endTime":337.58,"body":"on the green ellipse there."},{"startTime":337.58,"endTime":340.47,"body":"So I've got on both of these ellipses,"},{"startTime":340.47,"endTime":342.88,"body":"I've got the foci are marked"},{"startTime":342.88,"endTime":345.05,"body":"slightly differently, and"},{"startTime":342.88,"endTime":345.05,"body":"I'll tell you why in a second."},{"startTime":345.05,"endTime":346.12,"body":"And on that green ellipse there,"},{"startTime":346.12,"endTime":347.52,"body":"there's a point on the"},{"startTime":346.12,"endTime":347.52,"body":"edge of the ellipse,"},{"startTime":347.52,"endTime":348.71,"body":"which I'll just-"},{"startTime":348.71,"endTime":350.54,"body":"For those in the room, I'm showing you,"},{"startTime":350.54,"endTime":352.2,"body":"and for those at home,"},{"startTime":352.2,"endTime":354.986,"body":"it's where the yellow line"},{"startTime":352.2,"endTime":354.986,"body":"meets the dotted line."},{"startTime":354.986,"endTime":355.84,"body":"There's a point on the green ellipse."},{"startTime":355.84,"endTime":357.8,"body":"It looks like it's on"},{"startTime":355.84,"endTime":357.8,"body":"the blue ellipse as well,"},{"startTime":357.8,"endTime":359.88,"body":"but that's not guaranteed yet."},{"startTime":359.88,"endTime":362.4,"body":"I want to explain why"},{"startTime":359.88,"endTime":362.4,"body":"it must always also be"},{"startTime":362.4,"endTime":363.54,"body":"on the blue ellipse."},{"startTime":363.54,"endTime":364.89,"body":"So what is this setup?"},{"startTime":364.89,"endTime":367.87,"body":"I've got my two gear ellipses,"},{"startTime":367.87,"endTime":371.15,"body":"and they have to sort of"},{"startTime":367.87,"endTime":371.15,"body":"turn around something."},{"startTime":371.15,"endTime":375.114,"body":"So the one focus of each"},{"startTime":371.15,"endTime":375.114,"body":"ellipse has an axle through it,"},{"startTime":375.114,"endTime":378.24,"body":"and that's what the"},{"startTime":375.114,"endTime":378.24,"body":"ellipses are turning around."},{"startTime":378.24,"endTime":380.26,"body":"So that's marked with these orange circles"},{"startTime":380.26,"endTime":382.69,"body":"and little red arrows there."},{"startTime":382.69,"endTime":384.7,"body":"And we will make it so that they are"},{"startTime":384.7,"endTime":386.82,"body":"this distance R apart."},{"startTime":386.82,"endTime":389.513,"body":"So that distance between"},{"startTime":386.82,"endTime":389.513,"body":"the two axles is R."},{"startTime":390.71,"endTime":392.31,"body":"Now, if you just had them like that,"},{"startTime":392.31,"endTime":393.86,"body":"just fixed around the axles,"},{"startTime":393.86,"endTime":394.99,"body":"there's no reason for the ellipses"},{"startTime":394.99,"endTime":396.18,"body":"to sort of touch each other."},{"startTime":396.18,"endTime":398.92,"body":"They might just flop downwards"},{"startTime":396.18,"endTime":398.92,"body":"to the bottom and be useless."},{"startTime":398.92,"endTime":400.63,"body":"So we have to do another thing,"},{"startTime":400.63,"endTime":402.28,"body":"which is that we have to have some,"},{"startTime":402.28,"endTime":404.1,"body":"a rigid thing like a rod"},{"startTime":404.1,"endTime":407.62,"body":"joining the other two"},{"startTime":404.1,"endTime":407.62,"body":"foci of those ellipses,"},{"startTime":407.62,"endTime":409.73,"body":"and that's what that"},{"startTime":407.62,"endTime":409.73,"body":"yellow line is signifying"},{"startTime":409.73,"endTime":411.41,"body":"in the diagram there."},{"startTime":411.41,"endTime":414.96,"body":"Okay, and that also is"},{"startTime":411.41,"endTime":414.96,"body":"going to be length R."},{"startTime":414.96,"endTime":416.71,"body":"Okay, so that's the setup."},{"startTime":416.71,"endTime":419.97,"body":"Now we imagine our point"},{"startTime":416.71,"endTime":419.97,"body":"that's on the green ellipse,"},{"startTime":419.97,"endTime":424.06,"body":"the point that's marked"},{"startTime":419.97,"endTime":424.06,"body":"where those lines intersect."},{"startTime":424.06,"endTime":426.9,"body":"Now I've said, let's"},{"startTime":424.06,"endTime":426.9,"body":"just say it's distance A"},{"startTime":426.9,"endTime":428.743,"body":"from one focus and B from the other."},{"startTime":430.47,"endTime":432.38,"body":"We know because it lies on the ellipse"},{"startTime":432.38,"endTime":434.63,"body":"that A plus B must equal R,"},{"startTime":434.63,"endTime":437.76,"body":"the sum of the distances"},{"startTime":434.63,"endTime":437.76,"body":"to the foci is constant R."},{"startTime":437.76,"endTime":440.044,"body":"So that point then, let's think about it"},{"startTime":440.044,"endTime":442.18,"body":"in relation to the blue ellipse."},{"startTime":442.18,"endTime":444.037,"body":"We want to show that it also"},{"startTime":442.18,"endTime":444.037,"body":"must lie on the blue ellipse,"},{"startTime":444.037,"endTime":445.587,"body":"and that will mean that these cogwheels"},{"startTime":445.587,"endTime":448.173,"body":"are always touching all"},{"startTime":445.587,"endTime":448.173,"body":"points in the motion."},{"startTime":449.33,"endTime":451.02,"body":"Well, first of all, let's"},{"startTime":449.33,"endTime":451.02,"body":"look at this dotted line"},{"startTime":451.02,"endTime":454.363,"body":"that joins the two, the two"},{"startTime":451.02,"endTime":454.363,"body":"foci where we have our axles."},{"startTime":455.31,"endTime":457.97,"body":"The bit that's in the green ellipse is B"},{"startTime":457.97,"endTime":460.96,"body":"and we know that total"},{"startTime":457.97,"endTime":460.96,"body":"distance between them is R"},{"startTime":460.96,"endTime":461.87,"body":"so the missing part,"},{"startTime":461.87,"endTime":463.46,"body":"the part that's in the blue ellipse is A,"},{"startTime":463.46,"endTime":465.47,"body":"A plus B is R."},{"startTime":465.47,"endTime":467.36,"body":"And the same thing"},{"startTime":465.47,"endTime":467.36,"body":"between the other two foci"},{"startTime":467.36,"endTime":470.52,"body":"where we've got this rod"},{"startTime":467.36,"endTime":470.52,"body":"joining them together,"},{"startTime":470.52,"endTime":472.54,"body":"the bit that's in the green ellipse is A,"},{"startTime":472.54,"endTime":474.2,"body":"the total distance is R."},{"startTime":474.2,"endTime":478.6,"body":"A plus B is R, so the missing"},{"startTime":474.2,"endTime":478.6,"body":"measurement is B there."},{"startTime":478.6,"endTime":481.29,"body":"So that point that we know"},{"startTime":478.6,"endTime":481.29,"body":"is on the green ellipse,"},{"startTime":481.29,"endTime":484.05,"body":"the sum of its distances from the foci"},{"startTime":484.05,"endTime":487.85,"body":"of the blue ellipse is"},{"startTime":484.05,"endTime":487.85,"body":"also A plus B, which is R."},{"startTime":487.85,"endTime":492.72,"body":"So it must mathematically lie"},{"startTime":487.85,"endTime":492.72,"body":"on the blue ellipse as well."},{"startTime":492.72,"endTime":495.53,"body":"So what this tells you is"},{"startTime":492.72,"endTime":495.53,"body":"that these elliptical gears"},{"startTime":495.53,"endTime":497.48,"body":"are always going to be"},{"startTime":495.53,"endTime":497.48,"body":"touching each other,"},{"startTime":497.48,"endTime":498.74,"body":"and that means that it'll be"},{"startTime":498.74,"endTime":500.66,"body":"a smoothly operating gear system"},{"startTime":500.66,"endTime":503.0,"body":"with never any gaps or jamming"},{"startTime":503.0,"endTime":504.78,"body":"or overlapping or anything horrible."},{"startTime":504.78,"endTime":506.42,"body":"So these will always be touching."},{"startTime":506.42,"endTime":508.08,"body":"And so when you set the blue,"},{"startTime":508.08,"endTime":510.436,"body":"let's say the blue one is the input gear."},{"startTime":510.436,"endTime":512.29,"body":"If you set that one rotating,"},{"startTime":512.29,"endTime":514.16,"body":"it will make the green one rotate"},{"startTime":514.16,"endTime":515.37,"body":"in the other direction, of course,"},{"startTime":515.37,"endTime":517.35,"body":"but it will start moving as well,"},{"startTime":517.35,"endTime":518.844,"body":"but here's the thing."},{"startTime":518.844,"endTime":521.87,"body":"If you turn the blue"},{"startTime":518.844,"endTime":521.87,"body":"one at a constant rate,"},{"startTime":521.87,"endTime":525.08,"body":"the green one will have a"},{"startTime":521.87,"endTime":525.08,"body":"variable rate of motion."},{"startTime":525.08,"endTime":528.61,"body":"And the reason is that"},{"startTime":525.08,"endTime":528.61,"body":"at parts of the motion,"},{"startTime":528.61,"endTime":533.61,"body":"the distance to the edge of"},{"startTime":528.61,"endTime":533.61,"body":"the blue gear wheel is large,"},{"startTime":535.106,"endTime":537.16,"body":"and the distance to the edge"},{"startTime":535.106,"endTime":537.16,"body":"of the green one is small."},{"startTime":537.16,"endTime":539.74,"body":"So it's like a large input gear"},{"startTime":539.74,"endTime":541.7,"body":"leading to a small output gear,"},{"startTime":541.7,"endTime":544.16,"body":"so then the green one"},{"startTime":541.7,"endTime":544.16,"body":"will be moving faster."},{"startTime":544.16,"endTime":545.81,"body":"But then at other points of the motion,"},{"startTime":545.81,"endTime":547.7,"body":"when we rotate a bit further around,"},{"startTime":547.7,"endTime":551.923,"body":"the distance in the blue"},{"startTime":547.7,"endTime":551.923,"body":"one is large, is small,"},{"startTime":553.03,"endTime":555.05,"body":"and the distance to"},{"startTime":553.03,"endTime":555.05,"body":"the green one is large,"},{"startTime":555.05,"endTime":557.36,"body":"and then the green one"},{"startTime":555.05,"endTime":557.36,"body":"will be moving slower."},{"startTime":557.36,"endTime":560.15,"body":"So if you rotate the blue"},{"startTime":557.36,"endTime":560.15,"body":"one at a constant rate,"},{"startTime":560.15,"endTime":563.96,"body":"the output gear will be"},{"startTime":560.15,"endTime":563.96,"body":"rotating at a variable rate."},{"startTime":563.96,"endTime":566.05,"body":"And that's a really, really helpful thing,"},{"startTime":566.05,"endTime":568.39,"body":"as I've said, in some applications."},{"startTime":568.39,"endTime":569.93,"body":"So already, just with the definition,"},{"startTime":569.93,"endTime":572.41,"body":"we are able to think about"},{"startTime":569.93,"endTime":572.41,"body":"these elliptical gears"},{"startTime":573.3,"endTime":575.19,"body":"as an application."},{"startTime":575.19,"endTime":576.76,"body":"To tell you a bit more about applications,"},{"startTime":576.76,"endTime":580.35,"body":"I want to tell you a property of ellipses,"},{"startTime":580.35,"endTime":583.22,"body":"and I'm not going to give you a proof"},{"startTime":583.22,"endTime":585.18,"body":"of what I'm about to tell you."},{"startTime":585.18,"endTime":586.8,"body":"Again, it's not too difficult."},{"startTime":586.8,"endTime":589.46,"body":"I have put the proof in the"},{"startTime":586.8,"endTime":589.46,"body":"transcript if you're interested."},{"startTime":589.46,"endTime":592.14,"body":"Need a little bit of"},{"startTime":589.46,"endTime":592.14,"body":"geometry, a little bit of-"},{"startTime":592.14,"endTime":593.53,"body":"The sort of thing you"},{"startTime":592.14,"endTime":593.53,"body":"might have done in school"},{"startTime":593.53,"endTime":595.82,"body":"with similar triangles"},{"startTime":593.53,"endTime":595.82,"body":"and things like this."},{"startTime":595.82,"endTime":597.93,"body":"But I'll tell you the property."},{"startTime":597.93,"endTime":598.903,"body":"Here's an ellipse."},{"startTime":600.19,"endTime":602.67,"body":"I've drawn a tangent to"},{"startTime":600.19,"endTime":602.67,"body":"the ellipse at a point,"},{"startTime":602.67,"endTime":603.95,"body":"a random point, P."},{"startTime":603.95,"endTime":606.773,"body":"So the tangent is a line"},{"startTime":603.95,"endTime":606.773,"body":"that just touches a curve."},{"startTime":607.61,"endTime":609.37,"body":"So I've drawn a tangent at this point, P,"},{"startTime":609.37,"endTime":612.02,"body":"and then I've drawn the"},{"startTime":609.37,"endTime":612.02,"body":"lines from P to the foci,"},{"startTime":612.02,"endTime":614.15,"body":"to F and to F dash."},{"startTime":614.15,"endTime":617.06,"body":"And the equal angle of property says"},{"startTime":617.06,"endTime":619.854,"body":"that the angle between the tangent"},{"startTime":619.854,"endTime":622.663,"body":"and each of these lines is equal."},{"startTime":624.58,"endTime":627.28,"body":"This is very useful because,"},{"startTime":627.28,"endTime":629.41,"body":"if you remember for the"},{"startTime":627.28,"endTime":629.41,"body":"properties of light,"},{"startTime":629.41,"endTime":632.41,"body":"says that when you have a"},{"startTime":629.41,"endTime":632.41,"body":"light ray and it reflects,"},{"startTime":632.41,"endTime":633.81,"body":"the angle of incidence equals"},{"startTime":633.81,"endTime":634.75,"body":"the angle of reflection rate."},{"startTime":634.75,"endTime":636.68,"body":"You might remember that phrase."},{"startTime":636.68,"endTime":637.53,"body":"Well, what this tells you"},{"startTime":637.53,"endTime":640.13,"body":"is that if you were to"},{"startTime":637.53,"endTime":640.13,"body":"have an elliptical mirror"},{"startTime":640.13,"endTime":642.74,"body":"and you send a beam of"},{"startTime":640.13,"endTime":642.74,"body":"light from one focus,"},{"startTime":642.74,"endTime":644.33,"body":"it will bounce off the mirror"},{"startTime":644.33,"endTime":646.44,"body":"and pass through the other focus."},{"startTime":646.44,"endTime":648.84,"body":"Or, a bit more frivolously,"},{"startTime":646.44,"endTime":648.84,"body":"by the same token,"},{"startTime":648.84,"endTime":650.9,"body":"if you had an elliptical pool table"},{"startTime":650.9,"endTime":653.22,"body":"and you had a pocket at one focus,"},{"startTime":653.22,"endTime":655.03,"body":"and you put the ball at the other focus"},{"startTime":655.03,"endTime":658.79,"body":"and you hit it completely"},{"startTime":655.03,"endTime":658.79,"body":"randomly in any direction at all,"},{"startTime":658.79,"endTime":660.82,"body":"it's guaranteed to bounce off the cushion"},{"startTime":660.82,"endTime":662.32,"body":"and go through the pocket,"},{"startTime":662.32,"endTime":665.34,"body":"so go into the pocket"},{"startTime":662.32,"endTime":665.34,"body":"so you can always win."},{"startTime":665.34,"endTime":668.36,"body":"You can always win at pool"},{"startTime":665.34,"endTime":668.36,"body":"on an elliptical pool table."},{"startTime":668.36,"endTime":670.04,"body":"That isn't my second application."},{"startTime":670.04,"endTime":673.13,"body":"I have seen one of these"},{"startTime":670.04,"endTime":673.13,"body":"things in real life."},{"startTime":673.13,"endTime":675.01,"body":"Someone did decide to make one,"},{"startTime":675.01,"endTime":678.14,"body":"but yet that's not a"},{"startTime":675.01,"endTime":678.14,"body":"very useful application."},{"startTime":678.14,"endTime":680.27,"body":"However, here is a useful application"},{"startTime":680.27,"endTime":682.07,"body":"and it's an application to medicine,"},{"startTime":683.05,"endTime":684.8,"body":"and it's to do with kidney stones."},{"startTime":684.8,"endTime":689.377,"body":"So kidney stones are sort"},{"startTime":684.8,"endTime":689.377,"body":"of hard crystalline growths"},{"startTime":690.65,"endTime":692.77,"body":"that can appear in your kidney,"},{"startTime":692.77,"endTime":694.34,"body":"and they can be very painful"},{"startTime":694.34,"endTime":695.83,"body":"and to get rid of them, you know,"},{"startTime":695.83,"endTime":697.48,"body":"one option might be to do surgery,"},{"startTime":697.48,"endTime":699.07,"body":"but of course that it's invasive,"},{"startTime":699.07,"endTime":701.67,"body":"there are risks, it's not ideal."},{"startTime":701.67,"endTime":704.89,"body":"So there's a treatment for"},{"startTime":701.67,"endTime":704.89,"body":"kidney stones called lithotripsy,"},{"startTime":704.89,"endTime":707.53,"body":"which bombards the kidney stones"},{"startTime":707.53,"endTime":709.27,"body":"with high energy sound waves,"},{"startTime":709.27,"endTime":711.46,"body":"and literally sort of shatters them apart,"},{"startTime":711.46,"endTime":714.143,"body":"breaks them apart, they are shaken apart"},{"startTime":714.143,"endTime":715.78,"body":"into tiny little pieces,"},{"startTime":715.78,"endTime":718.95,"body":"which can then pass"},{"startTime":715.78,"endTime":718.95,"body":"harmlessly out of the body."},{"startTime":718.95,"endTime":720.982,"body":"But of course, you don't want just fire"},{"startTime":720.982,"endTime":724.84,"body":"randomly high energy"},{"startTime":720.982,"endTime":724.84,"body":"sound waves at your body."},{"startTime":724.84,"endTime":727.28,"body":"You know, you don't want to"},{"startTime":724.84,"endTime":727.28,"body":"damage the healthy tissue."},{"startTime":727.28,"endTime":730.25,"body":"So what you'd like to do"},{"startTime":727.28,"endTime":730.25,"body":"is just focus those waves"},{"startTime":730.25,"endTime":734.14,"body":"only on the bit that you want"},{"startTime":730.25,"endTime":734.14,"body":"to break up, the kidney stone."},{"startTime":734.14,"endTime":736.687,"body":"So we use this property of the ellipse,"},{"startTime":736.687,"endTime":738.61,"body":"the equal angle property"},{"startTime":738.61,"endTime":741.17,"body":"in a machine called a lithotripter"},{"startTime":741.17,"endTime":743.25,"body":"And here's the setup."},{"startTime":743.25,"endTime":745.56,"body":"So what you do is you have your emitter"},{"startTime":745.56,"endTime":749.9,"body":"of high energy sound waves"},{"startTime":745.56,"endTime":749.9,"body":"at one focus of an ellipse."},{"startTime":749.9,"endTime":752.893,"body":"And then you have, I mean,"},{"startTime":749.9,"endTime":752.893,"body":"not the whole ellipse,"},{"startTime":752.893,"endTime":753.726,"body":"'cause that would hit your body,"},{"startTime":753.726,"endTime":757.04,"body":"but you have part of the"},{"startTime":753.726,"endTime":757.04,"body":"ellipse as a reflector."},{"startTime":757.04,"endTime":761.27,"body":"And so the emitter beams"},{"startTime":757.04,"endTime":761.27,"body":"out these sound waves,"},{"startTime":761.27,"endTime":763.69,"body":"they reflect off the elliptical reflector,"},{"startTime":763.69,"endTime":766.49,"body":"and they are focused at the"},{"startTime":763.69,"endTime":766.49,"body":"other focus of the ellipse,"},{"startTime":766.49,"endTime":768.15,"body":"which you have carefully ensured"},{"startTime":768.15,"endTime":769.93,"body":"is where the kidney stone is."},{"startTime":769.93,"endTime":773.87,"body":"So it's this non-invasive"},{"startTime":769.93,"endTime":773.87,"body":"treatment for kidney stones,"},{"startTime":773.87,"endTime":776.19,"body":"which uses the mathematical"},{"startTime":773.87,"endTime":776.19,"body":"properties in the ellipse."},{"startTime":776.19,"endTime":779.85,"body":"That's really quite nice."},{"startTime":779.85,"endTime":784.44,"body":"Okay, so there's another way"},{"startTime":779.85,"endTime":784.44,"body":"we can think about ellipses."},{"startTime":784.44,"endTime":785.273,"body":"I've given you the definition."},{"startTime":785.273,"endTime":787.25,"body":"We've got this kind of way of drawing it"},{"startTime":787.25,"endTime":788.67,"body":"with two pins and a piece of paper,"},{"startTime":788.67,"endTime":790.75,"body":"or I've seen people do this with-"},{"startTime":790.75,"endTime":792.05,"body":"You can put pegs and then rope,"},{"startTime":792.05,"endTime":794.23,"body":"and then it'll help you"},{"startTime":792.05,"endTime":794.23,"body":"to make an elliptical lawn"},{"startTime":794.23,"endTime":795.773,"body":"for your garden, very nice."},{"startTime":796.62,"endTime":798.43,"body":"But there's another way that"},{"startTime":796.62,"endTime":798.43,"body":"you can perhaps not draw,"},{"startTime":798.43,"endTime":800.393,"body":"but construct an ellipse,"},{"startTime":801.25,"endTime":803.88,"body":"and to tell you about it,"},{"startTime":803.88,"endTime":805.47,"body":"I'm going to show you."},{"startTime":805.47,"endTime":807.51,"body":"Here's our diagram of the ellipse again,"},{"startTime":807.51,"endTime":809.79,"body":"with a, you know, generic point,"},{"startTime":809.79,"endTime":812.177,"body":"distance to the foci, the"},{"startTime":809.79,"endTime":812.177,"body":"sum of those distances,"},{"startTime":812.177,"endTime":813.64,"body":"there's a constant R."},{"startTime":813.64,"endTime":815.86,"body":"So what I'm going to do is"},{"startTime":813.64,"endTime":815.86,"body":"I'm going to take the line"},{"startTime":815.86,"endTime":818.84,"body":"that goes from one focus,"},{"startTime":815.86,"endTime":818.84,"body":"F dash on the diagram to P,"},{"startTime":818.84,"endTime":820.84,"body":"and I'm just going to extend it out"},{"startTime":820.84,"endTime":822.78,"body":"until we reach a point X"},{"startTime":822.78,"endTime":825.52,"body":"that's the same distance from P"},{"startTime":825.52,"endTime":826.353,"body":"as P is from F."},{"startTime":826.353,"endTime":828.74,"body":"So PX equals PF."},{"startTime":828.74,"endTime":831.22,"body":"That's how I'm going"},{"startTime":828.74,"endTime":831.22,"body":"to create this point X."},{"startTime":831.22,"endTime":833.43,"body":"So what that tells you,"},{"startTime":833.43,"endTime":838.03,"body":"this line we've now got from"},{"startTime":833.43,"endTime":838.03,"body":"the focus F dash up to X,"},{"startTime":838.03,"endTime":840.04,"body":"the total length of that line must be R,"},{"startTime":840.04,"endTime":843.26,"body":"because you've got the"},{"startTime":840.04,"endTime":843.26,"body":"bit from F dash to P"},{"startTime":843.26,"endTime":844.81,"body":"plus the bit from P to X,"},{"startTime":844.81,"endTime":846.82,"body":"but we've constructed that on purpose"},{"startTime":846.82,"endTime":850.31,"body":"so that that length is equal"},{"startTime":846.82,"endTime":850.31,"body":"to the length from P to F."},{"startTime":850.31,"endTime":854.61,"body":"So the total length, the"},{"startTime":850.31,"endTime":854.61,"body":"distance from F dash to X is R,"},{"startTime":854.61,"endTime":857.293,"body":"and that did not depend on"},{"startTime":854.61,"endTime":857.293,"body":"our choice of the point P."},{"startTime":858.47,"endTime":859.95,"body":"We get R every time."},{"startTime":859.95,"endTime":863.87,"body":"So these points X actually lie on a circle"},{"startTime":865.77,"endTime":869.04,"body":"whose center is F dash"},{"startTime":865.77,"endTime":869.04,"body":"and whose radius is R."},{"startTime":870.419,"endTime":872.29,"body":"And you can do this for all"},{"startTime":870.419,"endTime":872.29,"body":"the points P on the ellipse"},{"startTime":872.29,"endTime":873.38,"body":"and get all these points X."},{"startTime":873.38,"endTime":878.38,"body":"So you can make a circle in"},{"startTime":873.38,"endTime":878.38,"body":"theory out of these points X."},{"startTime":879.319,"endTime":882.27,"body":"That gives us a way to"},{"startTime":879.319,"endTime":882.27,"body":"construct an ellipse."},{"startTime":882.27,"endTime":883.32,"body":"It's really quite cute."},{"startTime":883.32,"endTime":884.82,"body":"Here's what you do."},{"startTime":884.82,"endTime":885.85,"body":"Imagine this was just, you know,"},{"startTime":885.85,"endTime":888.04,"body":"we printed this picture"},{"startTime":885.85,"endTime":888.04,"body":"onto a piece of paper,"},{"startTime":888.04,"endTime":891.35,"body":"and you folded, you"},{"startTime":888.04,"endTime":891.35,"body":"physically folded the paper"},{"startTime":891.35,"endTime":894.15,"body":"so that the point X"},{"startTime":891.35,"endTime":894.15,"body":"goes down to the point F"},{"startTime":894.15,"endTime":895.11,"body":"and you make a fold line."},{"startTime":895.11,"endTime":897.0,"body":"I've drawn it dotted on the diagram."},{"startTime":897.0,"endTime":899.52,"body":"Make a nice fold line and unfold, okay?"},{"startTime":899.52,"endTime":901.21,"body":"Now that fold line,"},{"startTime":901.21,"endTime":902.83,"body":"if you think about the"},{"startTime":901.21,"endTime":902.83,"body":"symmetry of the situation,"},{"startTime":902.83,"endTime":906.22,"body":"when you fold things,"},{"startTime":902.83,"endTime":906.22,"body":"it's like a mirror line."},{"startTime":906.22,"endTime":908.64,"body":"So everything on each"},{"startTime":906.22,"endTime":908.64,"body":"side is going to match up,"},{"startTime":908.64,"endTime":909.633,"body":"going to be equal."},{"startTime":910.725,"endTime":913.83,"body":"And so for example, everything"},{"startTime":910.725,"endTime":913.83,"body":"that's on that fold line"},{"startTime":913.83,"endTime":917.3,"body":"must be an equal distance from"},{"startTime":913.83,"endTime":917.3,"body":"the point X and the point F."},{"startTime":917.3,"endTime":919.28,"body":"So in geometry we call that line"},{"startTime":919.28,"endTime":922.6,"body":"the perpendicular bisector of the line XF."},{"startTime":922.6,"endTime":925.88,"body":"It's all the points that are"},{"startTime":922.6,"endTime":925.88,"body":"equidistant from X and F."},{"startTime":925.88,"endTime":928.33,"body":"And we know that P, our"},{"startTime":925.88,"endTime":928.33,"body":"point in the ellipse,"},{"startTime":928.33,"endTime":930.52,"body":"is exactly one of those points, right?"},{"startTime":930.52,"endTime":933.53,"body":"It's the same to distance"},{"startTime":930.52,"endTime":933.53,"body":"from X as it is from F."},{"startTime":933.53,"endTime":937.24,"body":"So what we're going to do is"},{"startTime":933.53,"endTime":937.24,"body":"we're going to get a circle,"},{"startTime":937.24,"endTime":939.81,"body":"get a circular piece of"},{"startTime":937.24,"endTime":939.81,"body":"paper, cut out a circle,"},{"startTime":939.81,"endTime":941.24,"body":"we're going to mark a point,"},{"startTime":941.24,"endTime":942.85,"body":"so that circle, the center of it"},{"startTime":942.85,"endTime":945.83,"body":"is going to be one focus of"},{"startTime":942.85,"endTime":945.83,"body":"our ellipse, that F dash."},{"startTime":945.83,"endTime":947.18,"body":"The radius will be our R"},{"startTime":947.18,"endTime":949.02,"body":"and we're going to need"},{"startTime":947.18,"endTime":949.02,"body":"to have another focus."},{"startTime":949.02,"endTime":952.28,"body":"So we'll mark a little"},{"startTime":949.02,"endTime":952.28,"body":"point on our piece of paper,"},{"startTime":952.28,"endTime":955.7,"body":"and then we're going to take"},{"startTime":952.28,"endTime":955.7,"body":"points on the circumference."},{"startTime":955.7,"endTime":959.6,"body":"I'm going to show you this"},{"startTime":955.7,"endTime":959.6,"body":"now, me starting to do this."},{"startTime":959.6,"endTime":962.23,"body":"So you're going to take"},{"startTime":959.6,"endTime":962.23,"body":"points on the circumference"},{"startTime":962.23,"endTime":964.904,"body":"and we're going to fold them in"},{"startTime":964.904,"endTime":968.31,"body":"so that they just touch that point"},{"startTime":968.31,"endTime":970.29,"body":"that's going to be one of our foci."},{"startTime":970.29,"endTime":971.363,"body":"I called it S."},{"startTime":973.16,"endTime":975.07,"body":"Dunno why, it was F"},{"startTime":973.16,"endTime":975.07,"body":"before, but now it's S."},{"startTime":975.07,"endTime":976.21,"body":"So we're going to do this."},{"startTime":976.21,"endTime":978.64,"body":"Now, I'm not going to do"},{"startTime":976.21,"endTime":978.64,"body":"infinitely many folds,"},{"startTime":978.64,"endTime":980.49,"body":"but I'm just showing you the first two"},{"startTime":980.49,"endTime":981.94,"body":"very slowly so you can see what I'm doing."},{"startTime":981.94,"endTime":986.15,"body":"You fold in, make a nice fold"},{"startTime":981.94,"endTime":986.15,"body":"line, fold back out again."},{"startTime":986.15,"endTime":990.03,"body":"All right, and then if we"},{"startTime":986.15,"endTime":990.03,"body":"can go onto the next video,"},{"startTime":990.03,"endTime":992.11,"body":"I've sped up the process a bit,"},{"startTime":992.11,"endTime":994.533,"body":"so now I'm doing it quickly."},{"startTime":995.73,"endTime":997.09,"body":"Look how quickly my hands are moving."},{"startTime":997.09,"endTime":999.67,"body":"So I'm going all the way"},{"startTime":997.09,"endTime":999.67,"body":"around the circle then."},{"startTime":999.67,"endTime":1002.693,"body":"That, okay, is what you end up with."},{"startTime":1003.57,"endTime":1005.647,"body":"And the person who taught me to do this"},{"startTime":1005.647,"endTime":1007.68,"body":"was a teacher at some master classes"},{"startTime":1007.68,"endTime":1009.89,"body":"I went to when I was a kid"},{"startTime":1007.68,"endTime":1009.89,"body":"and he was called Terry Herd."},{"startTime":1009.89,"endTime":1011.24,"body":"He was a great inspiration."},{"startTime":1011.24,"endTime":1013.89,"body":"And he said, if you this for yourself,"},{"startTime":1013.89,"endTime":1016.3,"body":"it's better to do it when"},{"startTime":1013.89,"endTime":1016.3,"body":"your hands are slightly dirty,"},{"startTime":1016.3,"endTime":1019.62,"body":"because then you sort of"},{"startTime":1016.3,"endTime":1019.62,"body":"leave a better impression"},{"startTime":1019.62,"endTime":1020.74,"body":"of the folds that you're making."},{"startTime":1020.74,"endTime":1022.57,"body":"But I think my hands were"},{"startTime":1020.74,"endTime":1022.57,"body":"maybe a bit too clean,"},{"startTime":1022.57,"endTime":1025.4,"body":"but I hope you can see there"},{"startTime":1022.57,"endTime":1025.4,"body":"that we are creating something"},{"startTime":1025.4,"endTime":1027.89,"body":"that looks like an ellipse"},{"startTime":1025.4,"endTime":1027.89,"body":"there out of these."},{"startTime":1027.89,"endTime":1030.32,"body":"So I think I really love that way"},{"startTime":1030.32,"endTime":1033.293,"body":"of folding an ellipse out of a circle."},{"startTime":1034.95,"endTime":1038.48,"body":"Okay, the reason in my"},{"startTime":1034.95,"endTime":1038.48,"body":"mind when I did this,"},{"startTime":1038.48,"endTime":1040.03,"body":"I called that focus S"},{"startTime":1040.03,"endTime":1042.71,"body":"is because I had in mind"},{"startTime":1040.03,"endTime":1042.71,"body":"the third application"},{"startTime":1042.71,"endTime":1043.84,"body":"that I want to mention,"},{"startTime":1043.84,"endTime":1045.4,"body":"which we probably all know"},{"startTime":1045.4,"endTime":1049.46,"body":"that the planets move in"},{"startTime":1045.4,"endTime":1049.46,"body":"elliptical orbits around the sun."},{"startTime":1049.46,"endTime":1051.38,"body":"And again, that the proof of that,"},{"startTime":1051.38,"endTime":1052.83,"body":"the algebra behind that"},{"startTime":1052.83,"endTime":1054.95,"body":"is a little time"},{"startTime":1052.83,"endTime":1054.95,"body":"consuming for one lecture,"},{"startTime":1054.95,"endTime":1056.32,"body":"so I won't do that for you,"},{"startTime":1056.32,"endTime":1060.61,"body":"but Isaac Newton did"},{"startTime":1056.32,"endTime":1060.61,"body":"it, Isaac Newton did it,"},{"startTime":1060.61,"endTime":1062.16,"body":"and he is commemorated,"},{"startTime":1062.16,"endTime":1064.48,"body":"or was in the olden days"},{"startTime":1064.48,"endTime":1068.223,"body":"before we had pound"},{"startTime":1064.48,"endTime":1068.223,"body":"coins, we had pound notes."},{"startTime":1069.431,"endTime":1072.47,"body":"And it has to say specimen on it,"},{"startTime":1072.47,"endTime":1075.17,"body":"because there are laws about"},{"startTime":1072.47,"endTime":1075.17,"body":"showing pictures of bank notes,"},{"startTime":1075.17,"endTime":1076.93,"body":"even old ones, so that's why,"},{"startTime":1076.93,"endTime":1079.45,"body":"but this is a picture"},{"startTime":1076.93,"endTime":1079.45,"body":"of the old pound note,"},{"startTime":1079.45,"endTime":1083.076,"body":"and it commemorates Isaac"},{"startTime":1079.45,"endTime":1083.076,"body":"Newton's work on gravitation,"},{"startTime":1083.076,"endTime":1087.347,"body":"on the astronomy, and it"},{"startTime":1083.076,"endTime":1087.347,"body":"shows the elliptical orbits"},{"startTime":1087.347,"endTime":1089.42,"body":"some of the elliptical orbits,"},{"startTime":1089.42,"endTime":1092.01,"body":"but there is a slight error here in this."},{"startTime":1092.01,"endTime":1094.77,"body":"I dunno if you are"},{"startTime":1092.01,"endTime":1094.77,"body":"spotting this right now,"},{"startTime":1094.77,"endTime":1099.61,"body":"but the sun is at the center"},{"startTime":1094.77,"endTime":1099.61,"body":"of some of those ellipses,"},{"startTime":1099.61,"endTime":1101.36,"body":"not at a focus."},{"startTime":1101.36,"endTime":1103.199,"body":"That's slightly embarrassing moment"},{"startTime":1103.199,"endTime":1105.28,"body":"for mathematics on bank notes,"},{"startTime":1105.28,"endTime":1107.79,"body":"but still anyway, let's just be pleased"},{"startTime":1107.79,"endTime":1109.851,"body":"that there was a"},{"startTime":1107.79,"endTime":1109.851,"body":"mathematic on a bank note,"},{"startTime":1109.851,"endTime":1111.38,"body":"and that has happened"},{"startTime":1109.851,"endTime":1111.38,"body":"a few times since then."},{"startTime":1111.38,"endTime":1113.51,"body":"Florence Nightingale and"},{"startTime":1111.38,"endTime":1113.51,"body":"Alan Turing are some others,"},{"startTime":1113.51,"endTime":1115.54,"body":"mathematicians on bank notes."},{"startTime":1115.54,"endTime":1119.28,"body":"This is a place where ellipses, of course,"},{"startTime":1119.28,"endTime":1121.17,"body":"are absolutely vital and crucial"},{"startTime":1121.17,"endTime":1123.123,"body":"to our understanding of astronomy."},{"startTime":1124.947,"endTime":1126.53,"body":"You can derive this fact"},{"startTime":1126.53,"endTime":1130.303,"body":"from the inverse square law"},{"startTime":1126.53,"endTime":1130.303,"body":"of gravitational attraction."},{"startTime":1131.24,"endTime":1134.43,"body":"As I say, we won't do that"},{"startTime":1131.24,"endTime":1134.43,"body":"right now, but it can be done."},{"startTime":1134.43,"endTime":1136.865,"body":"But I thought I should at least show you"},{"startTime":1136.865,"endTime":1137.698,"body":"the equation of an ellipse,"},{"startTime":1137.698,"endTime":1140.45,"body":"or at least a form of the"},{"startTime":1137.698,"endTime":1140.45,"body":"equation of an ellipse."},{"startTime":1140.45,"endTime":1143.54,"body":"So if it's been a while since"},{"startTime":1140.45,"endTime":1143.54,"body":"you have done any algebra,"},{"startTime":1143.54,"endTime":1144.78,"body":"you're allowed to let your eyes"},{"startTime":1144.78,"endTime":1146.3,"body":"glaze over for a couple of minutes,"},{"startTime":1146.3,"endTime":1149.21,"body":"and then as long as they"},{"startTime":1146.3,"endTime":1149.21,"body":"unglaze again, we're all good."},{"startTime":1149.21,"endTime":1151.781,"body":"So just wanted to show you,"},{"startTime":1151.781,"endTime":1155.62,"body":"if you take the definition of an ellipse"},{"startTime":1155.62,"endTime":1159.36,"body":"and just, you know,"},{"startTime":1155.62,"endTime":1159.36,"body":"convert that into algebra,"},{"startTime":1159.36,"endTime":1161.21,"body":"so the sum of the distance is,"},{"startTime":1161.21,"endTime":1163.09,"body":"you can work out there's"},{"startTime":1161.21,"endTime":1163.09,"body":"an algebraic expression"},{"startTime":1163.09,"endTime":1165.0,"body":"for the distance between two points,"},{"startTime":1165.0,"endTime":1167.16,"body":"and you want the sum of the two distances"},{"startTime":1167.16,"endTime":1168.3,"body":"to be some constant thing."},{"startTime":1168.3,"endTime":1172.09,"body":"If you sort of put that"},{"startTime":1168.3,"endTime":1172.09,"body":"into a bit of algebra,"},{"startTime":1172.09,"endTime":1173.89,"body":"crank the handle, you can come out"},{"startTime":1173.89,"endTime":1175.55,"body":"and you get an equation"},{"startTime":1173.89,"endTime":1175.55,"body":"that looks a bit like this."},{"startTime":1175.55,"endTime":1178.27,"body":"So it's a quadratic expression,"},{"startTime":1178.27,"endTime":1180.42,"body":"otherwise it's got X"},{"startTime":1178.27,"endTime":1180.42,"body":"squares and Y squares in it."},{"startTime":1180.42,"endTime":1183.02,"body":"I just wanted you to see that."},{"startTime":1183.02,"endTime":1184.56,"body":"And the A and the B,"},{"startTime":1184.56,"endTime":1186.52,"body":"depending on your choice of A and B,"},{"startTime":1186.52,"endTime":1188.56,"body":"you can get different shape ellipses,"},{"startTime":1188.56,"endTime":1191.01,"body":"different shapes and"},{"startTime":1188.56,"endTime":1191.01,"body":"different sizes of ellipses."},{"startTime":1191.91,"endTime":1194.879,"body":"I've drawn in, so I've"},{"startTime":1191.91,"endTime":1194.879,"body":"made the choice here"},{"startTime":1194.879,"endTime":1196.85,"body":"for this equation to have"},{"startTime":1196.85,"endTime":1200.39,"body":"kind of the fattest,"},{"startTime":1196.85,"endTime":1200.39,"body":"widest bit of the ellipse,"},{"startTime":1200.39,"endTime":1202.71,"body":"which we call the major axis."},{"startTime":1202.71,"endTime":1204.73,"body":"That's going to be X axis,"},{"startTime":1204.73,"endTime":1207.02,"body":"and the thinnest bit of the ellipse,"},{"startTime":1207.02,"endTime":1209.2,"body":"the minor axis is the Y axis here."},{"startTime":1209.2,"endTime":1213.189,"body":"So A is at least as big as B right there."},{"startTime":1213.189,"endTime":1214.98,"body":"And I've drawn in the foci."},{"startTime":1214.98,"endTime":1217.21,"body":"I've called them C and minus C,"},{"startTime":1217.21,"endTime":1221.4,"body":"just because I want to talk for a second"},{"startTime":1221.4,"endTime":1224.03,"body":"about, you know, the R"},{"startTime":1221.4,"endTime":1224.03,"body":"that we've talked about,"},{"startTime":1224.03,"endTime":1225.04,"body":"this constant distance"},{"startTime":1225.04,"endTime":1228.48,"body":"which is the sum of the"},{"startTime":1225.04,"endTime":1228.48,"body":"distances to the foci,"},{"startTime":1228.48,"endTime":1230.65,"body":"that's nowhere to be seen on"},{"startTime":1228.48,"endTime":1230.65,"body":"this diagram it looks like."},{"startTime":1230.65,"endTime":1233.26,"body":"So you might be thinking,"},{"startTime":1230.65,"endTime":1233.26,"body":"\"Well, where is it?\""},{"startTime":1233.26,"endTime":1236.11,"body":"Well, it's there to be found."},{"startTime":1236.11,"endTime":1238.4,"body":"If you think about the"},{"startTime":1236.11,"endTime":1238.4,"body":"total width of the ellipse,"},{"startTime":1238.4,"endTime":1241.03,"body":"which be twice A, right?"},{"startTime":1241.03,"endTime":1243.89,"body":"The total width of the"},{"startTime":1241.03,"endTime":1243.89,"body":"ellipse at its widest."},{"startTime":1243.89,"endTime":1247.37,"body":"Now at that point A on the X axis there,"},{"startTime":1247.37,"endTime":1248.77,"body":"that is a point on the ellipse."},{"startTime":1248.77,"endTime":1253.77,"body":"So the sum of its distances"},{"startTime":1248.77,"endTime":1253.77,"body":"to the foci must be our R."},{"startTime":1254.04,"endTime":1255.54,"body":"So we can actually work that out,"},{"startTime":1255.54,"endTime":1257.54,"body":"the distance to the ellipse"},{"startTime":1257.54,"endTime":1262.36,"body":"and to the focus on the"},{"startTime":1257.54,"endTime":1262.36,"body":"far left will be A plus C"},{"startTime":1262.36,"endTime":1266.57,"body":"and the distance to the nearest"},{"startTime":1262.36,"endTime":1266.57,"body":"focus will be A minus C."},{"startTime":1266.57,"endTime":1270.2,"body":"So the sum of the"},{"startTime":1266.57,"endTime":1270.2,"body":"distances will be A plus C"},{"startTime":1270.2,"endTime":1273.75,"body":"added to A minus C, so"},{"startTime":1270.2,"endTime":1273.75,"body":"those C's cancel out"},{"startTime":1273.75,"endTime":1275.65,"body":"and you just get two A."},{"startTime":1275.65,"endTime":1277.38,"body":"So that is our R."},{"startTime":1277.38,"endTime":1281.0,"body":"So we can find that from the graph here."},{"startTime":1281.0,"endTime":1283.263,"body":"Our R is two A."},{"startTime":1284.22,"endTime":1286.85,"body":"Now I've said that by"},{"startTime":1284.22,"endTime":1286.85,"body":"varying the A and the B,"},{"startTime":1286.85,"endTime":1290.87,"body":"you can get different"},{"startTime":1286.85,"endTime":1290.87,"body":"shapes and sizes of ellipse,"},{"startTime":1290.87,"endTime":1292.95,"body":"and I want to explore that a little bit."},{"startTime":1292.95,"endTime":1295.19,"body":"If you think about a circle,"},{"startTime":1295.19,"endTime":1297.43,"body":"all circles are the same shape, right?"},{"startTime":1297.43,"endTime":1298.83,"body":"They can be different sizes,"},{"startTime":1298.83,"endTime":1300.16,"body":"but they're all the same shape."},{"startTime":1300.16,"endTime":1301.658,"body":"We know what a circle is like."},{"startTime":1301.658,"endTime":1303.31,"body":"You can have big circles"},{"startTime":1301.658,"endTime":1303.31,"body":"and small circles."},{"startTime":1303.31,"endTime":1306.38,"body":"But ellipses aren't like that."},{"startTime":1306.38,"endTime":1308.57,"body":"You can shrink and grow them, sure,"},{"startTime":1308.57,"endTime":1310.68,"body":"but also you can change the shape."},{"startTime":1310.68,"endTime":1313.41,"body":"You can have sort of long, slim ones,"},{"startTime":1313.41,"endTime":1315.724,"body":"or you can have slightly fatter ones"},{"startTime":1315.724,"endTime":1317.84,"body":"and you can have ones"},{"startTime":1315.724,"endTime":1317.84,"body":"that look almost circular."},{"startTime":1317.84,"endTime":1319.03,"body":"So there's something more."},{"startTime":1319.03,"endTime":1323.18,"body":"There has to be kind of"},{"startTime":1319.03,"endTime":1323.18,"body":"another parameter, I guess,"},{"startTime":1323.18,"endTime":1324.8,"body":"that we need in order to understand"},{"startTime":1324.8,"endTime":1326.41,"body":"what our ellipse looks like."},{"startTime":1326.41,"endTime":1328.5,"body":"With a circle, we can just get the radius,"},{"startTime":1328.5,"endTime":1332.4,"body":"but with an ellipse, you"},{"startTime":1328.5,"endTime":1332.4,"body":"need to actually specify"},{"startTime":1332.4,"endTime":1333.97,"body":"which ellipse we're talking about."},{"startTime":1333.97,"endTime":1335.82,"body":"We clearly need a little"},{"startTime":1333.97,"endTime":1335.82,"body":"bit more information,"},{"startTime":1335.82,"endTime":1337.08,"body":"some other parameter."},{"startTime":1337.08,"endTime":1339.23,"body":"Now that parameter that we use"},{"startTime":1339.23,"endTime":1342.19,"body":"is known as the"},{"startTime":1339.23,"endTime":1342.19,"body":"eccentricity of an ellipse."},{"startTime":1342.19,"endTime":1344.92,"body":"I want to just show you how varying it-"},{"startTime":1344.92,"endTime":1345.753,"body":"I'll tell you what it is,"},{"startTime":1345.753,"endTime":1347.097,"body":"and then show you how varying"},{"startTime":1348.042,"endTime":1349.84,"body":"it can vary the kind of shape we get."},{"startTime":1349.84,"endTime":1351.07,"body":"So the eccentricity,"},{"startTime":1351.07,"endTime":1354.19,"body":"if you think about what"},{"startTime":1351.07,"endTime":1354.19,"body":"that word might mean,"},{"startTime":1354.19,"endTime":1357.64,"body":"it sort of means a bit how"},{"startTime":1354.19,"endTime":1357.64,"body":"far away from the center,"},{"startTime":1357.64,"endTime":1359.423,"body":"in some sense, that you are."},{"startTime":1360.42,"endTime":1363.07,"body":"And the definition of it is,"},{"startTime":1363.07,"endTime":1365.16,"body":"it's kind of how near the foci are"},{"startTime":1365.16,"endTime":1366.71,"body":"to the edge of the ellipse."},{"startTime":1366.71,"endTime":1368.97,"body":"So, you know, to be exact,"},{"startTime":1368.97,"endTime":1372.04,"body":"it's the distance between the foci"},{"startTime":1372.04,"endTime":1375.46,"body":"as a proportion of the"},{"startTime":1372.04,"endTime":1375.46,"body":"full width of the ellipse."},{"startTime":1375.46,"endTime":1377.5,"body":"So it's the distance between the foci"},{"startTime":1377.5,"endTime":1381.026,"body":"divided by our R, okay?"},{"startTime":1381.026,"endTime":1383.24,"body":"I will show you."},{"startTime":1383.24,"endTime":1388.24,"body":"So here's one where I"},{"startTime":1383.24,"endTime":1388.24,"body":"think I made it 4/5 here,"},{"startTime":1388.732,"endTime":1389.628,"body":"the eccentricity."},{"startTime":1389.628,"endTime":1391.168,"body":"So the distance between the foci"},{"startTime":1391.168,"endTime":1392.533,"body":"is 4/5 of the total width."},{"startTime":1393.89,"endTime":1396.82,"body":"You can see the eccentricity"},{"startTime":1393.89,"endTime":1396.82,"body":"is always less than one,"},{"startTime":1396.82,"endTime":1398.14,"body":"'cause you can never-"},{"startTime":1398.14,"endTime":1400.35,"body":"The foci are always inside the ellipse"},{"startTime":1400.35,"endTime":1401.95,"body":"and they're not ever on the ellipse."},{"startTime":1401.95,"endTime":1404.13,"body":"So the distance between the foci"},{"startTime":1404.13,"endTime":1405.77,"body":"is a proportion of the total width"},{"startTime":1405.77,"endTime":1407.57,"body":"is always less than one."},{"startTime":1407.57,"endTime":1410.32,"body":"As we move those foci"},{"startTime":1407.57,"endTime":1410.32,"body":"closer to the center,"},{"startTime":1410.32,"endTime":1412.49,"body":"so their distance is a smaller proportion,"},{"startTime":1412.49,"endTime":1414.67,"body":"I think this might have been 3/4."},{"startTime":1414.67,"endTime":1418.88,"body":"That's eccentricity of 3/4,"},{"startTime":1414.67,"endTime":1418.88,"body":"looking a bit fatter there."},{"startTime":1418.88,"endTime":1420.79,"body":"This is eccentricity of half,"},{"startTime":1420.79,"endTime":1422.85,"body":"looking really quite circular."},{"startTime":1422.85,"endTime":1425.106,"body":"And then, you know, they can actually-"},{"startTime":1425.106,"endTime":1428.52,"body":"You know, the endpoint of this"},{"startTime":1428.52,"endTime":1430.61,"body":"is where they actually coincide,"},{"startTime":1430.61,"endTime":1432.51,"body":"which is allowed, that can happen."},{"startTime":1432.51,"endTime":1436.46,"body":"If the foci coincide, that"},{"startTime":1432.51,"endTime":1436.46,"body":"means the eccentricity is zero"},{"startTime":1436.46,"endTime":1437.81,"body":"because the distance between them"},{"startTime":1437.81,"endTime":1440.82,"body":"divided by the total width is zero."},{"startTime":1440.82,"endTime":1443.12,"body":"And that is a circle, so"},{"startTime":1440.82,"endTime":1443.12,"body":"we recover a circle there,"},{"startTime":1443.12,"endTime":1445.113,"body":"a special case of an ellipse."},{"startTime":1446.68,"endTime":1449.98,"body":"So I've said the eccentricity"},{"startTime":1446.68,"endTime":1449.98,"body":"is always less than one."},{"startTime":1449.98,"endTime":1452.97,"body":"I want to show you what happens"},{"startTime":1452.97,"endTime":1455.3,"body":"as the eccentricity"},{"startTime":1452.97,"endTime":1455.3,"body":"gets larger and larger."},{"startTime":1455.3,"endTime":1457.94,"body":"And we'd sort of seen"},{"startTime":1455.3,"endTime":1457.94,"body":"a bit of a, you know,"},{"startTime":1457.94,"endTime":1460.27,"body":"perhaps theme ellipses somehow,"},{"startTime":1460.27,"endTime":1463.07,"body":"but a better way or a"},{"startTime":1460.27,"endTime":1463.07,"body":"good way of looking at"},{"startTime":1463.07,"endTime":1465.55,"body":"what happens as that"},{"startTime":1463.07,"endTime":1465.55,"body":"eccentricity gets higher"},{"startTime":1465.55,"endTime":1470.55,"body":"is to go to our kind of circle"},{"startTime":1465.55,"endTime":1470.55,"body":"idea that we had before."},{"startTime":1472.04,"endTime":1474.01,"body":"So remember here, we've got our ellipse"},{"startTime":1474.01,"endTime":1477.41,"body":"and we've extended the line"},{"startTime":1474.01,"endTime":1477.41,"body":"from one focus, F dash,"},{"startTime":1477.41,"endTime":1482.41,"body":"out to a point X, which"},{"startTime":1477.41,"endTime":1482.41,"body":"is so that PX equals PF,"},{"startTime":1482.746,"endTime":1485.65,"body":"and then those points X lie on a circle"},{"startTime":1485.65,"endTime":1487.71,"body":"whose center is the focus F dash"},{"startTime":1487.71,"endTime":1489.68,"body":"and whose radius is R."},{"startTime":1489.68,"endTime":1492.24,"body":"And when we've defined the"},{"startTime":1489.68,"endTime":1492.24,"body":"eccentricity of an ellipse"},{"startTime":1492.24,"endTime":1494.38,"body":"as the distance between the foci,"},{"startTime":1494.38,"endTime":1498.29,"body":"so the distance between"},{"startTime":1494.38,"endTime":1498.29,"body":"the foci divided by our R,"},{"startTime":1498.29,"endTime":1500.083,"body":"which is the radius of this circle."},{"startTime":1501.36,"endTime":1504.51,"body":"So what I'm going to do is"},{"startTime":1501.36,"endTime":1504.51,"body":"show you a series of ellipses"},{"startTime":1504.51,"endTime":1505.82,"body":"where the eccentricity is getting"},{"startTime":1505.82,"endTime":1507.26,"body":"higher and higher and higher"},{"startTime":1507.26,"endTime":1510.62,"body":"and approaching one,"},{"startTime":1507.26,"endTime":1510.62,"body":"never quite reaching it."},{"startTime":1510.62,"endTime":1512.53,"body":"So this is what I'm going to do,"},{"startTime":1512.53,"endTime":1514.28,"body":"the way I'm going to do it."},{"startTime":1514.28,"endTime":1518.31,"body":"So I'm going to get a"},{"startTime":1514.28,"endTime":1518.31,"body":"radius R of this big circle,"},{"startTime":1518.31,"endTime":1522.46,"body":"and I'm going to make the"},{"startTime":1518.31,"endTime":1522.46,"body":"distance between the foci"},{"startTime":1522.46,"endTime":1524.38,"body":"be just one less than that."},{"startTime":1524.38,"endTime":1527.502,"body":"So in other words, that"},{"startTime":1524.38,"endTime":1527.502,"body":"focus F is just distance one"},{"startTime":1527.502,"endTime":1530.5,"body":"from the circumference of the circle."},{"startTime":1530.5,"endTime":1532.75,"body":"And then if we, if we do that,"},{"startTime":1532.75,"endTime":1536.75,"body":"the eccentricity then will"},{"startTime":1532.75,"endTime":1536.75,"body":"be R minus one over R,"},{"startTime":1536.75,"endTime":1539.55,"body":"which if you cancel the R's"},{"startTime":1536.75,"endTime":1539.55,"body":"from the top and the bottom"},{"startTime":1539.55,"endTime":1541.49,"body":"is one minus one over R."},{"startTime":1541.49,"endTime":1543.19,"body":"So as we increase R"},{"startTime":1543.19,"endTime":1545.59,"body":"and R gets closer and closer to infinity,"},{"startTime":1545.59,"endTime":1546.97,"body":"getting larger and larger and larger,"},{"startTime":1546.97,"endTime":1549.56,"body":"this eccentricity will get"},{"startTime":1546.97,"endTime":1549.56,"body":"closer and closer to one"},{"startTime":1549.56,"endTime":1551.91,"body":"and in the limit, so in the limiting case,"},{"startTime":1551.91,"endTime":1553.98,"body":"there won't be an ellipse anymore"},{"startTime":1553.98,"endTime":1557.69,"body":"because the foci cannot"},{"startTime":1553.98,"endTime":1557.69,"body":"lie actually on the circle,"},{"startTime":1557.69,"endTime":1561.003,"body":"but we'll see what we"},{"startTime":1557.69,"endTime":1561.003,"body":"approach in the limit."},{"startTime":1562.09,"endTime":1564.79,"body":"So as R tends to infinity,"},{"startTime":1564.79,"endTime":1568.77,"body":"in the limit, we'll get a"},{"startTime":1564.79,"endTime":1568.77,"body":"circle of infinite radius,"},{"startTime":1568.77,"endTime":1570.84,"body":"which, you know, think what might be,"},{"startTime":1570.84,"endTime":1572.85,"body":"and somehow eccentricity one."},{"startTime":1572.85,"endTime":1575.71,"body":"So that won't be an ellipse"},{"startTime":1572.85,"endTime":1575.71,"body":"anymore, but what will it be?"},{"startTime":1575.71,"endTime":1576.6,"body":"Well, let's have a look."},{"startTime":1576.6,"endTime":1578.12,"body":"So I've drawn a few of these things"},{"startTime":1578.12,"endTime":1580.41,"body":"in pretty rainbow colors"},{"startTime":1578.12,"endTime":1580.41,"body":"'cause I like them,"},{"startTime":1580.41,"endTime":1582.97,"body":"and the big white blob is that focus F,"},{"startTime":1582.97,"endTime":1584.47,"body":"the one that isn't the"},{"startTime":1582.97,"endTime":1584.47,"body":"center of the circle,"},{"startTime":1584.47,"endTime":1586.91,"body":"the one that is the other"},{"startTime":1584.47,"endTime":1586.91,"body":"one, the left-hand side."},{"startTime":1586.91,"endTime":1588.36,"body":"And you can see, we start off"},{"startTime":1589.51,"endTime":1591.81,"body":"with something that's a circle"},{"startTime":1591.81,"endTime":1593.63,"body":"when R equals two, I guess,"},{"startTime":1593.63,"endTime":1595.57,"body":"and then we just keep going and going,"},{"startTime":1595.57,"endTime":1597.12,"body":"and in the picture, you know,"},{"startTime":1597.12,"endTime":1598.9,"body":"the right-hand edge of the ellipse"},{"startTime":1598.9,"endTime":1601.18,"body":"vanishes off the edge of the screen"},{"startTime":1601.18,"endTime":1603.577,"body":"as we increase our value of R"},{"startTime":1603.577,"endTime":1605.69,"body":"and increase the eccentricity."},{"startTime":1605.69,"endTime":1608.92,"body":"But there is it seems to be"},{"startTime":1605.69,"endTime":1608.92,"body":"a sort of limiting curve,"},{"startTime":1608.92,"endTime":1612.86,"body":"which you can see, we're"},{"startTime":1608.92,"endTime":1612.86,"body":"going up incrementally,"},{"startTime":1612.86,"endTime":1615.75,"body":"but by the time we get"},{"startTime":1612.86,"endTime":1615.75,"body":"to this purple curve,"},{"startTime":1615.75,"endTime":1618.79,"body":"which is the kind of"},{"startTime":1615.75,"endTime":1618.79,"body":"the extreme more case,"},{"startTime":1618.79,"endTime":1621.08,"body":"it's starting to look like a curve"},{"startTime":1621.08,"endTime":1623.11,"body":"which we have already seen in this talk,"},{"startTime":1623.11,"endTime":1625.003,"body":"and that was the parabola,"},{"startTime":1625.97,"endTime":1629.4,"body":"and that is what the limiting case is."},{"startTime":1629.4,"endTime":1632.21,"body":"So we have these ellipses"},{"startTime":1629.4,"endTime":1632.21,"body":"and in the limit,"},{"startTime":1632.21,"endTime":1634.34,"body":"I mean, you can sort"},{"startTime":1632.21,"endTime":1634.34,"body":"of think of a parabola"},{"startTime":1634.34,"endTime":1636.75,"body":"as an ellipse with eccentricity one."},{"startTime":1636.75,"endTime":1638.45,"body":"It isn't an ellipse, but that's the limit"},{"startTime":1638.45,"endTime":1641.443,"body":"and you can perhaps see"},{"startTime":1638.45,"endTime":1641.443,"body":"why we get this happening."},{"startTime":1642.384,"endTime":1646.043,"body":"So just to remind us how"},{"startTime":1642.384,"endTime":1646.043,"body":"we got to this point,"},{"startTime":1647.0,"endTime":1649.83,"body":"we are saying that in this limiting case,"},{"startTime":1649.83,"endTime":1651.89,"body":"we actually are looking"},{"startTime":1649.83,"endTime":1651.89,"body":"at that green circle"},{"startTime":1651.89,"endTime":1653.93,"body":"as a circle of infinite radius."},{"startTime":1653.93,"endTime":1656.27,"body":"And, you know, you can't have"},{"startTime":1653.93,"endTime":1656.27,"body":"a circle of infinite radius,"},{"startTime":1656.27,"endTime":1658.547,"body":"but if you could, it"},{"startTime":1656.27,"endTime":1658.547,"body":"would be a straight line."},{"startTime":1658.547,"endTime":1661.1,"body":"The curve is getting less"},{"startTime":1658.547,"endTime":1661.1,"body":"and less and less and less,"},{"startTime":1661.1,"endTime":1663.87,"body":"and so you end up in the"},{"startTime":1661.1,"endTime":1663.87,"body":"limit with a straight line."},{"startTime":1663.87,"endTime":1666.43,"body":"But you still have that property"},{"startTime":1666.43,"endTime":1669.5,"body":"of this PX equals PF."},{"startTime":1669.5,"endTime":1670.57,"body":"You've kept that all along"},{"startTime":1670.57,"endTime":1672.39,"body":"and you still have that in the limit,"},{"startTime":1672.39,"endTime":1677.003,"body":"and that is the way we define a parabola."},{"startTime":1677.003,"endTime":1678.2,"body":"There's a straight line,"},{"startTime":1678.2,"endTime":1681.166,"body":"we call it the directrix"},{"startTime":1678.2,"endTime":1681.166,"body":"and there's a focus,"},{"startTime":1681.166,"endTime":1682.28,"body":"there's just one focus"},{"startTime":1681.166,"endTime":1682.28,"body":"'cause the other one"},{"startTime":1682.28,"endTime":1685.27,"body":"zoomed off to infinity and the distance,"},{"startTime":1685.27,"endTime":1686.66,"body":"the parabola is the center point"},{"startTime":1686.66,"endTime":1688.24,"body":"whose distance from that straight line"},{"startTime":1688.24,"endTime":1691.33,"body":"is equal to the distance from the focus."},{"startTime":1691.33,"endTime":1694.313,"body":"So that's the definition of a parabola."},{"startTime":1696.118,"endTime":1699.24,"body":"I'm going to show you how"},{"startTime":1696.118,"endTime":1699.24,"body":"we can fold a parabola now."},{"startTime":1699.24,"endTime":1702.33,"body":"This time we're folding a"},{"startTime":1699.24,"endTime":1702.33,"body":"straight line, not a circle."},{"startTime":1702.33,"endTime":1704.62,"body":"So we get just a normal piece of paper"},{"startTime":1704.62,"endTime":1708.02,"body":"and we fold from the edge"},{"startTime":1704.62,"endTime":1708.02,"body":"and you'll see it happen."},{"startTime":1708.02,"endTime":1710.88,"body":"We just fold points from"},{"startTime":1708.02,"endTime":1710.88,"body":"the edge to the center,"},{"startTime":1710.88,"endTime":1711.713,"body":"not to the center,"},{"startTime":1711.713,"endTime":1714.62,"body":"to the focus that I've"},{"startTime":1711.713,"endTime":1714.62,"body":"just drawn on there."},{"startTime":1714.62,"endTime":1716.85,"body":"I think we might have had a"},{"startTime":1714.62,"endTime":1716.85,"body":"sneak peek of this one earlier."},{"startTime":1716.85,"endTime":1717.683,"body":"There it is."},{"startTime":1719.23,"endTime":1721.933,"body":"So you can see that's,"},{"startTime":1723.18,"endTime":1724.49,"body":"that shape is a parabola"},{"startTime":1724.49,"endTime":1726.09,"body":"and we've made it by folding this time"},{"startTime":1726.09,"endTime":1729.093,"body":"from the straight line"},{"startTime":1726.09,"endTime":1729.093,"body":"that's the edge of our paper."},{"startTime":1731.0,"endTime":1733.583,"body":"So that's kind of a cool"},{"startTime":1731.0,"endTime":1733.583,"body":"way to make a parabola."},{"startTime":1734.93,"endTime":1735.93,"body":"I won't go through all again."},{"startTime":1735.93,"endTime":1738.4,"body":"I won't go through some algebra,"},{"startTime":1738.4,"endTime":1739.84,"body":"but you can do some algebra here."},{"startTime":1739.84,"endTime":1742.59,"body":"Again, it's just taking a point"},{"startTime":1742.59,"endTime":1745.15,"body":"and using the fact that its"},{"startTime":1742.59,"endTime":1745.15,"body":"distance from a straight line"},{"startTime":1745.15,"endTime":1748.0,"body":"is equal to its distance from"},{"startTime":1745.15,"endTime":1748.0,"body":"a particular given point."},{"startTime":1748.0,"endTime":1751.62,"body":"I have chosen to put my axes in such a way"},{"startTime":1751.62,"endTime":1755.653,"body":"that the focus lies on"},{"startTime":1751.62,"endTime":1755.653,"body":"the X axis, it's point A,"},{"startTime":1756.749,"endTime":1761.37,"body":"and the directrix is on the"},{"startTime":1756.749,"endTime":1761.37,"body":"other side of the Y axis"},{"startTime":1761.37,"endTime":1764.69,"body":"and halfway between, right, is the Y axis."},{"startTime":1764.69,"endTime":1768.14,"body":"So the directrix is the"},{"startTime":1764.69,"endTime":1768.14,"body":"line X equals minus A"},{"startTime":1768.14,"endTime":1769.05,"body":"and the focus is A."},{"startTime":1769.05,"endTime":1771.7,"body":"If you do that, you get the equation"},{"startTime":1771.7,"endTime":1773.57,"body":"Y squared equals four AX,"},{"startTime":1773.57,"endTime":1778.023,"body":"and by varying A, you can"},{"startTime":1773.57,"endTime":1778.023,"body":"shrink or grow the parabola."},{"startTime":1779.086,"endTime":1782.73,"body":"Now you may have drawn a"},{"startTime":1779.086,"endTime":1782.73,"body":"parabola yourself at school."},{"startTime":1782.73,"endTime":1785.45,"body":"Perhaps if you ever drew"},{"startTime":1782.73,"endTime":1785.45,"body":"graphs and you plotted points,"},{"startTime":1785.45,"endTime":1788.67,"body":"then probably the first"},{"startTime":1785.45,"endTime":1788.67,"body":"curve you drew using that"},{"startTime":1788.67,"endTime":1791.063,"body":"was Y equals X squared."},{"startTime":1791.063,"endTime":1793.72,"body":"That is also a parabola, but it's just,"},{"startTime":1793.72,"endTime":1794.553,"body":"it's flipped over,"},{"startTime":1794.553,"endTime":1797.463,"body":"so we've interchanged"},{"startTime":1794.553,"endTime":1797.463,"body":"the roles of X and Y,"},{"startTime":1798.333,"endTime":1801.26,"body":"and in that special case,"},{"startTime":1798.333,"endTime":1801.26,"body":"you know, A is a quarter."},{"startTime":1801.26,"endTime":1804.31,"body":"But it's exactly the"},{"startTime":1801.26,"endTime":1804.31,"body":"same reasoning, really,"},{"startTime":1804.31,"endTime":1807.3,"body":"and so that Y equals X"},{"startTime":1804.31,"endTime":1807.3,"body":"squared, that is a parabola,"},{"startTime":1807.3,"endTime":1808.851,"body":"and this is kind of why."},{"startTime":1808.851,"endTime":1811.51,"body":"If you want to try and do"},{"startTime":1808.851,"endTime":1811.51,"body":"those few lines of algebra,"},{"startTime":1811.51,"endTime":1814.52,"body":"it's a fun exercise to do."},{"startTime":1814.52,"endTime":1817.61,"body":"Now, parabolas have an"},{"startTime":1814.52,"endTime":1817.61,"body":"equal angle property"},{"startTime":1817.61,"endTime":1818.78,"body":"just like ellipses,"},{"startTime":1818.78,"endTime":1820.16,"body":"and it's the one we might expect"},{"startTime":1820.16,"endTime":1822.35,"body":"because we've done this"},{"startTime":1820.16,"endTime":1822.35,"body":"sort of limiting process."},{"startTime":1822.35,"endTime":1823.68,"body":"Remember that somehow the parabola"},{"startTime":1823.68,"endTime":1826.29,"body":"has this focus way over infinity."},{"startTime":1826.29,"endTime":1830.831,"body":"So rays of light coming"},{"startTime":1826.29,"endTime":1830.831,"body":"from the other focus,"},{"startTime":1830.831,"endTime":1831.85,"body":"which, I mean, isn't really"},{"startTime":1830.831,"endTime":1831.85,"body":"there, it's infinity,"},{"startTime":1831.85,"endTime":1833.1,"body":"will be coming in and they'll be"},{"startTime":1833.1,"endTime":1835.3,"body":"kind of parallel lines like this,"},{"startTime":1835.3,"endTime":1837.97,"body":"and they will be"},{"startTime":1835.3,"endTime":1837.97,"body":"reflected off the parabola"},{"startTime":1837.97,"endTime":1842.07,"body":"and pass through its focus there."},{"startTime":1842.07,"endTime":1846.72,"body":"So this gives us an"},{"startTime":1842.07,"endTime":1846.72,"body":"application of parabolas,"},{"startTime":1846.72,"endTime":1849.12,"body":"or I guess usually it's"},{"startTime":1846.72,"endTime":1849.12,"body":"paraboloids, right?"},{"startTime":1849.12,"endTime":1852.0,"body":"Which let me just show you the picture."},{"startTime":1852.0,"endTime":1855.33,"body":"You take a parabola and you"},{"startTime":1852.0,"endTime":1855.33,"body":"just rotate it around its axis"},{"startTime":1855.33,"endTime":1857.21,"body":"so you get this surface."},{"startTime":1857.21,"endTime":1860.61,"body":"Light comes in from a distant"},{"startTime":1857.21,"endTime":1860.61,"body":"star, say for a telescope,"},{"startTime":1860.61,"endTime":1862.48,"body":"so far away that we can essentially assume"},{"startTime":1862.48,"endTime":1865.15,"body":"it's coming in from infinity"},{"startTime":1862.48,"endTime":1865.15,"body":"and the rays are parallel,"},{"startTime":1865.15,"endTime":1867.92,"body":"and it bounces off the parabolic reflector"},{"startTime":1867.92,"endTime":1869.52,"body":"and passes through the focus."},{"startTime":1869.52,"endTime":1873.8,"body":"So that focuses a line gives"},{"startTime":1869.52,"endTime":1873.8,"body":"us a really good, clear image,"},{"startTime":1873.8,"endTime":1876.173,"body":"same technology in"},{"startTime":1873.8,"endTime":1876.173,"body":"satellite dishes as well."},{"startTime":1877.6,"endTime":1879.573,"body":"So that's the first application."},{"startTime":1879.573,"endTime":1880.51,"body":"I promised you three"},{"startTime":1879.573,"endTime":1880.51,"body":"applications of each thing."},{"startTime":1880.51,"endTime":1882.15,"body":"So what might the next one be?"},{"startTime":1882.15,"endTime":1886.098,"body":"Well, is it the Archimedes death ray?"},{"startTime":1886.098,"endTime":1888.671,"body":"I apologize, I was being very frivolous."},{"startTime":1888.671,"endTime":1891.51,"body":"Archimedes did not fire"},{"startTime":1888.671,"endTime":1891.51,"body":"laser beams out of his eyes,"},{"startTime":1891.51,"endTime":1893.11,"body":"let's just get that clear."},{"startTime":1893.11,"endTime":1896.19,"body":"But there's a story about Archimedes,"},{"startTime":1896.19,"endTime":1898.563,"body":"a great mathematician and scientist."},{"startTime":1899.4,"endTime":1902.08,"body":"He was from Syracuse and"},{"startTime":1899.4,"endTime":1902.08,"body":"Syracuse was under threat"},{"startTime":1902.08,"endTime":1904.15,"body":"of invasion by the Romans."},{"startTime":1904.15,"endTime":1905.81,"body":"And story goes that one night,"},{"startTime":1905.81,"endTime":1909.17,"body":"the Roman fleet sailed"},{"startTime":1905.81,"endTime":1909.17,"body":"into the bay at Syracuse"},{"startTime":1909.17,"endTime":1911.54,"body":"and you know, \"Oh no,"},{"startTime":1909.17,"endTime":1911.54,"body":"what are we going to do?"},{"startTime":1911.54,"endTime":1914.53,"body":"Let's fetch our brainiest person.\""},{"startTime":1914.53,"endTime":1917.4,"body":"And what Archimedes did,"},{"startTime":1914.53,"endTime":1917.4,"body":"he had a solution to this,"},{"startTime":1917.4,"endTime":1919.94,"body":"a very powerful weapon."},{"startTime":1919.94,"endTime":1922.1,"body":"He got parabolic reflectors"},{"startTime":1922.1,"endTime":1924.16,"body":"and reflected the rays of the sun"},{"startTime":1924.16,"endTime":1927.3,"body":"so that they focused onto the ships"},{"startTime":1927.3,"endTime":1931.633,"body":"and the ships burst into flames"},{"startTime":1927.3,"endTime":1931.633,"body":"and problem solved, right?"},{"startTime":1932.55,"endTime":1937.24,"body":"Now, one or two teeny tiny"},{"startTime":1932.55,"endTime":1937.24,"body":"weeny issues with that."},{"startTime":1937.24,"endTime":1939.805,"body":"Number one, the technology at the time"},{"startTime":1939.805,"endTime":1942.93,"body":"was not absolutely amazing at making"},{"startTime":1942.93,"endTime":1945.72,"body":"incredibly reflective paraboloid shapes."},{"startTime":1945.72,"endTime":1947.49,"body":"That's really quite hard to do."},{"startTime":1947.49,"endTime":1950.344,"body":"It's not at all clear that"},{"startTime":1947.49,"endTime":1950.344,"body":"it would've been possible"},{"startTime":1950.344,"endTime":1953.02,"body":"to do it in such a way that"},{"startTime":1950.344,"endTime":1953.02,"body":"you could generate enough heat"},{"startTime":1953.02,"endTime":1955.35,"body":"to set things on fire at a distance."},{"startTime":1955.35,"endTime":1956.183,"body":"That's number one."},{"startTime":1956.183,"endTime":1959.04,"body":"Number two, contemporary accounts"},{"startTime":1959.04,"endTime":1961.57,"body":"did not bother to mention"},{"startTime":1959.04,"endTime":1961.57,"body":"this amazing thing"},{"startTime":1961.57,"endTime":1963.01,"body":"that would've been absolutely astonishing."},{"startTime":1963.01,"endTime":1966.46,"body":"The first account that"},{"startTime":1963.01,"endTime":1966.46,"body":"mentions it is 350 years later."},{"startTime":1966.46,"endTime":1967.87,"body":"Galen writes about it."},{"startTime":1967.87,"endTime":1969.743,"body":"So that's a bit suspicious."},{"startTime":1971.13,"endTime":1973.6,"body":"Number three, if this was so fantastic,"},{"startTime":1973.6,"endTime":1976.98,"body":"why has no one else ever done"},{"startTime":1973.6,"endTime":1976.98,"body":"it in the history of warfare?"},{"startTime":1976.98,"endTime":1978.2,"body":"That's number three."},{"startTime":1978.2,"endTime":1980.88,"body":"And finally, how did my story begin?"},{"startTime":1980.88,"endTime":1981.833,"body":"One night."},{"startTime":1983.126,"endTime":1984.28,"body":"(crowd laughing)"},{"startTime":1984.28,"endTime":1985.728,"body":"There you go."},{"startTime":1985.728,"endTime":1986.561,"body":"The clue was in the first word."},{"startTime":1986.561,"endTime":1989.67,"body":"So this is not the second"},{"startTime":1986.561,"endTime":1989.67,"body":"application of parabolas"},{"startTime":1989.67,"endTime":1990.503,"body":"as far as we know."},{"startTime":1990.503,"endTime":1992.54,"body":"However, there is another application"},{"startTime":1992.54,"endTime":1995.35,"body":"and it's done by reversing"},{"startTime":1992.54,"endTime":1995.35,"body":"the process essentially"},{"startTime":1995.35,"endTime":1997.55,"body":"that we have talked about."},{"startTime":1997.55,"endTime":1999.65,"body":"If instead of light coming"},{"startTime":1999.65,"endTime":2001.18,"body":"into the parabola and being focused,"},{"startTime":2001.18,"endTime":2004.76,"body":"if you have a light source,"},{"startTime":2001.18,"endTime":2004.76,"body":"a bulb at the focus,"},{"startTime":2004.76,"endTime":2007.54,"body":"then rays from it will come"},{"startTime":2004.76,"endTime":2007.54,"body":"out and they'll be reflected"},{"startTime":2007.54,"endTime":2008.91,"body":"into a parallel beam of light,"},{"startTime":2008.91,"endTime":2010.85,"body":"a nice strong beam of light."},{"startTime":2010.85,"endTime":2012.59,"body":"So you see this in torches,"},{"startTime":2012.59,"endTime":2016.55,"body":"you see it in car headlamps"},{"startTime":2012.59,"endTime":2016.55,"body":"or motorbike headlamps."},{"startTime":2016.55,"endTime":2018.62,"body":"If you've got a really old vintage car,"},{"startTime":2018.62,"endTime":2021.53,"body":"sometimes the headlamps"},{"startTime":2018.62,"endTime":2021.53,"body":"are actually external"},{"startTime":2021.53,"endTime":2023.05,"body":"to the body work of the car"},{"startTime":2023.05,"endTime":2025.83,"body":"and you can actually really"},{"startTime":2023.05,"endTime":2025.83,"body":"see that nice parabola shape."},{"startTime":2025.83,"endTime":2028.293,"body":"So always good to go"},{"startTime":2025.83,"endTime":2028.293,"body":"out parabola spotting."},{"startTime":2029.31,"endTime":2031.206,"body":"So that's the second application."},{"startTime":2031.206,"endTime":2036.02,"body":"The third one really as"},{"startTime":2031.206,"endTime":2036.02,"body":"it relates to what we said"},{"startTime":2036.02,"endTime":2039.096,"body":"about gravity causing the"},{"startTime":2036.02,"endTime":2039.096,"body":"planets to orbit the sun"},{"startTime":2039.096,"endTime":2041.53,"body":"with ecliptic orbits."},{"startTime":2041.53,"endTime":2044.61,"body":"Now, if you have a situation where"},{"startTime":2044.61,"endTime":2046.85,"body":"the two bodies that you're studying"},{"startTime":2046.85,"endTime":2049.22,"body":"are vastly different sizes,"},{"startTime":2049.22,"endTime":2053.24,"body":"for example, a whole"},{"startTime":2049.22,"endTime":2053.24,"body":"planet Earth and a petal,"},{"startTime":2053.24,"endTime":2056.34,"body":"then actually, although the motion"},{"startTime":2056.34,"endTime":2059.015,"body":"is strictly speaking, still in ellipse,"},{"startTime":2059.015,"endTime":2061.41,"body":"it's sufficiently an extreme one"},{"startTime":2061.41,"endTime":2065.47,"body":"that a parabola is a very,"},{"startTime":2061.41,"endTime":2065.47,"body":"very good approximation to it."},{"startTime":2065.47,"endTime":2069.65,"body":"And that's what we tend to"},{"startTime":2065.47,"endTime":2069.65,"body":"use as our approximation"},{"startTime":2069.65,"endTime":2071.69,"body":"when we are studying, you know,"},{"startTime":2071.69,"endTime":2075.51,"body":"I could talk about"},{"startTime":2071.69,"endTime":2075.51,"body":"ballistics or cannon balls"},{"startTime":2075.51,"endTime":2077.25,"body":"and warfare and things,"},{"startTime":2075.51,"endTime":2077.25,"body":"which I don't like that,"},{"startTime":2077.25,"endTime":2078.63,"body":"so instead, I'm going to use the example"},{"startTime":2078.63,"endTime":2080.183,"body":"of lovely fountains,"},{"startTime":2081.018,"endTime":2083.84,"body":"where you have water coming out"},{"startTime":2083.84,"endTime":2087.1,"body":"and it's fired out of a"},{"startTime":2083.84,"endTime":2087.1,"body":"pipe or something like this,"},{"startTime":2087.1,"endTime":2089.65,"body":"and it follows a parabolic trajectory"},{"startTime":2089.65,"endTime":2092.51,"body":"to an extremely good"},{"startTime":2089.65,"endTime":2092.51,"body":"level of approximation."},{"startTime":2092.51,"endTime":2095.949,"body":"And you can see some of"},{"startTime":2092.51,"endTime":2095.949,"body":"these beautiful things."},{"startTime":2095.949,"endTime":2098.64,"body":"You can see them in gardens"},{"startTime":2095.949,"endTime":2098.64,"body":"or all over the place"},{"startTime":2098.64,"endTime":2100.797,"body":"and lovely parabolas there to be seen."},{"startTime":2100.797,"endTime":2102.2,"body":"And if you had a waterfall,"},{"startTime":2102.2,"endTime":2104.69,"body":"you probably get kind of"},{"startTime":2102.2,"endTime":2104.69,"body":"half a parabola, you know?"},{"startTime":2104.69,"endTime":2106.16,"body":"So you can see those,"},{"startTime":2106.16,"endTime":2108.48,"body":"and I mean, they're very, very beautiful,"},{"startTime":2108.48,"endTime":2110.01,"body":"but if you know the mathematics,"},{"startTime":2110.01,"endTime":2111.19,"body":"you can go a bit further"},{"startTime":2111.19,"endTime":2113.59,"body":"and you can know exactly where"},{"startTime":2113.59,"endTime":2115.5,"body":"the other end of the parabola will be,"},{"startTime":2115.5,"endTime":2117.23,"body":"and that means, I dunno"},{"startTime":2115.5,"endTime":2117.23,"body":"if you've seen these,"},{"startTime":2117.23,"endTime":2119.31,"body":"but you can sometimes create these effects"},{"startTime":2119.31,"endTime":2121.79,"body":"where you have a series of kind of holes"},{"startTime":2121.79,"endTime":2124.88,"body":"through which water is"},{"startTime":2121.79,"endTime":2124.88,"body":"pumped out in little jets"},{"startTime":2124.88,"endTime":2127.39,"body":"and it'll go up and it'll"},{"startTime":2124.88,"endTime":2127.39,"body":"land on the next one,"},{"startTime":2127.39,"endTime":2128.993,"body":"and then a jet will come out of that one,"},{"startTime":2128.993,"endTime":2130.39,"body":"a little pulse of water,"},{"startTime":2130.39,"endTime":2131.4,"body":"so it looks like the water's"},{"startTime":2131.4,"endTime":2132.71,"body":"kind of bouncing along the ground,"},{"startTime":2132.71,"endTime":2135.07,"body":"and it's a very, very beautiful effect."},{"startTime":2135.07,"endTime":2138.1,"body":"So this is an application"},{"startTime":2135.07,"endTime":2138.1,"body":"then of parabolas"},{"startTime":2138.1,"endTime":2141.25,"body":"that these trajectories of"},{"startTime":2138.1,"endTime":2141.25,"body":"essentially very small things"},{"startTime":2141.25,"endTime":2142.79,"body":"in relation to planet Earth,"},{"startTime":2142.79,"endTime":2147.53,"body":"to a good approximation"},{"startTime":2142.79,"endTime":2147.53,"body":"follow a parabolic trajectory."},{"startTime":2147.53,"endTime":2151.09,"body":"So we talked about two of"},{"startTime":2147.53,"endTime":2151.09,"body":"the three conic sections"},{"startTime":2151.09,"endTime":2155.35,"body":"and the final one that's"},{"startTime":2151.09,"endTime":2155.35,"body":"left is the hyperbola."},{"startTime":2155.35,"endTime":2157.93,"body":"And hyperbolas, if you remember,"},{"startTime":2157.93,"endTime":2160.397,"body":"we're imagining our cone"},{"startTime":2157.93,"endTime":2160.397,"body":"and we're slicing through it"},{"startTime":2160.397,"endTime":2162.89,"body":"and sort of shallow"},{"startTime":2160.397,"endTime":2162.89,"body":"angles, we got ellipses."},{"startTime":2162.89,"endTime":2164.84,"body":"Then at this critical"},{"startTime":2162.89,"endTime":2164.84,"body":"point where the slope"},{"startTime":2164.84,"endTime":2167.96,"body":"was equal to the slope of"},{"startTime":2164.84,"endTime":2167.96,"body":"the cone, we got parabolas,"},{"startTime":2167.96,"endTime":2169.8,"body":"and if you then go a little bit further,"},{"startTime":2169.8,"endTime":2171.25,"body":"we cut both halves of the cone"},{"startTime":2171.25,"endTime":2173.273,"body":"and you start to get hyperbolas."},{"startTime":2174.647,"endTime":2176.47,"body":"And so here's a picture of this."},{"startTime":2176.47,"endTime":2179.13,"body":"And on the right you see"},{"startTime":2179.13,"endTime":2181.58,"body":"somewhere where we see"},{"startTime":2179.13,"endTime":2181.58,"body":"hyperbolas in the home."},{"startTime":2181.58,"endTime":2186.06,"body":"I'm all in favor of spotting"},{"startTime":2181.58,"endTime":2186.06,"body":"things in everyday life."},{"startTime":2186.06,"endTime":2187.51,"body":"What you've got on the right there"},{"startTime":2187.51,"endTime":2191.72,"body":"is a perfectly normal lampshade,"},{"startTime":2187.51,"endTime":2191.72,"body":"a cylindrical lampshade."},{"startTime":2191.72,"endTime":2193.83,"body":"We think what's happening"},{"startTime":2191.72,"endTime":2193.83,"body":"here with the light,"},{"startTime":2193.83,"endTime":2195.08,"body":"you've got a bulb in the middle,"},{"startTime":2195.08,"endTime":2196.96,"body":"and of course, the"},{"startTime":2195.08,"endTime":2196.96,"body":"light's going everywhere,"},{"startTime":2196.96,"endTime":2198.48,"body":"but the bit that can actually escape"},{"startTime":2198.48,"endTime":2200.73,"body":"from the top and the"},{"startTime":2198.48,"endTime":2200.73,"body":"bottom of the lampshade,"},{"startTime":2201.579,"endTime":2203.15,"body":"you've got the top a circle there,"},{"startTime":2203.15,"endTime":2206.595,"body":"so what's escaping is"},{"startTime":2203.15,"endTime":2206.595,"body":"a cone of light, right?"},{"startTime":2206.595,"endTime":2208.18,"body":"So you get a cone of light"},{"startTime":2208.18,"endTime":2210.49,"body":"and then that cone of light,"},{"startTime":2210.49,"endTime":2212.4,"body":"what will the shadow look"},{"startTime":2210.49,"endTime":2212.4,"body":"like on the vertical wall?"},{"startTime":2212.4,"endTime":2215.66,"body":"Well, it's slicing that"},{"startTime":2212.4,"endTime":2215.66,"body":"cone of light vertically."},{"startTime":2215.66,"endTime":2219.617,"body":"So you exactly get a"},{"startTime":2215.66,"endTime":2219.617,"body":"hyperbola in the shadow there."},{"startTime":2219.617,"endTime":2220.89,"body":"That's lovely."},{"startTime":2220.89,"endTime":2224.633,"body":"So we do see hyperbolas all around us."},{"startTime":2227.28,"endTime":2228.61,"body":"The equation of a hyperbola,"},{"startTime":2228.61,"endTime":2230.92,"body":"I'm not going to go into too much detail,"},{"startTime":2230.92,"endTime":2233.427,"body":"but we can define a hyperbola,"},{"startTime":2234.299,"endTime":2235.94,"body":"and it's sort of a \"Through"},{"startTime":2234.299,"endTime":2235.94,"body":"the Looking Glass\" thing."},{"startTime":2235.94,"endTime":2238.32,"body":"With an ellipse, there are two foci,"},{"startTime":2238.32,"endTime":2240.04,"body":"and the points on the ellipse,"},{"startTime":2240.04,"endTime":2243.67,"body":"the sum of the distances"},{"startTime":2240.04,"endTime":2243.67,"body":"to the foci is constant."},{"startTime":2243.67,"endTime":2246.98,"body":"With a hyperbola, it's the"},{"startTime":2243.67,"endTime":2246.98,"body":"difference of the distances"},{"startTime":2246.98,"endTime":2248.17,"body":"that is constant."},{"startTime":2248.17,"endTime":2251.72,"body":"And that's why you get two"},{"startTime":2248.17,"endTime":2251.72,"body":"branches because on one branch,"},{"startTime":2251.72,"endTime":2253.61,"body":"you know, one focus is closer"},{"startTime":2253.61,"endTime":2255.83,"body":"and on the other branch,"},{"startTime":2253.61,"endTime":2255.83,"body":"the other focus is closer,"},{"startTime":2255.83,"endTime":2258.7,"body":"but we only care about the"},{"startTime":2255.83,"endTime":2258.7,"body":"size of that difference."},{"startTime":2258.7,"endTime":2261.28,"body":"So if the distance to one of them is one"},{"startTime":2261.28,"endTime":2263.28,"body":"and the other one is seven,"},{"startTime":2261.28,"endTime":2263.28,"body":"then the difference is six."},{"startTime":2263.28,"endTime":2265.18,"body":"We don't care if it's plus"},{"startTime":2263.28,"endTime":2265.18,"body":"six or minus six, right?"},{"startTime":2265.18,"endTime":2268.14,"body":"So it's the difference of the distances"},{"startTime":2268.14,"endTime":2269.6,"body":"with a hyperbola that is constant"},{"startTime":2269.6,"endTime":2271.263,"body":"and that sort of reflect-"},{"startTime":2271.263,"endTime":2272.92,"body":"You've gone out the other"},{"startTime":2271.263,"endTime":2272.92,"body":"side through parabola"},{"startTime":2272.92,"endTime":2274.07,"body":"and out the other side."},{"startTime":2275.71,"endTime":2278.08,"body":"Hyperbolas have an equal"},{"startTime":2275.71,"endTime":2278.08,"body":"angle property as well,"},{"startTime":2278.08,"endTime":2279.85,"body":"and I'll show you that here."},{"startTime":2279.85,"endTime":2282.93,"body":"If you've got light"},{"startTime":2279.85,"endTime":2282.93,"body":"that is directed towards"},{"startTime":2282.93,"endTime":2286.03,"body":"one focus of the hyperbola,"},{"startTime":2286.03,"endTime":2288.93,"body":"and then you make that be a mirror,"},{"startTime":2288.93,"endTime":2290.99,"body":"so it's coming in from the"},{"startTime":2288.93,"endTime":2290.99,"body":"other side of the focus,"},{"startTime":2290.99,"endTime":2292.12,"body":"there's the focus coming"},{"startTime":2290.99,"endTime":2292.12,"body":"from the other side,"},{"startTime":2292.12,"endTime":2295.75,"body":"it'll be reflected off"},{"startTime":2292.12,"endTime":2295.75,"body":"towards the other focus."},{"startTime":2295.75,"endTime":2297.51,"body":"You want to throw away the"},{"startTime":2295.75,"endTime":2297.51,"body":"other half of the parabola,"},{"startTime":2297.51,"endTime":2300.62,"body":"of the hyperbola, if you"},{"startTime":2297.51,"endTime":2300.62,"body":"want it to reach that point,"},{"startTime":2300.62,"endTime":2302.843,"body":"but that's where we go."},{"startTime":2304.2,"endTime":2307.21,"body":"This can be used, this property,"},{"startTime":2307.21,"endTime":2310.13,"body":"to make telescopes that"},{"startTime":2307.21,"endTime":2310.13,"body":"are slightly more compact"},{"startTime":2310.13,"endTime":2313.44,"body":"than the Newtonian telescopes"},{"startTime":2310.13,"endTime":2313.44,"body":"that we talked about before"},{"startTime":2313.44,"endTime":2315.41,"body":"that have just a parabolic reflector."},{"startTime":2315.41,"endTime":2317.91,"body":"I'm going to show you this"},{"startTime":2315.41,"endTime":2317.91,"body":"slightly complicated diagram,"},{"startTime":2317.91,"endTime":2319.98,"body":"but let's talk our way through it."},{"startTime":2319.98,"endTime":2321.56,"body":"So this is a kind of telescope"},{"startTime":2321.56,"endTime":2324.06,"body":"where you have a parabolic mirror,"},{"startTime":2324.06,"endTime":2327.94,"body":"and so light comes in from"},{"startTime":2324.06,"endTime":2327.94,"body":"a distant star and it comes,"},{"startTime":2327.94,"endTime":2328.85,"body":"and the first thing that happens"},{"startTime":2328.85,"endTime":2331.05,"body":"is it hits a parabolic mirror"},{"startTime":2331.05,"endTime":2333.033,"body":"just like in a normal telescope."},{"startTime":2334.12,"endTime":2335.68,"body":"By the way, I've shown you kind of the bit"},{"startTime":2335.68,"endTime":2336.62,"body":"where the action is happening,"},{"startTime":2336.62,"endTime":2338.3,"body":"but you should imagine"},{"startTime":2336.62,"endTime":2338.3,"body":"the parabolic mirror"},{"startTime":2338.3,"endTime":2339.81,"body":"kind of carrying on significantly"},{"startTime":2339.81,"endTime":2341.14,"body":"further in both directions."},{"startTime":2341.14,"endTime":2343.45,"body":"So it hits the parabolic mirror first."},{"startTime":2343.45,"endTime":2346.08,"body":"Then it bounces off that,"},{"startTime":2343.45,"endTime":2346.08,"body":"and of course, as we know,"},{"startTime":2346.08,"endTime":2350.02,"body":"it's going to head towards the"},{"startTime":2346.08,"endTime":2350.02,"body":"focus of the parabolic mirror,"},{"startTime":2350.02,"endTime":2352.898,"body":"but before it gets there,"},{"startTime":2350.02,"endTime":2352.898,"body":"it hits another mirror."},{"startTime":2352.898,"endTime":2357.898,"body":"And the second mirror that it"},{"startTime":2352.898,"endTime":2357.898,"body":"hits is part of a hyperbola."},{"startTime":2359.25,"endTime":2361.31,"body":"And remember, the property"},{"startTime":2359.25,"endTime":2361.31,"body":"we've just found out"},{"startTime":2361.31,"endTime":2363.37,"body":"is that light that's heading towards"},{"startTime":2363.37,"endTime":2366.26,"body":"one focus of a hyperbola will bounce off"},{"startTime":2366.26,"endTime":2368.61,"body":"and be directed towards the other focus."},{"startTime":2368.61,"endTime":2370.49,"body":"So what you do, you set things up"},{"startTime":2370.49,"endTime":2372.41,"body":"so that the focus of the parabola"},{"startTime":2372.41,"endTime":2375.483,"body":"is also one focus of a hyperbola."},{"startTime":2375.483,"endTime":2377.83,"body":"So that light hits the hyperbolic mirror,"},{"startTime":2377.83,"endTime":2379.987,"body":"it bounces off it"},{"startTime":2377.83,"endTime":2379.987,"body":"towards the second focus,"},{"startTime":2379.987,"endTime":2382.513,"body":"and that's where the viewer is."},{"startTime":2383.397,"endTime":2386.19,"body":"And this makes for a"},{"startTime":2383.397,"endTime":2386.19,"body":"much more compact design."},{"startTime":2386.19,"endTime":2387.95,"body":"So it's useful for telescopes"},{"startTime":2387.95,"endTime":2389.6,"body":"that you might be moving around maybe,"},{"startTime":2389.6,"endTime":2392.3,"body":"and you also do sometimes"},{"startTime":2389.6,"endTime":2392.3,"body":"see in satellite dishes."},{"startTime":2392.3,"endTime":2393.57,"body":"So it combines, this is great,"},{"startTime":2393.57,"endTime":2395.78,"body":"two conic sections instead of one,"},{"startTime":2395.78,"endTime":2398.45,"body":"so obviously it's my"},{"startTime":2395.78,"endTime":2398.45,"body":"favorite kind of telescope."},{"startTime":2398.45,"endTime":2402.4,"body":"Okay, so that's one use of hyperbolas."},{"startTime":2402.4,"endTime":2404.68,"body":"Now I haven't yet given you or told you"},{"startTime":2404.68,"endTime":2406.75,"body":"what the equation of a hyperbola might be."},{"startTime":2406.75,"endTime":2409.22,"body":"Now, there is quite a general"},{"startTime":2406.75,"endTime":2409.22,"body":"form for the equation,"},{"startTime":2409.22,"endTime":2411.37,"body":"but I wanted to give you"},{"startTime":2409.22,"endTime":2411.37,"body":"one particular example,"},{"startTime":2411.37,"endTime":2414.07,"body":"because again, you know,"},{"startTime":2411.37,"endTime":2414.07,"body":"if you've drawn curves,"},{"startTime":2414.07,"endTime":2419.05,"body":"at school, some moments ago and"},{"startTime":2414.07,"endTime":2419.05,"body":"you were drawing the curves,"},{"startTime":2419.05,"endTime":2421.3,"body":"you might have done Y"},{"startTime":2419.05,"endTime":2421.3,"body":"equals X squared parabola,"},{"startTime":2421.3,"endTime":2423.86,"body":"and then you might have"},{"startTime":2421.3,"endTime":2423.86,"body":"done Y equals one over X."},{"startTime":2423.86,"endTime":2426.88,"body":"That's another curve that"},{"startTime":2423.86,"endTime":2426.88,"body":"we commonly learn to draw"},{"startTime":2426.88,"endTime":2429.81,"body":"or practice drawing when we"},{"startTime":2426.88,"endTime":2429.81,"body":"learn how to draw graphs."},{"startTime":2429.81,"endTime":2433.49,"body":"So Y equal one over X, that's"},{"startTime":2429.81,"endTime":2433.49,"body":"an example of hyperbola."},{"startTime":2433.49,"endTime":2436.12,"body":"But also Y equals two over"},{"startTime":2433.49,"endTime":2436.12,"body":"X, Y equals three over X."},{"startTime":2436.12,"endTime":2438.15,"body":"You can scale it up, and in general,"},{"startTime":2438.15,"endTime":2443.15,"body":"if M is any number, Y equals"},{"startTime":2438.15,"endTime":2443.15,"body":"M over X will be a hyperbola."},{"startTime":2444.33,"endTime":2446.44,"body":"Now the heading of this slide"},{"startTime":2446.44,"endTime":2447.81,"body":"might be doubling the cube."},{"startTime":2447.81,"endTime":2449.09,"body":"What's that to do with anything?"},{"startTime":2449.09,"endTime":2453.19,"body":"I wanted to just mention"},{"startTime":2449.09,"endTime":2453.19,"body":"conics have been studied,"},{"startTime":2453.19,"endTime":2454.94,"body":"were first studied by the ancient Greeks,"},{"startTime":2454.94,"endTime":2457.31,"body":"people like Apollonius and others,"},{"startTime":2457.31,"endTime":2461.06,"body":"but it's believed that the"},{"startTime":2457.31,"endTime":2461.06,"body":"first person to study conics,"},{"startTime":2461.06,"endTime":2464.19,"body":"write about them was an"},{"startTime":2461.06,"endTime":2464.19,"body":"ancient Greek mathematician,"},{"startTime":2464.19,"endTime":2467.27,"body":"Menaechmus, and he was interested"},{"startTime":2467.27,"endTime":2469.68,"body":"in the problem of doubling the cube,"},{"startTime":2469.68,"endTime":2471.63,"body":"which is, you may recall,"},{"startTime":2471.63,"endTime":2474.48,"body":"there were these three kind"},{"startTime":2471.63,"endTime":2474.48,"body":"of intractable problems"},{"startTime":2474.48,"endTime":2476.1,"body":"of ancient Greek geometry,"},{"startTime":2476.1,"endTime":2479.47,"body":"which is all about straight"},{"startTime":2476.1,"endTime":2479.47,"body":"edge and compass constructions."},{"startTime":2479.47,"endTime":2481.09,"body":"So that there are only a very small number"},{"startTime":2481.09,"endTime":2482.46,"body":"of things you're allowed to do"},{"startTime":2482.46,"endTime":2484.74,"body":"in order to construct"},{"startTime":2482.46,"endTime":2484.74,"body":"your geometrical figures."},{"startTime":2484.74,"endTime":2486.614,"body":"You can draw lines between points,"},{"startTime":2486.614,"endTime":2488.623,"body":"you can draw circles,"},{"startTime":2489.5,"endTime":2491.05,"body":"and you can make the"},{"startTime":2489.5,"endTime":2491.05,"body":"intersections of those"},{"startTime":2491.05,"endTime":2492.92,"body":"and get new points, but that's about it."},{"startTime":2492.92,"endTime":2494.56,"body":"So with this very strict set of rules"},{"startTime":2494.56,"endTime":2498.15,"body":"of what you can do with"},{"startTime":2494.56,"endTime":2498.15,"body":"your constructions,"},{"startTime":2498.15,"endTime":2501.95,"body":"there were three problems"},{"startTime":2498.15,"endTime":2501.95,"body":"that were difficult"},{"startTime":2501.95,"endTime":2503.23,"body":"and no one could do."},{"startTime":2503.23,"endTime":2505.44,"body":"So trisecting the angle is one."},{"startTime":2505.44,"endTime":2507.35,"body":"If you've got an angle, can"},{"startTime":2505.44,"endTime":2507.35,"body":"you do some construction"},{"startTime":2507.35,"endTime":2509.45,"body":"to cut exactly into three equal angles?"},{"startTime":2509.45,"endTime":2511.6,"body":"You can bisect any angle,"},{"startTime":2511.6,"endTime":2513.71,"body":"but can you trisect any angle?"},{"startTime":2513.71,"endTime":2516.23,"body":"Squaring the circles,"},{"startTime":2513.71,"endTime":2516.23,"body":"probably the most famous."},{"startTime":2516.23,"endTime":2518.154,"body":"If you've got a circle,"},{"startTime":2518.154,"endTime":2520.15,"body":"can you construct a square"},{"startTime":2518.154,"endTime":2520.15,"body":"which has the same area?"},{"startTime":2520.15,"endTime":2522.28,"body":"And doubling the cube is the third."},{"startTime":2522.28,"endTime":2524.635,"body":"If you've got a cube,"},{"startTime":2522.28,"endTime":2524.635,"body":"can you construct a cube"},{"startTime":2524.635,"endTime":2527.763,"body":"which has exactly doubled the"},{"startTime":2524.635,"endTime":2527.763,"body":"volume of your original cube?"},{"startTime":2528.6,"endTime":2530.72,"body":"Now, all of these are"},{"startTime":2528.6,"endTime":2530.72,"body":"impossible it turns out."},{"startTime":2530.72,"endTime":2534.35,"body":"Many, many centuries elapsed"},{"startTime":2530.72,"endTime":2534.35,"body":"before it was finally proved."},{"startTime":2534.35,"endTime":2537.1,"body":"I mean, almost 2000 years"},{"startTime":2534.35,"endTime":2537.1,"body":"before it was finally proved"},{"startTime":2537.1,"endTime":2538.89,"body":"that all three of these could not be done"},{"startTime":2538.89,"endTime":2541.49,"body":"with the standard straight"},{"startTime":2538.89,"endTime":2541.49,"body":"edge and compass construction."},{"startTime":2541.49,"endTime":2545.23,"body":"But Menaechmus was investigating conics"},{"startTime":2546.12,"endTime":2549.16,"body":"and right, anachronism alert,"},{"startTime":2549.16,"endTime":2550.39,"body":"'cause everything I'm about to say"},{"startTime":2550.39,"endTime":2552.31,"body":"is absolutely not how Menaechmus"},{"startTime":2552.31,"endTime":2553.86,"body":"would've expressed anything."},{"startTime":2553.86,"endTime":2555.95,"body":"There was no algebra at that time,"},{"startTime":2555.95,"endTime":2557.5,"body":"there was no Y equals one over X,"},{"startTime":2557.5,"endTime":2561.27,"body":"but I'm going to explain to"},{"startTime":2557.5,"endTime":2561.27,"body":"you kind of the idea of this,"},{"startTime":2561.27,"endTime":2562.38,"body":"'cause I think it's really cool."},{"startTime":2562.38,"endTime":2565.22,"body":"If you slightly expand what you allow,"},{"startTime":2565.22,"endTime":2567.8,"body":"so you allow not just circles but conics,"},{"startTime":2567.8,"endTime":2570.86,"body":"let's see what we can see."},{"startTime":2570.86,"endTime":2573.91,"body":"So take a cube of volume V."},{"startTime":2573.91,"endTime":2576.3,"body":"I'm going to find the"},{"startTime":2573.91,"endTime":2576.3,"body":"intersection between two conics"},{"startTime":2576.3,"endTime":2577.87,"body":"carefully chosen."},{"startTime":2577.87,"endTime":2581.8,"body":"So one of them is the"},{"startTime":2577.87,"endTime":2581.8,"body":"hyperbola, Y equals two V over X"},{"startTime":2581.8,"endTime":2583.68,"body":"and the other one is"},{"startTime":2581.8,"endTime":2583.68,"body":"our favorite parabola,"},{"startTime":2583.68,"endTime":2585.27,"body":"Y equals X squared."},{"startTime":2585.27,"endTime":2588.19,"body":"They will meet each"},{"startTime":2585.27,"endTime":2588.19,"body":"other, they will intersect"},{"startTime":2588.19,"endTime":2589.34,"body":"when those two things are equal,"},{"startTime":2589.34,"endTime":2592.543,"body":"so when two V over X equals X squared."},{"startTime":2593.48,"endTime":2595.81,"body":"So you find that intersection point,"},{"startTime":2595.81,"endTime":2597.585,"body":"you just draw your two conics,"},{"startTime":2597.585,"endTime":2598.64,"body":"find the intersection point,"},{"startTime":2598.64,"endTime":2603.24,"body":"and at that point X,"},{"startTime":2598.64,"endTime":2603.24,"body":"the point X will satisfy"},{"startTime":2603.24,"endTime":2605.47,"body":"two V over X equals X squared."},{"startTime":2605.47,"endTime":2609.82,"body":"So if you multiply it by X,"},{"startTime":2605.47,"endTime":2609.82,"body":"you get X cubed equals two V."},{"startTime":2609.82,"endTime":2613.2,"body":"So now if you draw a cube,"},{"startTime":2609.82,"endTime":2613.2,"body":"you've got that X now."},{"startTime":2613.2,"endTime":2615.07,"body":"If you draw a cube of side X,"},{"startTime":2615.07,"endTime":2618.53,"body":"then its volume will be"},{"startTime":2615.07,"endTime":2618.53,"body":"X cubed, which is two V."},{"startTime":2618.53,"endTime":2620.07,"body":"So we've doubled the cube."},{"startTime":2620.07,"endTime":2623.16,"body":"Now I'm not saying that that is-"},{"startTime":2623.16,"endTime":2625.4,"body":"We haven't done it legally, right?"},{"startTime":2625.4,"endTime":2626.68,"body":"Because we've used conics."},{"startTime":2626.68,"endTime":2628.88,"body":"But it's a really interesting"},{"startTime":2626.68,"endTime":2628.88,"body":"thing that you can just,"},{"startTime":2628.88,"endTime":2629.9,"body":"if you expand a little bit"},{"startTime":2629.9,"endTime":2632.32,"body":"what you say you are allowed to use,"},{"startTime":2632.32,"endTime":2634.113,"body":"you can then double the cube."},{"startTime":2636.65,"endTime":2639.693,"body":"Okay, I want to show you a way,"},{"startTime":2641.54,"endTime":2643.34,"body":"a way that hyperbolas are used"},{"startTime":2644.25,"endTime":2646.59,"body":"again, in buildings that we might see."},{"startTime":2646.59,"endTime":2648.4,"body":"So the iconic cooling tower,"},{"startTime":2648.4,"endTime":2651.85,"body":"there's three of them on the"},{"startTime":2648.4,"endTime":2651.85,"body":"left there that we all see."},{"startTime":2651.85,"endTime":2653.06,"body":"That is a hyperboloid."},{"startTime":2653.06,"endTime":2656.46,"body":"So again, take a hyperbola,"},{"startTime":2653.06,"endTime":2656.46,"body":"rotate it around its axis,"},{"startTime":2656.46,"endTime":2659.22,"body":"and you get this surface"},{"startTime":2656.46,"endTime":2659.22,"body":"called a hyperboloid."},{"startTime":2659.22,"endTime":2661.85,"body":"Cooling towers, it's"},{"startTime":2659.22,"endTime":2661.85,"body":"interesting 'cause you know,"},{"startTime":2661.85,"endTime":2664.713,"body":"we've all grown up just"},{"startTime":2661.85,"endTime":2664.713,"body":"seeing them as iconic things."},{"startTime":2665.86,"endTime":2667.53,"body":"You sort of imagine they've"},{"startTime":2665.86,"endTime":2667.53,"body":"been around forever,"},{"startTime":2667.53,"endTime":2669.7,"body":"but actually the design,"},{"startTime":2667.53,"endTime":2669.7,"body":"the hyperbola design"},{"startTime":2669.7,"endTime":2673.34,"body":"was first suggested and built in 1918."},{"startTime":2673.34,"endTime":2674.93,"body":"So it's a century old,"},{"startTime":2674.93,"endTime":2677.51,"body":"but you sort of feel like"},{"startTime":2674.93,"endTime":2677.51,"body":"it's as old as time somehow."},{"startTime":2677.51,"endTime":2678.82,"body":"Well, maybe that is."},{"startTime":2678.82,"endTime":2680.76,"body":"But that's a cooling tower."},{"startTime":2680.76,"endTime":2682.19,"body":"Now why is this shape used?"},{"startTime":2682.19,"endTime":2683.763,"body":"Well, in a cooling tower,"},{"startTime":2685.63,"endTime":2687.88,"body":"in industry water is used as a coolant,"},{"startTime":2687.88,"endTime":2689.88,"body":"but of course, it then warms up"},{"startTime":2689.88,"endTime":2691.47,"body":"and you have to cool it down itself"},{"startTime":2691.47,"endTime":2692.72,"body":"before you can reuse it, right?"},{"startTime":2692.72,"endTime":2695.46,"body":"So water comes in part"},{"startTime":2692.72,"endTime":2695.46,"body":"of the way up the tower,"},{"startTime":2695.46,"endTime":2697.97,"body":"it goes onto something"},{"startTime":2695.46,"endTime":2697.97,"body":"called an exchange surface,"},{"startTime":2697.97,"endTime":2700.32,"body":"a little bit of the water evaporates,"},{"startTime":2700.32,"endTime":2702.714,"body":"and that creates a vacuum"},{"startTime":2700.32,"endTime":2702.714,"body":"at the base of the tower,"},{"startTime":2702.714,"endTime":2705.53,"body":"which kind of sucks the"},{"startTime":2702.714,"endTime":2705.53,"body":"rest of the water down"},{"startTime":2705.53,"endTime":2708.02,"body":"through the exchange surface, cooling it"},{"startTime":2708.02,"endTime":2711.13,"body":"and it goes into a reservoir"},{"startTime":2708.02,"endTime":2711.13,"body":"and then it can be reused."},{"startTime":2711.13,"endTime":2713.67,"body":"Only about 2% of the water"},{"startTime":2711.13,"endTime":2713.67,"body":"is lost as water vapor."},{"startTime":2713.67,"endTime":2715.93,"body":"So that stuff, that gas"},{"startTime":2713.67,"endTime":2715.93,"body":"coming out of the top"},{"startTime":2715.93,"endTime":2717.49,"body":"of the cooling tower is just water vapor."},{"startTime":2717.49,"endTime":2719.37,"body":"It's nothing horrible."},{"startTime":2719.37,"endTime":2720.65,"body":"Why this shape?"},{"startTime":2720.65,"endTime":2722.463,"body":"Well, it's really strong."},{"startTime":2723.49,"endTime":2728.22,"body":"The shape helps the flow"},{"startTime":2723.49,"endTime":2728.22,"body":"of the vapor in the tower,"},{"startTime":2728.22,"endTime":2730.95,"body":"and also it uses less material."},{"startTime":2730.95,"endTime":2733.57,"body":"So it's like a cylinder"},{"startTime":2730.95,"endTime":2733.57,"body":"that's been on a diet, right?"},{"startTime":2733.57,"endTime":2734.57,"body":"It uses less material"},{"startTime":2734.57,"endTime":2736.15,"body":"'cause it kind of goes in in the middle."},{"startTime":2736.15,"endTime":2738.71,"body":"So it's very efficient in that way."},{"startTime":2738.71,"endTime":2741.87,"body":"The hyperbola shape also has"},{"startTime":2738.71,"endTime":2741.87,"body":"a really brilliant property"},{"startTime":2741.87,"endTime":2744.06,"body":"in terms of, you know,"},{"startTime":2741.87,"endTime":2744.06,"body":"building costs, I guess."},{"startTime":2744.06,"endTime":2747.8,"body":"It looks great, but it's"},{"startTime":2744.06,"endTime":2747.8,"body":"what's called a ruled surface"},{"startTime":2747.8,"endTime":2751.34,"body":"and that means you can actually"},{"startTime":2747.8,"endTime":2751.34,"body":"make it with straight lines."},{"startTime":2751.34,"endTime":2753.61,"body":"And you can see that in"},{"startTime":2751.34,"endTime":2753.61,"body":"the building on the right,"},{"startTime":2753.61,"endTime":2756.52,"body":"which is the air traffic control tower"},{"startTime":2756.52,"endTime":2758.62,"body":"at Newcastle Airport, actually."},{"startTime":2758.62,"endTime":2759.76,"body":"And you could see that it's made-"},{"startTime":2759.76,"endTime":2761.92,"body":"The lattice of straight lines"},{"startTime":2761.92,"endTime":2766.64,"body":"are defining this curved"},{"startTime":2761.92,"endTime":2766.64,"body":"shape, which is really great,"},{"startTime":2766.64,"endTime":2768.85,"body":"and in terms of building costs,"},{"startTime":2768.85,"endTime":2772.03,"body":"it is cheaper to have straight line beams"},{"startTime":2772.03,"endTime":2774.02,"body":"than to make curved ones."},{"startTime":2774.02,"endTime":2775.66,"body":"So you see these shapes"},{"startTime":2775.66,"endTime":2777.7,"body":"and there is actually,"},{"startTime":2775.66,"endTime":2777.7,"body":"there's another kind of shape"},{"startTime":2777.7,"endTime":2778.75,"body":"which I haven't got a picture of,"},{"startTime":2778.75,"endTime":2780.5,"body":"but if you remember the London Olympics,"},{"startTime":2780.5,"endTime":2784.02,"body":"the velodrome there got the"},{"startTime":2780.5,"endTime":2784.02,"body":"nickname the Pringle, right?"},{"startTime":2784.02,"endTime":2787.14,"body":"Because of a well known snack"},{"startTime":2784.02,"endTime":2787.14,"body":"food that it looks like."},{"startTime":2787.14,"endTime":2788.92,"body":"That has a cross section."},{"startTime":2788.92,"endTime":2791.83,"body":"So in one direction its"},{"startTime":2788.92,"endTime":2791.83,"body":"cross section are hyperbolas,"},{"startTime":2791.83,"endTime":2794.15,"body":"and in the other direction"},{"startTime":2791.83,"endTime":2794.15,"body":"its cross are parabolas."},{"startTime":2794.15,"endTime":2796.16,"body":"So it is a hyperbolic paraboloid."},{"startTime":2796.16,"endTime":2798.84,"body":"So impress your friends and family"},{"startTime":2798.84,"endTime":2801.14,"body":"when they are eating certain snack foods"},{"startTime":2801.14,"endTime":2804.825,"body":"by telling them that they are"},{"startTime":2801.14,"endTime":2804.825,"body":"eating hyperbolic paraboloids."},{"startTime":2804.825,"endTime":2806.251,"body":"We must never say Pringles again."},{"startTime":2806.251,"endTime":2811.251,"body":"So that's how we see"},{"startTime":2806.251,"endTime":2811.251,"body":"hyperbolas in building."},{"startTime":2812.106,"endTime":2816.38,"body":"In the last few minutes,"},{"startTime":2812.106,"endTime":2816.38,"body":"I want to show you a way-"},{"startTime":2816.38,"endTime":2818.89,"body":"It's not really a practical"},{"startTime":2816.38,"endTime":2818.89,"body":"way of making hyperbolas,"},{"startTime":2818.89,"endTime":2820.679,"body":"but I quite like it anyway."},{"startTime":2820.679,"endTime":2822.95,"body":"I call it tidying rectangles."},{"startTime":2822.95,"endTime":2825.24,"body":"And the inspiration for this is"},{"startTime":2826.45,"endTime":2829.1,"body":"tidying up after small children"},{"startTime":2829.1,"endTime":2831.2,"body":"who have building blocks"},{"startTime":2829.1,"endTime":2831.2,"body":"all over the floor,"},{"startTime":2831.2,"endTime":2833.39,"body":"and just imagine that you happen to have"},{"startTime":2833.39,"endTime":2835.32,"body":"a rather strange set of building blocks"},{"startTime":2835.32,"endTime":2837.29,"body":"where they all have the"},{"startTime":2835.32,"endTime":2837.29,"body":"same area, all right?"},{"startTime":2837.29,"endTime":2839.9,"body":"So I picked 12, 'cause"},{"startTime":2837.29,"endTime":2839.9,"body":"it has lots of factors,"},{"startTime":2839.9,"endTime":2841.26,"body":"but you could pick any number"},{"startTime":2841.26,"endTime":2843.9,"body":"and then you can have these"},{"startTime":2841.26,"endTime":2843.9,"body":"various blocks, 12 times what?"},{"startTime":2843.9,"endTime":2847.54,"body":"12 by one or three by four, six by two."},{"startTime":2847.54,"endTime":2850.31,"body":"Let's not be prejudiced"},{"startTime":2847.54,"endTime":2850.31,"body":"against non whole numbers,"},{"startTime":2850.31,"endTime":2851.79,"body":"eight by one and a half."},{"startTime":2851.79,"endTime":2854.31,"body":"Anything you like as"},{"startTime":2851.79,"endTime":2854.31,"body":"long as the area is 12"},{"startTime":2855.367,"endTime":2856.46,"body":"and we're going to tidy them away."},{"startTime":2856.46,"endTime":2859.51,"body":"We're just going to put them"},{"startTime":2856.46,"endTime":2859.51,"body":"all in the corner nicely,"},{"startTime":2859.51,"endTime":2861.46,"body":"not all over the floor."},{"startTime":2861.46,"endTime":2863.24,"body":"And so here here's my corner,"},{"startTime":2863.24,"endTime":2865.2,"body":"this is the axes now graphed really,"},{"startTime":2865.2,"endTime":2866.59,"body":"and I'm going to tidy up my rectangles."},{"startTime":2866.59,"endTime":2869.36,"body":"So I'm just going to put them"},{"startTime":2866.59,"endTime":2869.36,"body":"all nicely into the corner."},{"startTime":2869.36,"endTime":2871.61,"body":"There they all go, very nice."},{"startTime":2871.61,"endTime":2873.65,"body":"And if you do this,"},{"startTime":2873.65,"endTime":2878.65,"body":"you start to see that looks"},{"startTime":2873.65,"endTime":2878.65,"body":"like some sort of curve maybe."},{"startTime":2878.87,"endTime":2882.01,"body":"Zero points at this"},{"startTime":2878.87,"endTime":2882.01,"body":"stage, given the context"},{"startTime":2882.01,"endTime":2883.51,"body":"of what I'm talking about."},{"startTime":2883.51,"endTime":2885.13,"body":"I mean, here's a hyperbola, right?"},{"startTime":2885.13,"endTime":2887.07,"body":"But let's see why it's a hyperbola."},{"startTime":2887.07,"endTime":2889.96,"body":"So if we think about any old rectangle"},{"startTime":2891.28,"endTime":2893.43,"body":"that we've got here,"},{"startTime":2893.43,"endTime":2895.75,"body":"the top right-hand"},{"startTime":2893.43,"endTime":2895.75,"body":"corner of the rectangle,"},{"startTime":2895.75,"endTime":2897.627,"body":"that's the bit where,"},{"startTime":2897.627,"endTime":2900.64,"body":"these are sort of making"},{"startTime":2897.627,"endTime":2900.64,"body":"all the curve there."},{"startTime":2900.64,"endTime":2905.64,"body":"So if you've got a rectangle"},{"startTime":2900.64,"endTime":2905.64,"body":"of width X and height Y,"},{"startTime":2905.84,"endTime":2909.52,"body":"then the top right-hand corner"},{"startTime":2905.84,"endTime":2909.52,"body":"will be the point XY, right?"},{"startTime":2909.52,"endTime":2912.47,"body":"And the area of that rectangle"},{"startTime":2909.52,"endTime":2912.47,"body":"will be width times height,"},{"startTime":2912.47,"endTime":2913.77,"body":"which is X times Y."},{"startTime":2913.77,"endTime":2916.63,"body":"And we've said in this"},{"startTime":2913.77,"endTime":2916.63,"body":"case, that will be 12."},{"startTime":2916.63,"endTime":2920.56,"body":"So all the points that"},{"startTime":2916.63,"endTime":2920.56,"body":"are making up this curve"},{"startTime":2920.56,"endTime":2923.13,"body":"have the property that X times Y is 12."},{"startTime":2923.13,"endTime":2925.47,"body":"So Y equals 12 over X."},{"startTime":2925.47,"endTime":2927.02,"body":"Well, that's exactly an example"},{"startTime":2927.02,"endTime":2929.06,"body":"of one of the hyperbolas I"},{"startTime":2927.02,"endTime":2929.06,"body":"mentioned earlier, right?"},{"startTime":2929.06,"endTime":2933.48,"body":"So tidying rectangles in this"},{"startTime":2929.06,"endTime":2933.48,"body":"way results in hyperbolas."},{"startTime":2933.48,"endTime":2935.71,"body":"I mean, okay, it's half a hyperbola."},{"startTime":2935.71,"endTime":2937.27,"body":"You'd have to add in rectangles"},{"startTime":2937.27,"endTime":2939.84,"body":"of dimension minus four by minus three"},{"startTime":2939.84,"endTime":2941.62,"body":"if you want the other half."},{"startTime":2941.62,"endTime":2944.38,"body":"That might be a little advanced"},{"startTime":2941.62,"endTime":2944.38,"body":"for child's building blocks."},{"startTime":2944.38,"endTime":2946.937,"body":"But anyway, this is how"},{"startTime":2944.38,"endTime":2946.937,"body":"you get this hyperbola."},{"startTime":2948.26,"endTime":2949.42,"body":"Okay."},{"startTime":2949.42,"endTime":2953.92,"body":"I've promised three examples"},{"startTime":2949.42,"endTime":2953.92,"body":"or three applications"},{"startTime":2953.92,"endTime":2956.02,"body":"of each conic and here is the final,"},{"startTime":2956.02,"endTime":2958.07,"body":"the third application of the hyperbola"},{"startTime":2958.07,"endTime":2959.223,"body":"and it's in economics."},{"startTime":2960.16,"endTime":2962.03,"body":"Now in economics, you have something"},{"startTime":2962.03,"endTime":2963.02,"body":"called the law of demand,"},{"startTime":2963.02,"endTime":2967.5,"body":"which says that usually"},{"startTime":2963.02,"endTime":2967.5,"body":"for goods or products,"},{"startTime":2967.5,"endTime":2969.6,"body":"when the price rises, the demand falls."},{"startTime":2969.6,"endTime":2971.72,"body":"If the price of cars suddenly doubles,"},{"startTime":2971.72,"endTime":2973.41,"body":"then fewer cars will be sold."},{"startTime":2973.41,"endTime":2975.76,"body":"There's one or two little"},{"startTime":2973.41,"endTime":2975.76,"body":"exceptions to this."},{"startTime":2975.76,"endTime":2980.76,"body":"For example, super high end luxury goods."},{"startTime":2980.78,"endTime":2984.05,"body":"Sometimes part of the appeal"},{"startTime":2980.78,"endTime":2984.05,"body":"is that they're very expensive"},{"startTime":2984.05,"endTime":2985.91,"body":"and almost nobody can afford them."},{"startTime":2985.91,"endTime":2987.91,"body":"So all the billionaires will want one."},{"startTime":2987.91,"endTime":2989.78,"body":"If you increase the price of those things,"},{"startTime":2989.78,"endTime":2992.6,"body":"then the cache goes up"},{"startTime":2992.6,"endTime":2994.77,"body":"and so maybe the demand goes up."},{"startTime":2994.77,"endTime":2996.98,"body":"Then all the trillionaires"},{"startTime":2994.77,"endTime":2996.98,"body":"will definitely want one,"},{"startTime":2996.98,"endTime":2998.6,"body":"but all the billionaires will as well."},{"startTime":2998.6,"endTime":3000.77,"body":"So that's an extreme example."},{"startTime":3000.77,"endTime":3002.69,"body":"At the other end of things,"},{"startTime":3002.69,"endTime":3005.27,"body":"if the price of something is quite low,"},{"startTime":3005.27,"endTime":3008.26,"body":"that can sometimes signal that"},{"startTime":3005.27,"endTime":3008.26,"body":"the quality is perhaps low."},{"startTime":3008.26,"endTime":3010.69,"body":"So if you lower the price even more,"},{"startTime":3010.69,"endTime":3015.69,"body":"that can be a bad signal and"},{"startTime":3010.69,"endTime":3015.69,"body":"perhaps the demand goes down."},{"startTime":3015.69,"endTime":3018.49,"body":"But outside of those two special cases,"},{"startTime":3018.49,"endTime":3020.29,"body":"in the vast majority of cases,"},{"startTime":3020.29,"endTime":3021.8,"body":"there's an inverse relationship"},{"startTime":3021.8,"endTime":3023.26,"body":"between the price and the demand,"},{"startTime":3023.26,"endTime":3026.27,"body":"and this is plotted on what's"},{"startTime":3023.26,"endTime":3026.27,"body":"called the demand curve."},{"startTime":3026.27,"endTime":3031.27,"body":"So it plots the price per item"},{"startTime":3026.27,"endTime":3031.27,"body":"against the quantity demand."},{"startTime":3031.699,"endTime":3034.98,"body":"In other words, how many you"},{"startTime":3031.699,"endTime":3034.98,"body":"can sell at that price point."},{"startTime":3034.98,"endTime":3038.32,"body":"And different goods have"},{"startTime":3034.98,"endTime":3038.32,"body":"different demand curves,"},{"startTime":3038.32,"endTime":3041.04,"body":"but one nice example is what, again,"},{"startTime":3041.04,"endTime":3044.92,"body":"you might call the"},{"startTime":3041.04,"endTime":3044.92,"body":"kids' sweet shop budget,"},{"startTime":3044.92,"endTime":3048.06,"body":"which is if I give a kid"},{"startTime":3044.92,"endTime":3048.06,"body":"50 P to spend on sweets,"},{"startTime":3048.06,"endTime":3050.46,"body":"they will be spending that 50 P, right?"},{"startTime":3050.46,"endTime":3051.76,"body":"Doesn't matter what the sweets cost."},{"startTime":3051.76,"endTime":3053.62,"body":"If sweets are five P, they buy 10,"},{"startTime":3053.62,"endTime":3056.43,"body":"if sweets are two P, they're"},{"startTime":3053.62,"endTime":3056.43,"body":"buying 25 sweets, right?"},{"startTime":3056.43,"endTime":3058.35,"body":"So they will spend their whole budget"},{"startTime":3058.35,"endTime":3060.59,"body":"and it doesn't matter what the price is,"},{"startTime":3060.59,"endTime":3062.37,"body":"they're just going to"},{"startTime":3060.59,"endTime":3062.37,"body":"spend all of the money."},{"startTime":3062.37,"endTime":3065.35,"body":"So if you take any point"},{"startTime":3062.37,"endTime":3065.35,"body":"on the demand curve,"},{"startTime":3065.35,"endTime":3066.7,"body":"so at this point here,"},{"startTime":3066.7,"endTime":3068.39,"body":"imagine drawing a rectangle underneath it,"},{"startTime":3068.39,"endTime":3070.52,"body":"so we were doing our tidying rectangles."},{"startTime":3070.52,"endTime":3072.193,"body":"A rectangle underneath there,"},{"startTime":3073.22,"endTime":3075.64,"body":"the kind of the width of that rectangle"},{"startTime":3075.64,"endTime":3077.86,"body":"will be the price per sweet,"},{"startTime":3077.86,"endTime":3079.13,"body":"and the height of the rectangle"},{"startTime":3079.13,"endTime":3081.28,"body":"will be how many sweets get sold."},{"startTime":3081.28,"endTime":3084.05,"body":"So the total spend on sweets,"},{"startTime":3084.05,"endTime":3085.33,"body":"it will be the price per sweet"},{"startTime":3085.33,"endTime":3087.763,"body":"multiplied by the number that gets sold."},{"startTime":3088.73,"endTime":3091.68,"body":"And that gives you the area"},{"startTime":3088.73,"endTime":3091.68,"body":"of the rectangle, right?"},{"startTime":3091.68,"endTime":3093.69,"body":"'Cause it's the width times the height."},{"startTime":3093.69,"endTime":3097.03,"body":"So in this case where you're"},{"startTime":3093.69,"endTime":3097.03,"body":"going to spend your whole budget,"},{"startTime":3097.03,"endTime":3099.85,"body":"the area of the rectangle is"},{"startTime":3097.03,"endTime":3099.85,"body":"going to be that whole budget,"},{"startTime":3099.85,"endTime":3103.7,"body":"in our case, the 50 P that"},{"startTime":3099.85,"endTime":3103.7,"body":"we had to spend on sweets."},{"startTime":3103.7,"endTime":3107.01,"body":"And that's true for any"},{"startTime":3103.7,"endTime":3107.01,"body":"point on this curve, right?"},{"startTime":3107.01,"endTime":3108.01,"body":"You're spending your whole budget."},{"startTime":3108.01,"endTime":3112.14,"body":"So the area of any rectangle"},{"startTime":3108.01,"endTime":3112.14,"body":"we draw under this curve"},{"startTime":3112.14,"endTime":3114.583,"body":"will be the total budget, our 50 P."},{"startTime":3115.6,"endTime":3118.36,"body":"And we've already seen"},{"startTime":3115.6,"endTime":3118.36,"body":"that if you do that,"},{"startTime":3118.36,"endTime":3120.23,"body":"if you pile up all your"},{"startTime":3118.36,"endTime":3120.23,"body":"rectangles like that,"},{"startTime":3120.23,"endTime":3121.66,"body":"you exactly get a hyperbola."},{"startTime":3121.66,"endTime":3124.86,"body":"So in that case where"},{"startTime":3121.66,"endTime":3124.86,"body":"you have a fixed budget,"},{"startTime":3124.86,"endTime":3127.507,"body":"the demand curve is exactly a hyperbola."},{"startTime":3128.7,"endTime":3130.95,"body":"And of course, you know,"},{"startTime":3128.7,"endTime":3130.95,"body":"this is a toy example,"},{"startTime":3130.95,"endTime":3134.69,"body":"but there are real life"},{"startTime":3130.95,"endTime":3134.69,"body":"situations where we would do this."},{"startTime":3134.69,"endTime":3137.08,"body":"I mean, even as adults we"},{"startTime":3134.69,"endTime":3137.08,"body":"don't have a sweets budget,"},{"startTime":3137.08,"endTime":3138.45,"body":"well, maybe we do,"},{"startTime":3138.45,"endTime":3140.85,"body":"but we might have something"},{"startTime":3138.45,"endTime":3140.85,"body":"like a budget for eating out"},{"startTime":3140.85,"endTime":3143.35,"body":"or a budget for holidays"},{"startTime":3140.85,"endTime":3143.35,"body":"or something like this"},{"startTime":3143.35,"endTime":3144.88,"body":"and we will spend that budget,"},{"startTime":3144.88,"endTime":3146.15,"body":"and if the cost of eating out goes up,"},{"startTime":3146.15,"endTime":3149.04,"body":"we eat out a bit less, but"},{"startTime":3146.15,"endTime":3149.04,"body":"we have a fixed budget."},{"startTime":3149.04,"endTime":3152.87,"body":"So this is a really important"},{"startTime":3149.04,"endTime":3152.87,"body":"example in economics,"},{"startTime":3152.87,"endTime":3155.72,"body":"and it's a hyperbola, which is great."},{"startTime":3155.72,"endTime":3160.72,"body":"So I hope you've enjoyed"},{"startTime":3155.72,"endTime":3160.72,"body":"this guided tour of conics."},{"startTime":3161.49,"endTime":3163.65,"body":"And, you know, it really highlights for me"},{"startTime":3163.65,"endTime":3167.17,"body":"the issue with any kind of pressure"},{"startTime":3167.17,"endTime":3170.23,"body":"to only study useful things."},{"startTime":3170.23,"endTime":3172.97,"body":"I mean, of course, I'm sure"},{"startTime":3170.23,"endTime":3172.97,"body":"we all agree in this room"},{"startTime":3172.97,"endTime":3175.52,"body":"that knowledge itself is worth having."},{"startTime":3175.52,"endTime":3178.05,"body":"You know, studying things"},{"startTime":3175.52,"endTime":3178.05,"body":"because they're interesting"},{"startTime":3178.05,"endTime":3180.49,"body":"and wonderful and beautiful,"},{"startTime":3178.05,"endTime":3180.49,"body":"that's already enough."},{"startTime":3180.49,"endTime":3183.997,"body":"But imagine if Menaechmus had"},{"startTime":3180.49,"endTime":3183.997,"body":"to say to a grant committee,"},{"startTime":3183.997,"endTime":3186.91,"body":"\"What are the applications"},{"startTime":3183.997,"endTime":3186.91,"body":"of these conics of yours?\""},{"startTime":3186.91,"endTime":3189.72,"body":"You know, you can't say,"},{"startTime":3186.91,"endTime":3189.72,"body":"\"Well, 2000 years from now,"},{"startTime":3189.72,"endTime":3191.31,"body":"they'll be for treating kidney stones,"},{"startTime":3191.31,"endTime":3193.1,"body":"seeing distant stars.\""},{"startTime":3193.1,"endTime":3194.84,"body":"You just never know, you never know."},{"startTime":3194.84,"endTime":3196.09,"body":"And so this is a plea,"},{"startTime":3196.09,"endTime":3198.48,"body":"let's all study things"},{"startTime":3196.09,"endTime":3198.48,"body":"for the love of them."},{"startTime":3198.48,"endTime":3200.06,"body":"Okay, and I will finish there"},{"startTime":3200.06,"endTime":3202.69,"body":"with a little plug for my next lecture,"},{"startTime":3202.69,"endTime":3204.58,"body":"the final one of this academic here,"},{"startTime":3204.58,"endTime":3205.8,"body":"more coming next year,"},{"startTime":3205.8,"endTime":3208.03,"body":"The Incredible Sine Wave and Its Uses"},{"startTime":3208.03,"endTime":3210.65,"body":"and that's on the 23rd of May,"},{"startTime":3208.03,"endTime":3210.65,"body":"so I hope to see you there."},{"startTime":3210.65,"endTime":3212.076,"body":"Thank you."},{"startTime":3212.076,"endTime":3215.47,"body":"(crowd applauds)"},{"startTime":3215.47,"endTime":3216.92,"body":"- Are there parallel conic sections"},{"startTime":3216.92,"endTime":3219.52,"body":"for fourth and higher dimensions?"},{"startTime":3219.52,"endTime":3220.353,"body":"- Yes."},{"startTime":3221.71,"endTime":3224.8,"body":"Yes, but I'm not aware of any"},{"startTime":3221.71,"endTime":3224.8,"body":"applications of them as yet,"},{"startTime":3224.8,"endTime":3228.1,"body":"although let's wait 2000"},{"startTime":3224.8,"endTime":3228.1,"body":"years and maybe we will."},{"startTime":3228.1,"endTime":3229.1,"body":"- Great."},{"startTime":3229.1,"endTime":3230.39,"body":"And the second one is,"},{"startTime":3230.39,"endTime":3233.27,"body":"is it true that there"},{"startTime":3230.39,"endTime":3233.27,"body":"is no analytical formula"},{"startTime":3233.27,"endTime":3236.58,"body":"for the length of an"},{"startTime":3233.27,"endTime":3236.58,"body":"ellipse, its circumference?"},{"startTime":3236.58,"endTime":3238.14,"body":"- Yeah, this is so weird."},{"startTime":3238.14,"endTime":3239.36,"body":"There's a really-"},{"startTime":3239.36,"endTime":3241.28,"body":"Right, so with a circle,"},{"startTime":3241.28,"endTime":3243.56,"body":"The area of the circle"},{"startTime":3241.28,"endTime":3243.56,"body":"is Pi R squared, right?"},{"startTime":3243.56,"endTime":3246.44,"body":"And the circumference two Pi R, easy."},{"startTime":3246.44,"endTime":3248.05,"body":"Ellipse, area of the ellipse"},{"startTime":3248.05,"endTime":3249.52,"body":"in terms of the A and B, I said,"},{"startTime":3249.52,"endTime":3251.95,"body":"it's Pi A plus B, right, squared?"},{"startTime":3251.95,"endTime":3254.23,"body":"And circumference, no formula."},{"startTime":3254.23,"endTime":3255.063,"body":"Well, there is a formula,"},{"startTime":3255.063,"endTime":3256.93,"body":"but it's got like integrals"},{"startTime":3255.063,"endTime":3256.93,"body":"and calculus stuff"},{"startTime":3256.93,"endTime":3257.94,"body":"and it's really hard."},{"startTime":3257.94,"endTime":3260.07,"body":"So that's a very, very odd"},{"startTime":3260.07,"endTime":3262.79,"body":"and curious thing about"},{"startTime":3260.07,"endTime":3262.79,"body":"ellipses, but yeah, that's true."},{"startTime":3262.79,"endTime":3264.09,"body":"- Thank you so much, Professor Hart."},{"startTime":3264.09,"endTime":3265.92,"body":"I'm sorry to bring these"},{"startTime":3264.09,"endTime":3265.92,"body":"questions to a close."},{"startTime":3265.92,"endTime":3267.27,"body":"It's just we've overrun a bit."},{"startTime":3267.27,"endTime":3269.67,"body":"Thank you very much for"},{"startTime":3267.27,"endTime":3269.67,"body":"the fascinating lecture."},{"startTime":3270.688,"endTime":3273.313,"body":"(crowd applauds)"}]}