{"version":"1.0.0","segments":[{"startTime":0.2,"endTime":3.73,"body":"- Proportion has long been"},{"startTime":0.2,"endTime":3.73,"body":"central to our ideas of form"},{"startTime":3.73,"endTime":5.83,"body":"and beauty, and today"},{"startTime":3.73,"endTime":5.83,"body":"I want to talk to you"},{"startTime":5.83,"endTime":8.33,"body":"about the mathematics of proportion."},{"startTime":8.33,"endTime":10.83,"body":"And along the way, we're going"},{"startTime":8.33,"endTime":10.83,"body":"to find out what links honeybees"},{"startTime":10.83,"endTime":13.01,"body":"to the architecture of Le Corbusier."},{"startTime":13.01,"endTime":18.01,"body":"Why is A4 paper like this"},{"startTime":13.01,"endTime":18.01,"body":"exactly 297 millimeters long?"},{"startTime":18.306,"endTime":22.16,"body":"And what can an ancient"},{"startTime":18.306,"endTime":22.16,"body":"Egypt scribe tell us"},{"startTime":22.16,"endTime":23.916,"body":"about baking cakes?"},{"startTime":23.916,"endTime":28.916,"body":"So let's begin at the beginning,"},{"startTime":23.916,"endTime":28.916,"body":"and this diagram here,"},{"startTime":29.76,"endTime":34.26,"body":"Leonardo DaVinci's famous"},{"startTime":29.76,"endTime":34.26,"body":"Vitruvian man named"},{"startTime":34.26,"endTime":36.093,"body":"for the Roman architect Vitruvius."},{"startTime":37.79,"endTime":41.88,"body":"Greek and Roman thinkers"},{"startTime":37.79,"endTime":41.88,"body":"were absolutely convinced"},{"startTime":41.88,"endTime":44.33,"body":"that proportion was key to beauty."},{"startTime":44.33,"endTime":46.87,"body":"Aristotle said so, and"},{"startTime":44.33,"endTime":46.87,"body":"along with symmetry,"},{"startTime":46.87,"endTime":49.77,"body":"which he also said was"},{"startTime":46.87,"endTime":49.77,"body":"very important, and he said"},{"startTime":49.77,"endTime":51.94,"body":"it's mathematics that helps"},{"startTime":49.77,"endTime":51.94,"body":"us understand these things."},{"startTime":51.94,"endTime":53.38,"body":"Of course, I agree."},{"startTime":53.38,"endTime":58.239,"body":"Vitruvius said that the"},{"startTime":53.38,"endTime":58.239,"body":"human body is put together"},{"startTime":58.239,"endTime":61.7,"body":"with fixed proportions, each"},{"startTime":58.239,"endTime":61.7,"body":"part corresponds to the whole"},{"startTime":61.7,"endTime":66.1,"body":"in certain fixed, beautiful"},{"startTime":61.7,"endTime":66.1,"body":"proportions, and he said that"},{"startTime":66.1,"endTime":68.868,"body":"therefore, the ideal"},{"startTime":66.1,"endTime":68.868,"body":"building should also have"},{"startTime":68.868,"endTime":71.944,"body":"fixed proportions, and that"},{"startTime":68.868,"endTime":71.944,"body":"would make it beautiful."},{"startTime":71.944,"endTime":75.14,"body":"But what do we mean by proportion?"},{"startTime":75.14,"endTime":79.19,"body":"Well, one of the reasons why"},{"startTime":75.14,"endTime":79.19,"body":"Vitruvius was so influential,"},{"startTime":79.19,"endTime":82.53,"body":"and throughout Western art"},{"startTime":79.19,"endTime":82.53,"body":"architecture, you know,"},{"startTime":82.53,"endTime":86.04,"body":"it's a name that you hear"},{"startTime":82.53,"endTime":86.04,"body":"over and over again, in part,"},{"startTime":86.04,"endTime":88.67,"body":"the translations of his work"},{"startTime":86.04,"endTime":88.67,"body":"from Latin into languages"},{"startTime":88.67,"endTime":91.642,"body":"like French and English in"},{"startTime":88.67,"endTime":91.642,"body":"the 16th and 17th centuries"},{"startTime":91.642,"endTime":94.66,"body":"helped his work to become more widespread."},{"startTime":94.66,"endTime":96.63,"body":"And one translation by Claude Piroux"},{"startTime":96.63,"endTime":100.7,"body":"in 17th century described"},{"startTime":96.63,"endTime":100.7,"body":"proportion by saying"},{"startTime":100.7,"endTime":103.66,"body":"it's all about relation, so"},{"startTime":100.7,"endTime":103.66,"body":"the relation, for example,"},{"startTime":103.66,"endTime":107.098,"body":"that windows of eight foot"},{"startTime":103.66,"endTime":107.098,"body":"high have with windows"},{"startTime":107.098,"endTime":109.067,"body":"of six foot high, when"},{"startTime":107.098,"endTime":109.067,"body":"the one of four foot wide"},{"startTime":109.067,"endTime":111.17,"body":"and the other are three foot wide."},{"startTime":111.17,"endTime":113.69,"body":"In other words, it's eight"},{"startTime":111.17,"endTime":113.69,"body":"to four being the same"},{"startTime":113.69,"endTime":115.37,"body":"as six to three."},{"startTime":115.37,"endTime":118.83,"body":"It's proportion, it's ratio"},{"startTime":115.37,"endTime":118.83,"body":"that we're talking about here."},{"startTime":118.83,"endTime":121.02,"body":"So that's what I want"},{"startTime":118.83,"endTime":121.02,"body":"to explain more about,"},{"startTime":121.02,"endTime":125.04,"body":"and explore today, is this"},{"startTime":121.02,"endTime":125.04,"body":"idea of ratio and proportion."},{"startTime":125.04,"endTime":128.34,"body":"And we're going to begin with"},{"startTime":125.04,"endTime":128.34,"body":"probably the most famous ratio"},{"startTime":128.34,"endTime":131.063,"body":"of all, the so-called golden ratio."},{"startTime":132.07,"endTime":136.12,"body":"Now, you read about the golden"},{"startTime":132.07,"endTime":136.12,"body":"ratio, not by that name,"},{"startTime":136.12,"endTime":138.87,"body":"in Greek texts like the work of Euclid,"},{"startTime":138.87,"endTime":142.09,"body":"most famous geometry textbook"},{"startTime":138.87,"endTime":142.09,"body":"of all time, I should think."},{"startTime":142.09,"endTime":147.09,"body":"In Euclid, often you encounter this ratio"},{"startTime":147.257,"endTime":150.55,"body":"when we are talking about regular shapes,"},{"startTime":150.55,"endTime":154.06,"body":"regular polygons, the Platonic"},{"startTime":150.55,"endTime":154.06,"body":"solids, things like this."},{"startTime":154.06,"endTime":155.25,"body":"So what's a regular polygon?"},{"startTime":155.25,"endTime":157.95,"body":"It's a shape where, you know,"},{"startTime":155.25,"endTime":157.95,"body":"it's made of straight lines,"},{"startTime":157.95,"endTime":159.93,"body":"and all the lines are equal,"},{"startTime":157.95,"endTime":159.93,"body":"and all the angles are equal,"},{"startTime":159.93,"endTime":163.7,"body":"so square, equilateral"},{"startTime":159.93,"endTime":163.7,"body":"triangle, that kind of thing."},{"startTime":163.7,"endTime":165.363,"body":"Now, if you are an ancient Greek geometer,"},{"startTime":165.363,"endTime":167.62,"body":"what you want to do, if you"},{"startTime":165.363,"endTime":167.62,"body":"have a shape that you like,"},{"startTime":167.62,"endTime":169.59,"body":"you want to be able to construct it using"},{"startTime":169.59,"endTime":171.47,"body":"just a strange edge and compasses."},{"startTime":171.47,"endTime":173.1,"body":"So you've go to these"},{"startTime":171.47,"endTime":173.1,"body":"geometric constructions,"},{"startTime":173.1,"endTime":174.89,"body":"and you may have learned to do things"},{"startTime":174.89,"endTime":177.94,"body":"like perpendicular bisectors,"},{"startTime":174.89,"endTime":177.94,"body":"and equilateral triangles"},{"startTime":177.94,"endTime":180.055,"body":"when you're at school doing geometry."},{"startTime":180.055,"endTime":183.15,"body":"But how do you draw a regular pentagon?"},{"startTime":183.15,"endTime":185.31,"body":"Well, without going into"},{"startTime":183.15,"endTime":185.31,"body":"details of how you do that,"},{"startTime":185.31,"endTime":187.64,"body":"it is possible, no guarantee"},{"startTime":185.31,"endTime":187.64,"body":"it would be possible,"},{"startTime":187.64,"endTime":191.33,"body":"but it is possible, but if"},{"startTime":187.64,"endTime":191.33,"body":"you're going to do this,"},{"startTime":191.33,"endTime":194.64,"body":"let's start with, you"},{"startTime":191.33,"endTime":194.64,"body":"know, an edge, a side A B,"},{"startTime":194.64,"endTime":196.8,"body":"let's say that has unit length one."},{"startTime":196.8,"endTime":200.32,"body":"How are you going to construct"},{"startTime":196.8,"endTime":200.32,"body":"the next vertex, vertex C?"},{"startTime":200.32,"endTime":203.88,"body":"Well, if you happen to know"},{"startTime":200.32,"endTime":203.88,"body":"how long the diagonal is,"},{"startTime":203.88,"endTime":205.97,"body":"and I've just given it a name X there,"},{"startTime":205.97,"endTime":207.49,"body":"if you happen to know that length,"},{"startTime":207.49,"endTime":210.04,"body":"then what you could do is"},{"startTime":207.49,"endTime":210.04,"body":"you could take a circle,"},{"startTime":210.04,"endTime":213.8,"body":"center it at A, radius X,"},{"startTime":210.04,"endTime":213.8,"body":"that's definitely going to go"},{"startTime":213.8,"endTime":215.91,"body":"through your point C,"},{"startTime":213.8,"endTime":215.91,"body":"and then you could take"},{"startTime":215.91,"endTime":219.37,"body":"another circle, center"},{"startTime":215.91,"endTime":219.37,"body":"it at B, radius one."},{"startTime":219.37,"endTime":221.54,"body":"That will definitely also"},{"startTime":219.37,"endTime":221.54,"body":"go through the point C,"},{"startTime":221.54,"endTime":223.9,"body":"so they'll intersect"},{"startTime":221.54,"endTime":223.9,"body":"at a point that can be"},{"startTime":223.9,"endTime":226.17,"body":"the next vertex of your pentagon, right?"},{"startTime":226.17,"endTime":228.9,"body":"I mean, actually isn't quite"},{"startTime":226.17,"endTime":228.9,"body":"the way you do it in real life,"},{"startTime":228.9,"endTime":232.558,"body":"but that would definitely allow"},{"startTime":228.9,"endTime":232.558,"body":"you to do this construction."},{"startTime":232.558,"endTime":235.27,"body":"So it depends on knowing"},{"startTime":232.558,"endTime":235.27,"body":"the length of this diagonal."},{"startTime":235.27,"endTime":237.382,"body":"So can we find the length of"},{"startTime":235.27,"endTime":237.382,"body":"diagonal of a regular pentagon?"},{"startTime":237.382,"endTime":242.331,"body":"You can, with through"},{"startTime":237.382,"endTime":242.331,"body":"the wonders of triangles,"},{"startTime":242.331,"endTime":243.96,"body":"which are the best"},{"startTime":242.331,"endTime":243.96,"body":"things, here's a pentagon,"},{"startTime":243.96,"endTime":245.81,"body":"and I've drawn in all the diagonals,"},{"startTime":245.81,"endTime":248.9,"body":"so it's got this kind of"},{"startTime":245.81,"endTime":248.9,"body":"pentagram design in it."},{"startTime":248.9,"endTime":252.07,"body":"And whenever two diagonals"},{"startTime":248.9,"endTime":252.07,"body":"cross each other,"},{"startTime":252.07,"endTime":254.07,"body":"they each cut each other"},{"startTime":254.07,"endTime":258.29,"body":"into a long bit, technical"},{"startTime":254.07,"endTime":258.29,"body":"term, and a short bit."},{"startTime":258.29,"endTime":261.08,"body":"I've put the long bit in a"},{"startTime":258.29,"endTime":261.08,"body":"sort of pale green color,"},{"startTime":261.08,"endTime":264.01,"body":"and the short bit is an orange color."},{"startTime":264.01,"endTime":267.23,"body":"And we are going to find the"},{"startTime":264.01,"endTime":267.23,"body":"whole length X by finding out"},{"startTime":267.23,"endTime":269.69,"body":"the length of the long bit, and"},{"startTime":267.23,"endTime":269.69,"body":"the length of the short bit,"},{"startTime":269.69,"endTime":271.27,"body":"and adding them together."},{"startTime":271.27,"endTime":274.55,"body":"And it's going to be three"},{"startTime":271.27,"endTime":274.55,"body":"pairs of triangles here,"},{"startTime":274.55,"endTime":276.12,"body":"so if it's been a while"},{"startTime":274.55,"endTime":276.12,"body":"since you've done geometry,"},{"startTime":276.12,"endTime":278.27,"body":"you can just sort of daydream"},{"startTime":276.12,"endTime":278.27,"body":"for a little, for a minute,"},{"startTime":278.27,"endTime":279.71,"body":"but don't really, though,"},{"startTime":279.71,"endTime":281.69,"body":"'cause it's very, very fun and exciting."},{"startTime":281.69,"endTime":283.621,"body":"So diagonal is the long"},{"startTime":281.69,"endTime":283.621,"body":"bit and the short bit,"},{"startTime":283.621,"endTime":285.6,"body":"and here's two triangles."},{"startTime":285.6,"endTime":286.87,"body":"They're the same triangle."},{"startTime":286.87,"endTime":288.06,"body":"They're congruent to each other."},{"startTime":288.06,"endTime":289.24,"body":"How do I know this?"},{"startTime":289.24,"endTime":291.15,"body":"Well, they're both isosceles right?"},{"startTime":291.15,"endTime":293.297,"body":"They've both got two equal sides."},{"startTime":293.297,"endTime":296.97,"body":"And the angle between those"},{"startTime":293.297,"endTime":296.97,"body":"two equal sides is the same,"},{"startTime":296.97,"endTime":299.26,"body":"because it's the angle between diagonals"},{"startTime":299.26,"endTime":301.09,"body":"of a regular pentagon."},{"startTime":301.09,"endTime":303.937,"body":"For the blue triangle, it's"},{"startTime":301.09,"endTime":303.937,"body":"the small, regular pentagon,"},{"startTime":303.937,"endTime":306.98,"body":"and for the red one,"},{"startTime":303.937,"endTime":306.98,"body":"it's the big pentagon,"},{"startTime":306.98,"endTime":309.79,"body":"so isosceles with the same"},{"startTime":306.98,"endTime":309.79,"body":"angle between the equal sides,"},{"startTime":309.79,"endTime":311.5,"body":"and they share the base,"},{"startTime":311.5,"endTime":313.43,"body":"so those things are congruent triangles."},{"startTime":313.43,"endTime":316.825,"body":"So in particular, that tells"},{"startTime":313.43,"endTime":316.825,"body":"us that there are two sides"},{"startTime":316.825,"endTime":319.01,"body":"that I've marked there with a little line,"},{"startTime":319.01,"endTime":320.784,"body":"those lines are equal in length."},{"startTime":320.784,"endTime":322.51,"body":"What is that line?"},{"startTime":322.51,"endTime":324.92,"body":"It's this short bit of a diagonal."},{"startTime":324.92,"endTime":326.46,"body":"So we've got that, okay."},{"startTime":326.46,"endTime":328.033,"body":"Now for two more triangles."},{"startTime":328.91,"endTime":330.02,"body":"They don't look like triangles,"},{"startTime":330.02,"endTime":331.93,"body":"'cause one of them's on top of the other,"},{"startTime":331.93,"endTime":333.35,"body":"but there's a green triangle here,"},{"startTime":333.35,"endTime":335.44,"body":"which is the whole height of the pentagon,"},{"startTime":335.44,"endTime":338.42,"body":"and it's an isosceles triangle"},{"startTime":338.42,"endTime":340.51,"body":"with the two sloped sides are equal there."},{"startTime":340.51,"endTime":342.48,"body":"They're the diagonals of the pentagon,"},{"startTime":342.48,"endTime":344.91,"body":"and then you've got the"},{"startTime":342.48,"endTime":344.91,"body":"base, which is a side"},{"startTime":344.91,"endTime":347.04,"body":"of the pentagon, that's"},{"startTime":344.91,"endTime":347.04,"body":"the green triangle,"},{"startTime":347.04,"endTime":349.73,"body":"and then the yellow one is similar to it."},{"startTime":349.73,"endTime":351.86,"body":"They're not congruent, but"},{"startTime":349.73,"endTime":351.86,"body":"they're similar triangles,"},{"startTime":351.86,"endTime":353.04,"body":"'cause again, it's isosceles."},{"startTime":353.04,"endTime":355.71,"body":"It's got to the two sloped"},{"startTime":353.04,"endTime":355.71,"body":"sides, though, this time"},{"startTime":355.71,"endTime":357.64,"body":"with the long bit of the diagonal,"},{"startTime":357.64,"endTime":360.193,"body":"and then the base is the"},{"startTime":357.64,"endTime":360.193,"body":"short bit of the diagonal."},{"startTime":361.34,"endTime":363.04,"body":"So those are similar triangles."},{"startTime":363.04,"endTime":364.61,"body":"What does that tell us?"},{"startTime":364.61,"endTime":366.14,"body":"It tells us that the ratio"},{"startTime":366.14,"endTime":368.47,"body":"of corresponding sides is the same."},{"startTime":368.47,"endTime":373.103,"body":"So if you take the green"},{"startTime":368.47,"endTime":373.103,"body":"triangle, it's short, it's base,"},{"startTime":373.103,"endTime":375.8,"body":"it's just the side of"},{"startTime":373.103,"endTime":375.8,"body":"the pentagon, it's one,"},{"startTime":375.8,"endTime":380.61,"body":"and the slope side is X, so"},{"startTime":375.8,"endTime":380.61,"body":"that one to X is the same"},{"startTime":380.61,"endTime":383.76,"body":"as the base of the yellow"},{"startTime":380.61,"endTime":383.76,"body":"triangle, which is the short bit"},{"startTime":383.76,"endTime":387.25,"body":"of the diagonal, to the"},{"startTime":383.76,"endTime":387.25,"body":"long bit of diagonal."},{"startTime":387.25,"endTime":389.89,"body":"So you've got this ratio going on."},{"startTime":389.89,"endTime":392.54,"body":"One to X is the same as short to long."},{"startTime":392.54,"endTime":396.2,"body":"So the division of the"},{"startTime":392.54,"endTime":396.2,"body":"diagonal has this nice property"},{"startTime":396.2,"endTime":397.98,"body":"that the proportion of it it's the same"},{"startTime":397.98,"endTime":400.81,"body":"as the side to the whole diagonal."},{"startTime":400.81,"endTime":402.93,"body":"We haven't found them"},{"startTime":400.81,"endTime":402.93,"body":"yet, but we're about to."},{"startTime":402.93,"endTime":405.2,"body":"Here's our final pair of triangles,"},{"startTime":405.2,"endTime":408.1,"body":"so a purple and orange triangle there."},{"startTime":408.1,"endTime":410.25,"body":"Again, they are the same as each other."},{"startTime":410.25,"endTime":414.74,"body":"They are, they share their base,"},{"startTime":410.25,"endTime":414.74,"body":"and they're both isosceles,"},{"startTime":414.74,"endTime":418.776,"body":"and they've both got the"},{"startTime":414.74,"endTime":418.776,"body":"internal angle of a pentagon"},{"startTime":418.776,"endTime":422.09,"body":"as their top angle"},{"startTime":418.776,"endTime":422.09,"body":"there, opposite the base."},{"startTime":422.09,"endTime":423.698,"body":"So those are congruent triangles,"},{"startTime":423.698,"endTime":427.54,"body":"and so all of the"},{"startTime":423.698,"endTime":427.54,"body":"corresponding sides match up."},{"startTime":427.54,"endTime":431.8,"body":"So those two sides that are"},{"startTime":427.54,"endTime":431.8,"body":"edges on the orange triangle,"},{"startTime":431.8,"endTime":435.24,"body":"they're sides of the pentagon,"},{"startTime":431.8,"endTime":435.24,"body":"those have length one,"},{"startTime":435.24,"endTime":438.04,"body":"'cause it's a pentagon with"},{"startTime":435.24,"endTime":438.04,"body":"side one, and so that means"},{"startTime":438.04,"endTime":441.51,"body":"the sides on that purple"},{"startTime":438.04,"endTime":441.51,"body":"triangle, which are the long bit"},{"startTime":441.51,"endTime":443.81,"body":"of the diagonal, they are also length one."},{"startTime":443.81,"endTime":445.54,"body":"So we know now that the long bit"},{"startTime":445.54,"endTime":449.096,"body":"of the diagonal is just"},{"startTime":445.54,"endTime":449.096,"body":"length one, and the short bit,"},{"startTime":449.096,"endTime":451.2,"body":"well, that's got to be X minus one,"},{"startTime":451.2,"endTime":452.86,"body":"'cause the whole thing is X."},{"startTime":452.86,"endTime":454.57,"body":"So what could we do now?"},{"startTime":454.57,"endTime":456.53,"body":"We now can get a relationship between"},{"startTime":456.53,"endTime":460.23,"body":"an equation involving X,"},{"startTime":456.53,"endTime":460.23,"body":"'cause this one to X equals"},{"startTime":460.23,"endTime":463.73,"body":"short to long gives us this,"},{"startTime":460.23,"endTime":463.73,"body":"turn it into a fraction,"},{"startTime":463.73,"endTime":467.41,"body":"one over X equals X minus one over one."},{"startTime":467.41,"endTime":469.88,"body":"And if you rearrange that a"},{"startTime":467.41,"endTime":469.88,"body":"bit to multiply on both sides"},{"startTime":469.88,"endTime":473.78,"body":"by X, for example, look,"},{"startTime":469.88,"endTime":473.78,"body":"there's a quadratic equation,"},{"startTime":473.78,"endTime":476.21,"body":"and we can solve quadratic"},{"startTime":473.78,"endTime":476.21,"body":"equations if we remember,"},{"startTime":476.21,"endTime":479.61,"body":"you know how to do that, but"},{"startTime":476.21,"endTime":479.61,"body":"that is a solvable problem."},{"startTime":479.61,"endTime":482.17,"body":"And if you solve that"},{"startTime":479.61,"endTime":482.17,"body":"problem, you get an answer,"},{"startTime":482.17,"endTime":486.87,"body":"that X is a half times root five plus one,"},{"startTime":486.87,"endTime":489.655,"body":"which is about 1.6, one eight-ish..."},{"startTime":489.655,"endTime":491.942,"body":"It's an irrational number."},{"startTime":491.942,"endTime":495.93,"body":"There's no fraction"},{"startTime":491.942,"endTime":495.93,"body":"that it is this exactly,"},{"startTime":495.93,"endTime":497.94,"body":"but approximately 1.618."},{"startTime":497.94,"endTime":501.22,"body":"And that we now call phi we"},{"startTime":497.94,"endTime":501.22,"body":"call that, it's a golden number,"},{"startTime":501.22,"endTime":506.05,"body":"so-called, and the golden"},{"startTime":501.22,"endTime":506.05,"body":"ratio is one to phi."},{"startTime":506.05,"endTime":509.927,"body":"So that is the length of the"},{"startTime":506.05,"endTime":509.927,"body":"diagonal of a regular pentagon,"},{"startTime":509.927,"endTime":512.763,"body":"and that's where you see it"},{"startTime":509.927,"endTime":512.763,"body":"in, for example, in Euclid."},{"startTime":514.88,"endTime":519.11,"body":"We've got a nice property"},{"startTime":514.88,"endTime":519.11,"body":"here that this line,"},{"startTime":519.11,"endTime":521.27,"body":"this number, the sort of X here,"},{"startTime":521.27,"endTime":522.55,"body":"or we're now calling it phi,"},{"startTime":522.55,"endTime":525.2,"body":"'cause that's the more common"},{"startTime":522.55,"endTime":525.2,"body":"name for the golden number."},{"startTime":525.2,"endTime":528.25,"body":"When you subdivide it into"},{"startTime":525.2,"endTime":528.25,"body":"a long bit and a short bit"},{"startTime":528.25,"endTime":531.08,"body":"in the way we have,"},{"startTime":528.25,"endTime":531.08,"body":"those numbers are related"},{"startTime":531.08,"endTime":533.14,"body":"in the same way that"},{"startTime":531.08,"endTime":533.14,"body":"the long bit is related"},{"startTime":533.14,"endTime":536.84,"body":"to the whole thing, so"},{"startTime":533.14,"endTime":536.84,"body":"one to X is just up here."},{"startTime":536.84,"endTime":539.07,"body":"We can say it now in terms of this phi."},{"startTime":539.07,"endTime":542.34,"body":"One to phi is the same"},{"startTime":539.07,"endTime":542.34,"body":"as phi minus one to one."},{"startTime":542.34,"endTime":547.24,"body":"So that's a kind of nice"},{"startTime":542.34,"endTime":547.24,"body":"equality of ratios there."},{"startTime":547.24,"endTime":549.08,"body":"That allows us to produce something"},{"startTime":549.08,"endTime":550.97,"body":"called a golden rectangle."},{"startTime":550.97,"endTime":554.35,"body":"Now, this is supposed to"},{"startTime":550.97,"endTime":554.35,"body":"be the, often claimed to be"},{"startTime":554.35,"endTime":556.57,"body":"the most beautiful rectangle there is."},{"startTime":556.57,"endTime":558.86,"body":"I'm not sure I necessarily believe that."},{"startTime":558.86,"endTime":560.277,"body":"I think there have been"},{"startTime":558.86,"endTime":560.277,"body":"various tests done,"},{"startTime":560.277,"endTime":562.03,"body":"you know, experiments, which,"},{"startTime":562.03,"endTime":564.27,"body":"tick which rectangle you"},{"startTime":562.03,"endTime":564.27,"body":"find the most beautiful,"},{"startTime":564.27,"endTime":566.181,"body":"and golden rectangles"},{"startTime":564.27,"endTime":566.181,"body":"don't necessarily win"},{"startTime":566.181,"endTime":567.76,"body":"such competitions."},{"startTime":567.76,"endTime":570.145,"body":"However, it does have a lovely property."},{"startTime":570.145,"endTime":572.37,"body":"If you take a rectangle with sides"},{"startTime":572.37,"endTime":574.67,"body":"in the proportion one to phi,"},{"startTime":574.67,"endTime":576.57,"body":"and you slice a square out of it,"},{"startTime":576.57,"endTime":580.16,"body":"get rid of a square from"},{"startTime":576.57,"endTime":580.16,"body":"that, then what's left,"},{"startTime":580.16,"endTime":582.49,"body":"that rectangle that's left over, well,"},{"startTime":582.49,"endTime":585.54,"body":"its sides are going to be,"},{"startTime":582.49,"endTime":585.54,"body":"its short side is going to be"},{"startTime":585.54,"endTime":588.74,"body":"phi minus one, and it's long"},{"startTime":585.54,"endTime":588.74,"body":"side is going to be one,"},{"startTime":588.74,"endTime":591.32,"body":"and we know, because I've put"},{"startTime":588.74,"endTime":591.32,"body":"a little golden crib sheet"},{"startTime":591.32,"endTime":592.85,"body":"up in the top right hand side to remind us"},{"startTime":592.85,"endTime":594.32,"body":"what we know about phi."},{"startTime":594.32,"endTime":597.78,"body":"We know that one to"},{"startTime":594.32,"endTime":597.78,"body":"phi, so the proportions"},{"startTime":597.78,"endTime":599.45,"body":"of the original rectangle,"},{"startTime":599.45,"endTime":601.923,"body":"is equal to phi minus one to one."},{"startTime":601.923,"endTime":605.55,"body":"So this smaller rectangle"},{"startTime":601.923,"endTime":605.55,"body":"we've produced is"},{"startTime":605.55,"endTime":608.05,"body":"exactly the same shape"},{"startTime":605.55,"endTime":608.05,"body":"as the original one."},{"startTime":608.05,"endTime":610.17,"body":"You've got another"},{"startTime":608.05,"endTime":610.17,"body":"smaller golden rectangle,"},{"startTime":610.17,"endTime":612.02,"body":"and you could keep cutting"},{"startTime":610.17,"endTime":612.02,"body":"squares out of this,"},{"startTime":612.02,"endTime":614.17,"body":"and each time you get another"},{"startTime":612.02,"endTime":614.17,"body":"smaller golden rectangle,"},{"startTime":614.17,"endTime":615.62,"body":"you keep going forever and ever,"},{"startTime":615.62,"endTime":619.033,"body":"so it's other beautiful"},{"startTime":615.62,"endTime":619.033,"body":"property of this rectangle."},{"startTime":620.44,"endTime":622.2,"body":"Okay, so that sets a few things"},{"startTime":622.2,"endTime":625.247,"body":"about this wonderful golden"},{"startTime":622.2,"endTime":625.247,"body":"number, or golden ratio."},{"startTime":625.247,"endTime":626.68,"body":"The golden number is phi,"},{"startTime":626.68,"endTime":628.59,"body":"and the golden ratio is one to phi,"},{"startTime":628.59,"endTime":631.903,"body":"so yeah, a number on"},{"startTime":628.59,"endTime":631.903,"body":"its own isn't a ratio."},{"startTime":631.903,"endTime":634.9,"body":"I want to tell you about the"},{"startTime":631.903,"endTime":634.9,"body":"link to the Fibonacci sequence,"},{"startTime":634.9,"endTime":637.68,"body":"because that's another"},{"startTime":634.9,"endTime":637.68,"body":"really cool property"},{"startTime":637.68,"endTime":639.54,"body":"of this golden number."},{"startTime":639.54,"endTime":642.54,"body":"I've put Fibonacci in quotation marks,"},{"startTime":642.54,"endTime":644.83,"body":"because it's not Fibonacci sequence."},{"startTime":644.83,"endTime":647.39,"body":"Like, he didn't invent"},{"startTime":644.83,"endTime":647.39,"body":"it, he popularized it."},{"startTime":647.39,"endTime":650.65,"body":"He put it in a book that"},{"startTime":647.39,"endTime":650.65,"body":"he'd wrote, and with a,"},{"startTime":650.65,"endTime":653.19,"body":"rather dare I say it,"},{"startTime":650.65,"endTime":653.19,"body":"slightly fatuous example"},{"startTime":653.19,"endTime":655.68,"body":"about rabbits' populations."},{"startTime":655.68,"endTime":657.73,"body":"They don't really, you know,"},{"startTime":655.68,"endTime":657.73,"body":"that's not really what happens"},{"startTime":657.73,"endTime":658.59,"body":"with rabbit populations,"},{"startTime":658.59,"endTime":662.1,"body":"however, he brought it, he popularized it."},{"startTime":662.1,"endTime":664.02,"body":"But this sequence had"},{"startTime":662.1,"endTime":664.02,"body":"been known, you know,"},{"startTime":664.02,"endTime":666.45,"body":"in India, and other places"},{"startTime":664.02,"endTime":666.45,"body":"for hundreds of years"},{"startTime":666.45,"endTime":669.16,"body":"before Fibonacci, however,"},{"startTime":666.45,"endTime":669.16,"body":"this is what it's known as,"},{"startTime":669.16,"endTime":671.8,"body":"so I thought I should"},{"startTime":669.16,"endTime":671.8,"body":"at least say that word."},{"startTime":671.8,"endTime":672.73,"body":"What is the sequence?"},{"startTime":672.73,"endTime":674.47,"body":"Well, I've put the first few terms here."},{"startTime":674.47,"endTime":676.096,"body":"You may or may not have seen it before,"},{"startTime":676.096,"endTime":679.88,"body":"that the sequence rule"},{"startTime":676.096,"endTime":679.88,"body":"is to make the next term,"},{"startTime":679.88,"endTime":681.58,"body":"you just add the previous two."},{"startTime":681.58,"endTime":684.55,"body":"So if you look, there's an eight up there,"},{"startTime":684.55,"endTime":688.173,"body":"eight is made from adding three"},{"startTime":684.55,"endTime":688.173,"body":"and five, and then five plus"},{"startTime":688.173,"endTime":691.377,"body":"eight is 13, eight plus"},{"startTime":688.173,"endTime":691.377,"body":"13 is 21, and so on."},{"startTime":691.377,"endTime":694.537,"body":"So the rule is, if you"},{"startTime":691.377,"endTime":694.537,"body":"wanted a formula for it,"},{"startTime":694.537,"endTime":697.887,"body":"you get the N plus first"},{"startTime":694.537,"endTime":697.887,"body":"term, F N plus one,"},{"startTime":697.887,"endTime":699.6,"body":"you take FN, the Nth term,"},{"startTime":699.6,"endTime":702.46,"body":"and then add the N minus first term."},{"startTime":702.46,"endTime":704.502,"body":"So that's the rule for making"},{"startTime":702.46,"endTime":704.502,"body":"the Fibonacci sequence,"},{"startTime":704.502,"endTime":708.98,"body":"start with two ones,"},{"startTime":704.502,"endTime":708.98,"body":"and then you just turn"},{"startTime":708.98,"endTime":710.55,"body":"the handle, and keep going."},{"startTime":710.55,"endTime":712.69,"body":"What's that got to do"},{"startTime":710.55,"endTime":712.69,"body":"with the golden number?"},{"startTime":712.69,"endTime":716.681,"body":"Well, if you take the"},{"startTime":712.69,"endTime":716.681,"body":"ratios of consecutive terms,"},{"startTime":716.681,"endTime":721.261,"body":"so you find out one over one,"},{"startTime":716.681,"endTime":721.261,"body":"two over one, three over two,"},{"startTime":721.261,"endTime":723.22,"body":"five over three, and so on,"},{"startTime":723.22,"endTime":726.52,"body":"so you work out all of these ratios."},{"startTime":726.52,"endTime":728.37,"body":"You don't have to do it"},{"startTime":726.52,"endTime":728.37,"body":"right now in your head,"},{"startTime":728.37,"endTime":731.7,"body":"but I've started to put"},{"startTime":728.37,"endTime":731.7,"body":"what this is, you know,"},{"startTime":731.7,"endTime":733.445,"body":"underneath them as a decimal."},{"startTime":733.445,"endTime":738.445,"body":"I've stopped at three decimal"},{"startTime":733.445,"endTime":738.445,"body":"places just, you know,"},{"startTime":738.64,"endTime":741.53,"body":"'cause of space, but you can"},{"startTime":738.64,"endTime":741.53,"body":"see what starts to happen."},{"startTime":741.53,"endTime":743.501,"body":"It moves around a little"},{"startTime":741.53,"endTime":743.501,"body":"bit at the beginning,"},{"startTime":743.501,"endTime":745.182,"body":"but it starts to settle down,"},{"startTime":745.182,"endTime":747.8,"body":"and by the sort of seventh or eighth term,"},{"startTime":747.8,"endTime":752.8,"body":"you are getting 1.618, 1.618,"},{"startTime":747.8,"endTime":752.8,"body":"1.618, and a few, you know,"},{"startTime":754.06,"endTime":757.18,"body":"infinitely many terms after"},{"startTime":754.06,"endTime":757.18,"body":"that, but I've put those."},{"startTime":757.18,"endTime":760.74,"body":"So it looks very much"},{"startTime":757.18,"endTime":760.74,"body":"as if this is settling"},{"startTime":760.74,"endTime":765.34,"body":"towards some limit, as"},{"startTime":760.74,"endTime":765.34,"body":"you know, your terms,"},{"startTime":765.34,"endTime":767.35,"body":"the number of terms get"},{"startTime":765.34,"endTime":767.35,"body":"higher and higher and higher,"},{"startTime":767.35,"endTime":769.1,"body":"you're getting closer and"},{"startTime":767.35,"endTime":769.1,"body":"closer, it looks like,"},{"startTime":769.1,"endTime":772.98,"body":"to a number which seems"},{"startTime":769.1,"endTime":772.98,"body":"to resemble our phi,"},{"startTime":772.98,"endTime":774.32,"body":"our golden number."},{"startTime":774.32,"endTime":778.03,"body":"So why might this be, why might this be?"},{"startTime":778.03,"endTime":781.388,"body":"Well, if it does, if"},{"startTime":778.03,"endTime":781.388,"body":"this is going to end up"},{"startTime":781.388,"endTime":783.37,"body":"at sort of getting closer"},{"startTime":781.388,"endTime":783.37,"body":"and closer and closer"},{"startTime":783.37,"endTime":785.68,"body":"to some limit, which"},{"startTime":783.37,"endTime":785.68,"body":"is what it looks like,"},{"startTime":785.68,"endTime":789.265,"body":"if that's true, then it means"},{"startTime":785.68,"endTime":789.265,"body":"once you get large enough,"},{"startTime":789.265,"endTime":791.76,"body":"every time you do this, you are very close"},{"startTime":791.76,"endTime":792.83,"body":"to whatever this number is."},{"startTime":792.83,"endTime":794.12,"body":"Let's call it L for the moment."},{"startTime":794.12,"endTime":796.1,"body":"We don't want to jump to any conclusions."},{"startTime":796.1,"endTime":798.92,"body":"So you've got sort of"},{"startTime":796.1,"endTime":798.92,"body":"the N plus first term,"},{"startTime":798.92,"endTime":800.87,"body":"FN plus one over FN."},{"startTime":800.87,"endTime":803.95,"body":"It's roughly the same"},{"startTime":800.87,"endTime":803.95,"body":"as FN over FN minus one,"},{"startTime":803.95,"endTime":805.98,"body":"which is roughly this L, whatever it is"},{"startTime":805.98,"endTime":807.65,"body":"we're hoping to find."},{"startTime":807.65,"endTime":811.26,"body":"So now all we need to do"},{"startTime":807.65,"endTime":811.26,"body":"is to take that equation"},{"startTime":811.26,"endTime":814.18,"body":"that gives us, or that"},{"startTime":811.26,"endTime":814.18,"body":"expression that gives us,"},{"startTime":814.18,"endTime":816.085,"body":"you know, the next term in the sequence."},{"startTime":816.085,"endTime":820.65,"body":"If I divide that through by"},{"startTime":816.085,"endTime":820.65,"body":"FN, you'll see what happens,"},{"startTime":820.65,"endTime":823.6,"body":"I've done it there, then"},{"startTime":820.65,"endTime":823.6,"body":"what you end up with is,"},{"startTime":823.6,"endTime":827.35,"body":"right, this ratio on the left,"},{"startTime":827.35,"endTime":829.12,"body":"you've got FN plus one over FN."},{"startTime":829.12,"endTime":831.32,"body":"That's equal to our L,"},{"startTime":829.12,"endTime":831.32,"body":"assuming approximately,"},{"startTime":831.32,"endTime":834.44,"body":"then FN over FN, well, that's just one,"},{"startTime":834.44,"endTime":837.51,"body":"and then FN minus one over FN, so that's"},{"startTime":837.51,"endTime":840.56,"body":"that L sort of the other"},{"startTime":837.51,"endTime":840.56,"body":"way up, it's one over L."},{"startTime":840.56,"endTime":842.24,"body":"So we get this expression,"},{"startTime":842.24,"endTime":844.744,"body":"L is equal to one plus one over L,"},{"startTime":844.744,"endTime":849.07,"body":"and that, I'll slightly"},{"startTime":844.744,"endTime":849.07,"body":"rearrange it, just so that"},{"startTime":849.07,"endTime":852.44,"body":"I can show you it's exactly"},{"startTime":849.07,"endTime":852.44,"body":"the same as the expression"},{"startTime":852.44,"endTime":854.95,"body":"one over phi equals phi"},{"startTime":852.44,"endTime":854.95,"body":"minus one that we know"},{"startTime":854.95,"endTime":857.24,"body":"about the golden ratio, the golden number."},{"startTime":857.24,"endTime":859.04,"body":"So one over L equal L minus one."},{"startTime":859.04,"endTime":862.19,"body":"So they, these two things"},{"startTime":859.04,"endTime":862.19,"body":"satisfy the exact same equation,"},{"startTime":862.19,"endTime":865.08,"body":"and so that's why when"},{"startTime":862.19,"endTime":865.08,"body":"you solve that equation,"},{"startTime":865.08,"endTime":867.23,"body":"you get the same answer that comes out."},{"startTime":867.23,"endTime":868.96,"body":"And it's true of the Fibonacci sequence."},{"startTime":868.96,"endTime":871.02,"body":"If you take the ratios"},{"startTime":868.96,"endTime":871.02,"body":"of consecutive terms,"},{"startTime":871.02,"endTime":874.653,"body":"it does end up closer and"},{"startTime":871.02,"endTime":874.653,"body":"closer to the golden number."},{"startTime":875.54,"endTime":877.59,"body":"So that's a really nice fact"},{"startTime":875.54,"endTime":877.59,"body":"about the Fibonacci sequence,"},{"startTime":877.59,"endTime":879.552,"body":"and about the golden number."},{"startTime":879.552,"endTime":881.81,"body":"We hear a lot about the golden ratio,"},{"startTime":881.81,"endTime":885.11,"body":"and how it's supposed to be,"},{"startTime":881.81,"endTime":885.11,"body":"you know, everywhere in nature,"},{"startTime":885.11,"endTime":886.75,"body":"and you know, the human body."},{"startTime":886.75,"endTime":889.31,"body":"A beautiful human face"},{"startTime":886.75,"endTime":889.31,"body":"is supposed to have loads"},{"startTime":889.31,"endTime":891.636,"body":"of golden ratios in it, and"},{"startTime":889.31,"endTime":891.636,"body":"golden, you know, everywhere."},{"startTime":891.636,"endTime":893.6,"body":"Let's think about that a little bit,"},{"startTime":893.6,"endTime":895.439,"body":"and where did that come from?"},{"startTime":895.439,"endTime":899.2,"body":"Part of where that comes"},{"startTime":895.439,"endTime":899.2,"body":"from, that belief came from is"},{"startTime":899.2,"endTime":902.78,"body":"not really the fault of Luca"},{"startTime":899.2,"endTime":902.78,"body":"Pacioli, who wrote a book"},{"startTime":902.78,"endTime":905.5,"body":"in 1509 called Divina Proportione"},{"startTime":905.5,"endTime":908.639,"body":"about the golden proportion,"},{"startTime":905.5,"endTime":908.639,"body":"the golden ratio,"},{"startTime":908.639,"endTime":912.9,"body":"but this is the start to"},{"startTime":908.639,"endTime":912.9,"body":"the trouble, I would say."},{"startTime":912.9,"endTime":913.733,"body":"So he wrote this book."},{"startTime":913.733,"endTime":916.84,"body":"This is hundreds of years before"},{"startTime":913.733,"endTime":916.84,"body":"the word golden ratio was,"},{"startTime":916.84,"endTime":918.85,"body":"the phrase was invented,"},{"startTime":918.85,"endTime":922.056,"body":"but he though phi was"},{"startTime":918.85,"endTime":922.056,"body":"amazing and wonderful."},{"startTime":922.056,"endTime":924.677,"body":"He wrote this book called"},{"startTime":922.056,"endTime":924.677,"body":"The Divine Proportion,"},{"startTime":924.677,"endTime":929.677,"body":"which a lot of it was about"},{"startTime":924.677,"endTime":929.677,"body":"the golden number, and he gave"},{"startTime":930.01,"endTime":932.83,"body":"lots of properties of it and"},{"startTime":930.01,"endTime":932.83,"body":"said, you know, it's fantastic."},{"startTime":932.83,"endTime":934.24,"body":"13 properties he gave."},{"startTime":934.24,"endTime":936.59,"body":"That's not cause 13 is a Fibonacci number,"},{"startTime":936.59,"endTime":938.08,"body":"but because he was trying to connect it"},{"startTime":938.08,"endTime":940.62,"body":"in all the ways he could"},{"startTime":938.08,"endTime":940.62,"body":"think of to the divine,"},{"startTime":940.62,"endTime":942.81,"body":"and in particular, the Christian God,"},{"startTime":942.81,"endTime":945.23,"body":"and there were 13 people"},{"startTime":942.81,"endTime":945.23,"body":"at the last supper, okay?"},{"startTime":945.23,"endTime":947.66,"body":"So therefore, he'd done 13 properties."},{"startTime":947.66,"endTime":950.639,"body":"I'm not sure which one"},{"startTime":947.66,"endTime":950.639,"body":"is Judas, I don't know."},{"startTime":950.639,"endTime":953.53,"body":"But what else did he say?"},{"startTime":953.53,"endTime":956.26,"body":"It's like the divine."},{"startTime":956.26,"endTime":957.757,"body":"Why is it?"},{"startTime":957.757,"endTime":959.52,"body":"So one reason is he said,"},{"startTime":957.757,"endTime":959.52,"body":"you know, I said earlier,"},{"startTime":959.52,"endTime":961.37,"body":"you couldn't represent it as a fraction,"},{"startTime":961.37,"endTime":964.54,"body":"so it's not that it's irrational."},{"startTime":964.54,"endTime":966.27,"body":"I mean, it is irrational, but that's not,"},{"startTime":966.27,"endTime":968.13,"body":"we're not saying that about the divine."},{"startTime":968.13,"endTime":970.3,"body":"What we're saying is you"},{"startTime":968.13,"endTime":970.3,"body":"can never fully know it."},{"startTime":970.3,"endTime":971.16,"body":"You can never fully know it,"},{"startTime":971.16,"endTime":973.61,"body":"because you can never fully"},{"startTime":971.16,"endTime":973.61,"body":"write it down as a fraction."},{"startTime":973.61,"endTime":977.539,"body":"It goes on forever, and so does God,"},{"startTime":977.539,"endTime":978.7,"body":"and you can't fully know God."},{"startTime":978.7,"endTime":979.9,"body":"What else?"},{"startTime":979.9,"endTime":983.65,"body":"When you define the golden"},{"startTime":979.9,"endTime":983.65,"body":"ratio, you do it in terms"},{"startTime":983.65,"endTime":986.61,"body":"of this, you know, the small"},{"startTime":983.65,"endTime":986.61,"body":"part to the large part is"},{"startTime":986.61,"endTime":988.78,"body":"the same ratio as the"},{"startTime":986.61,"endTime":988.78,"body":"large part to the whole."},{"startTime":988.78,"endTime":991.43,"body":"It's defined in terms of three quantities"},{"startTime":991.43,"endTime":993.85,"body":"that are all related to each other,"},{"startTime":993.85,"endTime":997.04,"body":"and so that's a bit like"},{"startTime":993.85,"endTime":997.04,"body":"the Holy Trinity, he says."},{"startTime":997.04,"endTime":998.76,"body":"So there, and then he gives lots"},{"startTime":998.76,"endTime":1001.903,"body":"of wonderful geometrical"},{"startTime":998.76,"endTime":1001.903,"body":"properties that I'm more qualified"},{"startTime":1001.903,"endTime":1004.93,"body":"to know what I think of,"},{"startTime":1004.93,"endTime":1007.1,"body":"and illustrated here is one of those."},{"startTime":1007.1,"endTime":1010.177,"body":"So you've got here that the"},{"startTime":1007.1,"endTime":1010.177,"body":"red lines are showing you"},{"startTime":1010.177,"endTime":1012.02,"body":"one of the Platonic solids"},{"startTime":1012.02,"endTime":1014.47,"body":"of the five Platonic"},{"startTime":1012.02,"endTime":1014.47,"body":"solids, the icosahedron."},{"startTime":1014.47,"endTime":1017.2,"body":"It's made of 20 equilateral triangles,"},{"startTime":1017.2,"endTime":1020.26,"body":"and the amazing, this"},{"startTime":1017.2,"endTime":1020.26,"body":"is a wonderful thing."},{"startTime":1020.26,"endTime":1025.26,"body":"You can put the vertices"},{"startTime":1020.26,"endTime":1025.26,"body":"of this into groups of four"},{"startTime":1025.295,"endTime":1030.295,"body":"that make up three mutually"},{"startTime":1025.295,"endTime":1030.295,"body":"perpendicular golden rectangles."},{"startTime":1030.74,"endTime":1033.52,"body":"And they're sitting"},{"startTime":1030.74,"endTime":1033.52,"body":"inside those 12 vertices,"},{"startTime":1033.52,"endTime":1036.831,"body":"make up three groups of"},{"startTime":1033.52,"endTime":1036.831,"body":"four vertices that define"},{"startTime":1036.831,"endTime":1039.33,"body":"these golden rectangles that are sitting"},{"startTime":1039.33,"endTime":1040.98,"body":"at all the right angles for each other"},{"startTime":1040.98,"endTime":1044.49,"body":"in the middle of the icosahedron,"},{"startTime":1040.98,"endTime":1044.49,"body":"which is really nice."},{"startTime":1044.49,"endTime":1046.44,"body":"So Luca Pacioli gives lots of properties."},{"startTime":1046.44,"endTime":1048.0,"body":"He said this is a wonderful thing,"},{"startTime":1048.0,"endTime":1051.23,"body":"the divine proportion, it's amazing."},{"startTime":1051.23,"endTime":1054.04,"body":"Now, also in his book, in this same book,"},{"startTime":1054.04,"endTime":1056.37,"body":"there are sections about architecture,"},{"startTime":1056.37,"endTime":1058.967,"body":"and so of course,"},{"startTime":1056.37,"endTime":1058.967,"body":"naturally you might think,"},{"startTime":1058.967,"endTime":1061.33,"body":"\"Well, so he's going to say"},{"startTime":1058.967,"endTime":1061.33,"body":"how the divine proportion,"},{"startTime":1061.33,"endTime":1065.3,"body":"how it's involved in"},{"startTime":1061.33,"endTime":1065.3,"body":"architecture,\" but no references"},{"startTime":1065.3,"endTime":1068.54,"body":"at all to the golden"},{"startTime":1065.3,"endTime":1068.54,"body":"ratio in those sections."},{"startTime":1068.54,"endTime":1071.47,"body":"It's just not mentioned"},{"startTime":1068.54,"endTime":1071.47,"body":"at all, which is curious."},{"startTime":1071.47,"endTime":1074.45,"body":"So I have to think,"},{"startTime":1071.47,"endTime":1074.45,"body":"\"Oh, okay, how, why do?\""},{"startTime":1074.45,"endTime":1076.06,"body":"You can sort of see why"},{"startTime":1074.45,"endTime":1076.06,"body":"it might have been thought"},{"startTime":1076.06,"endTime":1077.93,"body":"that they would, but if"},{"startTime":1076.06,"endTime":1077.93,"body":"you look at the book,"},{"startTime":1077.93,"endTime":1080.45,"body":"it's not in there at all in"},{"startTime":1077.93,"endTime":1080.45,"body":"the sections on architecture."},{"startTime":1080.45,"endTime":1084.52,"body":"And these words, golden"},{"startTime":1080.45,"endTime":1084.52,"body":"ratio, golden number, phi,"},{"startTime":1084.52,"endTime":1088.42,"body":"that symbol, these are"},{"startTime":1084.52,"endTime":1088.42,"body":"first used, not by Pacioli,"},{"startTime":1088.42,"endTime":1090.81,"body":"not by Vitruvius, not by Leonardo,"},{"startTime":1090.81,"endTime":1093.73,"body":"but in the late 19th century by a chap"},{"startTime":1093.73,"endTime":1096.77,"body":"called Adolf Zeising,"},{"startTime":1093.73,"endTime":1096.77,"body":"who wrote a book saying,"},{"startTime":1096.77,"endTime":1098.24,"body":"you know, this is an amazing thing,"},{"startTime":1098.24,"endTime":1102.044,"body":"and it's all over nature,"},{"startTime":1098.24,"endTime":1102.044,"body":"and art, and architecture."},{"startTime":1102.044,"endTime":1105.67,"body":"It's, I would say it's sort"},{"startTime":1102.044,"endTime":1105.67,"body":"of a bit in these things,"},{"startTime":1105.67,"endTime":1108.16,"body":"but perhaps not to the"},{"startTime":1105.67,"endTime":1108.16,"body":"extent that he was claiming."},{"startTime":1108.16,"endTime":1111.69,"body":"For example, that phi, why is"},{"startTime":1108.16,"endTime":1111.69,"body":"it, why did he call it phi?"},{"startTime":1111.69,"endTime":1114.16,"body":"That's the first letter"},{"startTime":1111.69,"endTime":1114.16,"body":"in the name Phidias,"},{"startTime":1114.16,"endTime":1116.28,"body":"the architect who designed the Parthenon,"},{"startTime":1116.28,"endTime":1120.03,"body":"but there's no record of any, you know,"},{"startTime":1120.03,"endTime":1122.3,"body":"anyone writing that we are going to use"},{"startTime":1122.3,"endTime":1124.66,"body":"this particular ratio in the design."},{"startTime":1124.66,"endTime":1129.51,"body":"So let's have a little"},{"startTime":1124.66,"endTime":1129.51,"body":"look, then, at architecture."},{"startTime":1129.51,"endTime":1131.56,"body":"Is the golden ratio used in architecture?"},{"startTime":1131.56,"endTime":1134.1,"body":"If it is, where is it used?"},{"startTime":1134.1,"endTime":1136.69,"body":"So there are lots and lots of ratios"},{"startTime":1136.69,"endTime":1138.68,"body":"and proportions in architecture."},{"startTime":1138.68,"endTime":1143.54,"body":"It's absolutely critical and"},{"startTime":1138.68,"endTime":1143.54,"body":"important to our understanding"},{"startTime":1143.54,"endTime":1145.83,"body":"of form and design, and, you know,"},{"startTime":1145.83,"endTime":1149.48,"body":"architects are constantly choosing ratios"},{"startTime":1149.48,"endTime":1152.89,"body":"such that they can get, you"},{"startTime":1149.48,"endTime":1152.89,"body":"know, nicely shaped buildings,"},{"startTime":1152.89,"endTime":1156.04,"body":"but not so much the golden"},{"startTime":1152.89,"endTime":1156.04,"body":"ratio, certainly not in the work"},{"startTime":1156.04,"endTime":1158.04,"body":"we talked about Vitruvius,"},{"startTime":1156.04,"endTime":1158.04,"body":"the Roman architect."},{"startTime":1158.04,"endTime":1161.69,"body":"He wrote in De Architectura,"},{"startTime":1158.04,"endTime":1161.69,"body":"he wrote a kind of manual"},{"startTime":1161.69,"endTime":1164.4,"body":"for how to build all the"},{"startTime":1161.69,"endTime":1164.4,"body":"possible kinds of temple"},{"startTime":1164.4,"endTime":1166.021,"body":"you might want to build, for example,"},{"startTime":1166.021,"endTime":1169.01,"body":"and very, very detailed instructions."},{"startTime":1169.01,"endTime":1170.76,"body":"You know, he was absolutely clear"},{"startTime":1170.76,"endTime":1173.49,"body":"that there are right ways"},{"startTime":1170.76,"endTime":1173.49,"body":"and wrong ways. (chuckles)"},{"startTime":1173.49,"endTime":1174.79,"body":"Here are the rules you need."},{"startTime":1174.79,"endTime":1177.89,"body":"Here are the proportions of the, you know,"},{"startTime":1177.89,"endTime":1180.339,"body":"six different kinds of"},{"startTime":1177.89,"endTime":1180.339,"body":"temple, or whatever."},{"startTime":1180.339,"endTime":1182.64,"body":"This is how many columns wide."},{"startTime":1182.64,"endTime":1185.27,"body":"This is how many columns"},{"startTime":1182.64,"endTime":1185.27,"body":"deep they should be,"},{"startTime":1185.27,"endTime":1187.48,"body":"and this is the gap between each column."},{"startTime":1187.48,"endTime":1190.61,"body":"So not about how, you"},{"startTime":1187.48,"endTime":1190.61,"body":"know, your column should be"},{"startTime":1190.61,"endTime":1192.03,"body":"three cubits wide, or whatever."},{"startTime":1192.03,"endTime":1194.96,"body":"It's what your column"},{"startTime":1192.03,"endTime":1194.96,"body":"width is, and then the gap"},{"startTime":1194.96,"endTime":1197.25,"body":"between the columns in"},{"startTime":1194.96,"endTime":1197.25,"body":"terms of the column width,"},{"startTime":1197.25,"endTime":1199.167,"body":"and then how many of the columns you have,"},{"startTime":1199.167,"endTime":1201.33,"body":"and so you can work out."},{"startTime":1201.33,"endTime":1202.917,"body":"He doesn't explicitly say that, you know,"},{"startTime":1202.917,"endTime":1205.971,"body":"the Eustyle temple is in"},{"startTime":1202.917,"endTime":1205.971,"body":"this ratio of rectangle,"},{"startTime":1205.971,"endTime":1209.16,"body":"but he explains what the"},{"startTime":1205.971,"endTime":1209.16,"body":"columns, how many columns,"},{"startTime":1209.16,"endTime":1211.2,"body":"and how wide, how the gaps should be."},{"startTime":1211.2,"endTime":1212.81,"body":"And you look out what the ratios then are"},{"startTime":1212.81,"endTime":1216.2,"body":"for the rectangles, the"},{"startTime":1212.81,"endTime":1216.2,"body":"footprints of these temples,"},{"startTime":1216.2,"endTime":1219.189,"body":"and they're not, none"},{"startTime":1216.2,"endTime":1219.189,"body":"of them is golden ratio,"},{"startTime":1219.189,"endTime":1220.9,"body":"I'm pretty sure."},{"startTime":1220.9,"endTime":1222.27,"body":"So I've worked out a few of them."},{"startTime":1222.27,"endTime":1223.97,"body":"Didn't find any golden ratios."},{"startTime":1223.97,"endTime":1226.37,"body":"I think the issue is, and"},{"startTime":1223.97,"endTime":1226.37,"body":"it's the same sort of thing"},{"startTime":1226.37,"endTime":1230.21,"body":"that happens with the"},{"startTime":1226.37,"endTime":1230.21,"body":"claims about the human face."},{"startTime":1230.21,"endTime":1232.433,"body":"There are lots of numbers that you can,"},{"startTime":1232.433,"endTime":1233.57,"body":"that you could measure."},{"startTime":1233.57,"endTime":1237.01,"body":"There are lots of measurements"},{"startTime":1233.57,"endTime":1237.01,"body":"you could take of a temple."},{"startTime":1237.01,"endTime":1239.89,"body":"And if you take enough"},{"startTime":1237.01,"endTime":1239.89,"body":"measurements, and you're willing"},{"startTime":1239.89,"endTime":1241.197,"body":"enough to approximate and say,"},{"startTime":1241.197,"endTime":1244.951,"body":"\"Oh, that's roughly 1.618-ish,\" you know,"},{"startTime":1244.951,"endTime":1249.951,"body":"\"That 1.7 was nearly"},{"startTime":1244.951,"endTime":1249.951,"body":"1.618,\" then you can find"},{"startTime":1250.03,"endTime":1253.45,"body":"golden ratios, but there's"},{"startTime":1250.03,"endTime":1253.45,"body":"not really, I would say,"},{"startTime":1253.45,"endTime":1255.99,"body":"enough evidence that the Greeks"},{"startTime":1255.99,"endTime":1257.872,"body":"and Roman architects"},{"startTime":1255.99,"endTime":1257.872,"body":"were consciously using"},{"startTime":1257.872,"endTime":1260.04,"body":"this wonderful ratio."},{"startTime":1260.04,"endTime":1261.61,"body":"They had other wonderful ratios."},{"startTime":1261.61,"endTime":1265.031,"body":"So that's the state of play, I would say,"},{"startTime":1265.031,"endTime":1266.12,"body":"with the Greek and Roman architecture."},{"startTime":1266.12,"endTime":1268.891,"body":"So is there any use of the"},{"startTime":1266.12,"endTime":1268.891,"body":"golden ratio in architecture?"},{"startTime":1268.891,"endTime":1270.21,"body":"Yes, there is."},{"startTime":1270.21,"endTime":1271.49,"body":"Yes, it was, luckily."},{"startTime":1271.49,"endTime":1272.323,"body":"Yes, there is."},{"startTime":1272.323,"endTime":1274.8,"body":"Never ask a question like"},{"startTime":1272.323,"endTime":1274.8,"body":"that if the answer isn't yes."},{"startTime":1274.8,"endTime":1278.96,"body":"Yes, I want to give you probably"},{"startTime":1274.8,"endTime":1278.96,"body":"the most important example"},{"startTime":1278.96,"endTime":1282.643,"body":"of its use, and that's by the"},{"startTime":1278.96,"endTime":1282.643,"body":"Swiss architect Le Corbusier."},{"startTime":1283.69,"endTime":1288.045,"body":"This is a commemorative coin"},{"startTime":1283.69,"endTime":1288.045,"body":"produced by the Swiss mint,"},{"startTime":1288.045,"endTime":1290.75,"body":"representing his Modulor system,"},{"startTime":1290.75,"endTime":1291.71,"body":"which I want to talk to you about."},{"startTime":1291.71,"endTime":1296.71,"body":"So Le Corbusier, he is well"},{"startTime":1291.71,"endTime":1296.71,"body":"known, for good or ill,"},{"startTime":1299.18,"endTime":1302.928,"body":"for having a very firm belief system"},{"startTime":1302.928,"endTime":1305.74,"body":"about what architecture"},{"startTime":1302.928,"endTime":1305.74,"body":"should be, let's say."},{"startTime":1305.74,"endTime":1308.52,"body":"Notoriously wanted to"},{"startTime":1305.74,"endTime":1308.52,"body":"not down half of Paris,"},{"startTime":1308.52,"endTime":1311.84,"body":"and replace what had been there"},{"startTime":1308.52,"endTime":1311.84,"body":"with 60-story tower blocks,"},{"startTime":1311.84,"endTime":1313.37,"body":"because that was efficient,"},{"startTime":1311.84,"endTime":1313.37,"body":"and that was the right,"},{"startTime":1313.37,"endTime":1315.39,"body":"you know, that was the thing you should do"},{"startTime":1315.39,"endTime":1316.98,"body":"with the wonderful new building materials."},{"startTime":1316.98,"endTime":1320.839,"body":"So he was able to take advantage"},{"startTime":1320.839,"endTime":1325.69,"body":"of some new building materials,"},{"startTime":1320.839,"endTime":1325.69,"body":"like reinforced concrete,"},{"startTime":1325.69,"endTime":1330.69,"body":"to have more freedom to design"},{"startTime":1325.69,"endTime":1330.69,"body":"structures and buildings."},{"startTime":1331.11,"endTime":1333.3,"body":"No longer did you have"},{"startTime":1331.11,"endTime":1333.3,"body":"to have a wall there,"},{"startTime":1333.3,"endTime":1334.69,"body":"because otherwise the house falls down."},{"startTime":1334.69,"endTime":1337.82,"body":"You could have it as open"},{"startTime":1334.69,"endTime":1337.82,"body":"plan as you wanted to,"},{"startTime":1337.82,"endTime":1342.01,"body":"and you could put walls"},{"startTime":1337.82,"endTime":1342.01,"body":"where they were best to serve"},{"startTime":1342.01,"endTime":1344.91,"body":"the people who were using"},{"startTime":1342.01,"endTime":1344.91,"body":"or inhabiting that building."},{"startTime":1344.91,"endTime":1348.39,"body":"So he famously said a house"},{"startTime":1344.91,"endTime":1348.39,"body":"is a machine for living in."},{"startTime":1348.39,"endTime":1349.4,"body":"That's what it should be."},{"startTime":1349.4,"endTime":1354.23,"body":"That's, we're making this"},{"startTime":1349.4,"endTime":1354.23,"body":"machine to serve this purpose"},{"startTime":1354.23,"endTime":1357.82,"body":"of being inhabited by humans,"},{"startTime":1354.23,"endTime":1357.82,"body":"and so it shouldn't have stuff"},{"startTime":1357.82,"endTime":1359.779,"body":"that isn't important to that."},{"startTime":1359.779,"endTime":1362.4,"body":"It should have precisely"},{"startTime":1359.779,"endTime":1362.4,"body":"the things it needs."},{"startTime":1362.4,"endTime":1364.39,"body":"So he was all sort of, you"},{"startTime":1362.4,"endTime":1364.39,"body":"know, let's be efficient."},{"startTime":1364.39,"endTime":1366.03,"body":"Let's almost like let's, you know,"},{"startTime":1366.03,"endTime":1369.52,"body":"we've got the model T for"},{"startTime":1366.03,"endTime":1369.52,"body":"cars at that time, you know,"},{"startTime":1369.52,"endTime":1370.9,"body":"let's have a production line for houses,"},{"startTime":1370.9,"endTime":1373.86,"body":"and let's make them, let's"},{"startTime":1370.9,"endTime":1373.86,"body":"think of them in this way."},{"startTime":1373.86,"endTime":1376.88,"body":"He wasn't too keen on ornament,"},{"startTime":1376.88,"endTime":1379.52,"body":"and things like this, decor, no."},{"startTime":1379.52,"endTime":1382.48,"body":"He said art was all right,"},{"startTime":1379.52,"endTime":1382.48,"body":"but just sort of have"},{"startTime":1382.48,"endTime":1384.96,"body":"nice, little, fancy, decorative touches,"},{"startTime":1384.96,"endTime":1387.44,"body":"and, you know, Corinthian"},{"startTime":1384.96,"endTime":1387.44,"body":"columns and things."},{"startTime":1387.44,"endTime":1390.403,"body":"No, we've got this for living in."},{"startTime":1392.53,"endTime":1396.88,"body":"So one of his frustrations,"},{"startTime":1392.53,"endTime":1396.88,"body":"or the things he was unhappy"},{"startTime":1396.88,"endTime":1401.88,"body":"about was the system of"},{"startTime":1396.88,"endTime":1401.88,"body":"measurements that was included"},{"startTime":1401.92,"endTime":1404.38,"body":"in the metric system, and it's because"},{"startTime":1404.38,"endTime":1407.82,"body":"it's not really to do with human scales."},{"startTime":1407.82,"endTime":1411.1,"body":"The meter, unlike so many"},{"startTime":1407.82,"endTime":1411.1,"body":"other unit of measurements"},{"startTime":1411.1,"endTime":1413.013,"body":"that we had, have had at other times,"},{"startTime":1413.013,"endTime":1416.13,"body":"it's nothing to do with the"},{"startTime":1413.013,"endTime":1416.13,"body":"human body and human scales."},{"startTime":1416.13,"endTime":1417.73,"body":"The meter is defined"},{"startTime":1416.13,"endTime":1417.73,"body":"in terms of, you know,"},{"startTime":1417.73,"endTime":1420.04,"body":"millionths of the"},{"startTime":1417.73,"endTime":1420.04,"body":"circumference of the earth,"},{"startTime":1420.04,"endTime":1423.32,"body":"which okay, fine, but you"},{"startTime":1420.04,"endTime":1423.32,"body":"know, if you're a human,"},{"startTime":1423.32,"endTime":1425.97,"body":"actually, a foot or,"},{"startTime":1423.32,"endTime":1425.97,"body":"you know, even a cubit,"},{"startTime":1425.97,"endTime":1427.54,"body":"something like that"},{"startTime":1425.97,"endTime":1427.54,"body":"that's defined in terms"},{"startTime":1427.54,"endTime":1431.91,"body":"of the human body, feels,"},{"startTime":1427.54,"endTime":1431.91,"body":"to him, more natural."},{"startTime":1431.91,"endTime":1434.04,"body":"And what he wanted to do was"},{"startTime":1431.91,"endTime":1434.04,"body":"to take that idea of having"},{"startTime":1434.04,"endTime":1436.6,"body":"human scales, things that"},{"startTime":1434.04,"endTime":1436.6,"body":"are relevant to humans,"},{"startTime":1436.6,"endTime":1438.45,"body":"and to create, as he said,"},{"startTime":1438.45,"endTime":1441.69,"body":"a range of harmonious measurements"},{"startTime":1438.45,"endTime":1441.69,"body":"to suit the human scale,"},{"startTime":1441.69,"endTime":1445.12,"body":"universally applicable to"},{"startTime":1441.69,"endTime":1445.12,"body":"architecture and to mechanics."},{"startTime":1445.12,"endTime":1447.01,"body":"So the idea is you've got this scale"},{"startTime":1447.01,"endTime":1449.89,"body":"of measurements that works with humans."},{"startTime":1449.89,"endTime":1451.81,"body":"You know, these are the"},{"startTime":1449.89,"endTime":1451.81,"body":"things we're building"},{"startTime":1451.81,"endTime":1455.73,"body":"the buildings for, from the"},{"startTime":1451.81,"endTime":1455.73,"body":"smallest scale, from, you know,"},{"startTime":1455.73,"endTime":1458.872,"body":"like, the door handle in"},{"startTime":1455.73,"endTime":1458.872,"body":"your home, to the furniture,"},{"startTime":1458.872,"endTime":1463.19,"body":"to the room, to the house, to"},{"startTime":1458.872,"endTime":1463.19,"body":"the city, all of those scales,"},{"startTime":1463.19,"endTime":1466.93,"body":"there should be something in"},{"startTime":1463.19,"endTime":1466.93,"body":"this range of measurements"},{"startTime":1466.93,"endTime":1469.29,"body":"that will suit, and then"},{"startTime":1466.93,"endTime":1469.29,"body":"they'll all harmoniously work"},{"startTime":1469.29,"endTime":1471.25,"body":"with each other, and everything will be"},{"startTime":1471.25,"endTime":1472.75,"body":"beautiful and wonderful."},{"startTime":1472.75,"endTime":1473.92,"body":"So it's all very good."},{"startTime":1473.92,"endTime":1477.39,"body":"So what is this scale he's"},{"startTime":1473.92,"endTime":1477.39,"body":"got, this Modulor system?"},{"startTime":1477.39,"endTime":1482.39,"body":"So it's based on the height"},{"startTime":1477.39,"endTime":1482.39,"body":"of a man, 1.83 meters,"},{"startTime":1482.56,"endTime":1485.23,"body":"and then with a raised arm,"},{"startTime":1482.56,"endTime":1485.23,"body":"so that brings the total"},{"startTime":1485.23,"endTime":1489.01,"body":"up to 2.23 meters, which is 89 inches."},{"startTime":1489.01,"endTime":1491.856,"body":"89 is, I'm sure you've"},{"startTime":1489.01,"endTime":1491.856,"body":"spotted, a Fibonacci number,"},{"startTime":1491.856,"endTime":1495.26,"body":"so that's where that comes from."},{"startTime":1495.26,"endTime":1497.08,"body":"What about this 1.83 meters?"},{"startTime":1497.08,"endTime":1500.56,"body":"Well, if you are, if"},{"startTime":1497.08,"endTime":1500.56,"body":"you're good at converting"},{"startTime":1500.56,"endTime":1502.42,"body":"between meters, and feet, and inches,"},{"startTime":1502.42,"endTime":1503.253,"body":"you might know what that is."},{"startTime":1503.253,"endTime":1508.253,"body":"Corbusier said he chose"},{"startTime":1503.253,"endTime":1508.253,"body":"that height for the man,"},{"startTime":1509.01,"endTime":1512.74,"body":"because in English novels,"},{"startTime":1509.01,"endTime":1512.74,"body":"the good-looking men,"},{"startTime":1512.74,"endTime":1515.91,"body":"who are usually policemen,"},{"startTime":1512.74,"endTime":1515.91,"body":"are six foot tall."},{"startTime":1515.91,"endTime":1519.52,"body":"So I don't know how to"},{"startTime":1515.91,"endTime":1519.52,"body":"deconstruct that, (chuckles)"},{"startTime":1519.52,"endTime":1521.13,"body":"but anyway, that's what he said."},{"startTime":1521.13,"endTime":1523.69,"body":"And so 1.83 meters is six foot tall."},{"startTime":1523.69,"endTime":1527.44,"body":"So we can have a discussion about whether,"},{"startTime":1527.44,"endTime":1530.223,"body":"how great or otherwise it might"},{"startTime":1527.44,"endTime":1530.223,"body":"be to put your whole design"},{"startTime":1530.223,"endTime":1534.07,"body":"based on a man who is six"},{"startTime":1530.223,"endTime":1534.07,"body":"foot tall, but there we are."},{"startTime":1534.07,"endTime":1535.48,"body":"That's where it came from."},{"startTime":1535.48,"endTime":1537.94,"body":"And you can see, I mean, you"},{"startTime":1535.48,"endTime":1537.94,"body":"can see in the design there are"},{"startTime":1537.94,"endTime":1539.63,"body":"things that look like golden rectangles,"},{"startTime":1539.63,"endTime":1541.17,"body":"and they are golden rectangles."},{"startTime":1541.17,"endTime":1544.683,"body":"So you take his whole height,"},{"startTime":1541.17,"endTime":1544.683,"body":"halfway up is the navel,"},{"startTime":1544.683,"endTime":1547.89,"body":"and then if you go right to"},{"startTime":1544.683,"endTime":1547.89,"body":"the top of upstretched arm,"},{"startTime":1547.89,"endTime":1549.79,"body":"that's a golden rectangle."},{"startTime":1549.79,"endTime":1553.16,"body":"But up to the point of the top"},{"startTime":1549.79,"endTime":1553.16,"body":"of the head, that's a square."},{"startTime":1553.16,"endTime":1555.25,"body":"So if you take a square out,"},{"startTime":1553.16,"endTime":1555.25,"body":"you get a rectangle left,"},{"startTime":1555.25,"endTime":1556.71,"body":"a golden rectangle left."},{"startTime":1556.71,"endTime":1558.69,"body":"And so then there's all this whole range"},{"startTime":1558.69,"endTime":1560.92,"body":"of golden rectangles"},{"startTime":1558.69,"endTime":1560.92,"body":"getting smaller and smaller."},{"startTime":1560.92,"endTime":1562.74,"body":"You can get them bigger"},{"startTime":1560.92,"endTime":1562.74,"body":"and bigger, and combined"},{"startTime":1562.74,"endTime":1565.3,"body":"with squares, they all mix"},{"startTime":1562.74,"endTime":1565.3,"body":"in beautifully together,"},{"startTime":1565.3,"endTime":1567.64,"body":"and you get this range of decreasing"},{"startTime":1567.64,"endTime":1570.57,"body":"and increasing measurements"},{"startTime":1567.64,"endTime":1570.57,"body":"that you can use."},{"startTime":1570.57,"endTime":1574.93,"body":"I want to show you just a couple"},{"startTime":1570.57,"endTime":1574.93,"body":"of the buildings or interiors"},{"startTime":1574.93,"endTime":1576.98,"body":"that he designed using this system."},{"startTime":1576.98,"endTime":1581.719,"body":"And I will mention"},{"startTime":1576.98,"endTime":1581.719,"body":"Modulor, that name, module,"},{"startTime":1581.719,"endTime":1583.86,"body":"and or, the French for gold."},{"startTime":1583.86,"endTime":1585.6,"body":"So that's, the clue is in the name."},{"startTime":1585.6,"endTime":1587.25,"body":"The golden ratio is everywhere,"},{"startTime":1587.25,"endTime":1588.96,"body":"and golden rectangles are everywhere."},{"startTime":1588.96,"endTime":1590.03,"body":"So this is an example."},{"startTime":1590.03,"endTime":1592.58,"body":"This is an apartment block in Marseille,"},{"startTime":1592.58,"endTime":1596.24,"body":"one of several that he"},{"startTime":1592.58,"endTime":1596.24,"body":"designed, and you can, you know,"},{"startTime":1596.24,"endTime":1598.64,"body":"just have a look, spot"},{"startTime":1596.24,"endTime":1598.64,"body":"some golden rectangles,"},{"startTime":1598.64,"endTime":1602.03,"body":"the tiles on the floor, on the ceiling."},{"startTime":1602.03,"endTime":1604.84,"body":"There's some windows that"},{"startTime":1602.03,"endTime":1604.84,"body":"are golden rectangles."},{"startTime":1604.84,"endTime":1606.907,"body":"So there's a few just spot there,"},{"startTime":1606.907,"endTime":1608.727,"body":"and then this is the"},{"startTime":1606.907,"endTime":1608.727,"body":"kitchen of one of these,"},{"startTime":1608.727,"endTime":1611.79,"body":"and you can look at those"},{"startTime":1608.727,"endTime":1611.79,"body":"cushions, golden rectangles."},{"startTime":1611.79,"endTime":1613.77,"body":"You've got the windows are as well."},{"startTime":1613.77,"endTime":1615.68,"body":"The little kind of the grid shapes"},{"startTime":1615.68,"endTime":1618.702,"body":"in that ventilation holes there,"},{"startTime":1615.68,"endTime":1618.702,"body":"they're golden rectangles."},{"startTime":1618.702,"endTime":1620.27,"body":"You've got some squares in there,"},{"startTime":1620.27,"endTime":1623.55,"body":"so the furniture as well,"},{"startTime":1620.27,"endTime":1623.55,"body":"golden rectangles everywhere."},{"startTime":1623.55,"endTime":1628.55,"body":"So I think for me, you know,"},{"startTime":1623.55,"endTime":1628.55,"body":"sort of pleasing to look at,"},{"startTime":1628.91,"endTime":1631.89,"body":"and when you, whether or not"},{"startTime":1628.91,"endTime":1631.89,"body":"you think golden rectangles"},{"startTime":1631.89,"endTime":1634.591,"body":"themselves are particularly"},{"startTime":1631.89,"endTime":1634.591,"body":"the most beautiful thing,"},{"startTime":1634.591,"endTime":1637.39,"body":"it's still pleasing that"},{"startTime":1634.591,"endTime":1637.39,"body":"you can have a system"},{"startTime":1637.39,"endTime":1638.96,"body":"where everything aligns"},{"startTime":1637.39,"endTime":1638.96,"body":"with everything else."},{"startTime":1638.96,"endTime":1641.469,"body":"You've got this shape repeated"},{"startTime":1638.96,"endTime":1641.469,"body":"at big and small scales."},{"startTime":1641.469,"endTime":1643.89,"body":"You can create smaller"},{"startTime":1641.469,"endTime":1643.89,"body":"and smaller, or bigger"},{"startTime":1643.89,"endTime":1647.06,"body":"and bigger golden rectangles"},{"startTime":1643.89,"endTime":1647.06,"body":"by taking squares away."},{"startTime":1647.06,"endTime":1649.97,"body":"So I think that's what"},{"startTime":1647.06,"endTime":1649.97,"body":"helps to make it harmonious."},{"startTime":1649.97,"endTime":1651.62,"body":"It all fits together beautifully."},{"startTime":1652.63,"endTime":1654.667,"body":"So that's architecture."},{"startTime":1654.667,"endTime":1658.873,"body":"We also hear claims about"},{"startTime":1654.667,"endTime":1658.873,"body":"the golden ratio in nature."},{"startTime":1659.807,"endTime":1661.97,"body":"Is the golden ratio everywhere in nature?"},{"startTime":1661.97,"endTime":1665.56,"body":"Well, again, I'd say ratios"},{"startTime":1661.97,"endTime":1665.56,"body":"are everywhere in nature."},{"startTime":1665.56,"endTime":1668.02,"body":"Some of them are golden,"},{"startTime":1665.56,"endTime":1668.02,"body":"some of them aren't,"},{"startTime":1668.02,"endTime":1668.95,"body":"and that's okay."},{"startTime":1668.95,"endTime":1670.69,"body":"It's all still interesting,"},{"startTime":1670.69,"endTime":1673.37,"body":"but I want to talk a little bit"},{"startTime":1670.69,"endTime":1673.37,"body":"about where they may appear,"},{"startTime":1673.37,"endTime":1676.37,"body":"where you do, perhaps, get"},{"startTime":1673.37,"endTime":1676.37,"body":"the golden ratio in nature."},{"startTime":1676.37,"endTime":1678.26,"body":"And first I will just say,"},{"startTime":1676.37,"endTime":1678.26,"body":"you know, I've mentioned"},{"startTime":1678.26,"endTime":1680.47,"body":"about the human face, and"},{"startTime":1678.26,"endTime":1680.47,"body":"how it's supposed to have"},{"startTime":1680.47,"endTime":1683.52,"body":"lots of golden rectangles"},{"startTime":1680.47,"endTime":1683.52,"body":"or golden ratios in it."},{"startTime":1683.52,"endTime":1686.48,"body":"And just like I said about"},{"startTime":1683.52,"endTime":1686.48,"body":"buildings, there are lots"},{"startTime":1686.48,"endTime":1689.75,"body":"of measurements you could"},{"startTime":1686.48,"endTime":1689.75,"body":"take of the human face,"},{"startTime":1689.75,"endTime":1692.76,"body":"and there are lots of human"},{"startTime":1689.75,"endTime":1692.76,"body":"faces, and who gets to decide"},{"startTime":1692.76,"endTime":1694.12,"body":"what's beautiful and what isn't?"},{"startTime":1694.12,"endTime":1695.67,"body":"And who gets to decide"},{"startTime":1694.12,"endTime":1695.67,"body":"which measurements are"},{"startTime":1695.67,"endTime":1697.74,"body":"the most important, and which aren't?"},{"startTime":1697.74,"endTime":1700.823,"body":"So one, just one tiny example I found."},{"startTime":1701.658,"endTime":1705.37,"body":"There's a claim that in the,"},{"startTime":1701.658,"endTime":1705.37,"body":"you know, in a beautiful face,"},{"startTime":1705.37,"endTime":1708.08,"body":"the distance, the ratio of"},{"startTime":1705.37,"endTime":1708.08,"body":"the distance from the center"},{"startTime":1708.08,"endTime":1710.084,"body":"of your nose to the start of your eye,"},{"startTime":1710.084,"endTime":1714.71,"body":"that distance, compared"},{"startTime":1710.084,"endTime":1714.71,"body":"to the width of your eye,"},{"startTime":1714.71,"endTime":1717.69,"body":"that's supposed to be the"},{"startTime":1714.71,"endTime":1717.69,"body":"golden ratio, one to phi."},{"startTime":1717.69,"endTime":1720.91,"body":"So the width of your eye is"},{"startTime":1717.69,"endTime":1720.91,"body":"supposed to be about 1.618 times"},{"startTime":1720.91,"endTime":1722.39,"body":"the distance from the center of your nose"},{"startTime":1722.39,"endTime":1723.69,"body":"to the start of your eye."},{"startTime":1723.69,"endTime":1725.63,"body":"Okay, I didn't measure my own face."},{"startTime":1725.63,"endTime":1727.505,"body":"You know I don't want to"},{"startTime":1725.63,"endTime":1727.505,"body":"set myself up for failure."},{"startTime":1727.505,"endTime":1730.31,"body":"Maybe it was the golden ratio, I dunno."},{"startTime":1730.31,"endTime":1732.17,"body":"Instead, I looked at"},{"startTime":1730.31,"endTime":1732.17,"body":"Audrey Hepburn's face."},{"startTime":1732.17,"endTime":1735.46,"body":"I thought, if it doesn't work"},{"startTime":1732.17,"endTime":1735.46,"body":"for her, then it doesn't work."},{"startTime":1735.46,"endTime":1739.41,"body":"And for her, that measurement,"},{"startTime":1735.46,"endTime":1739.41,"body":"that ratio was one to 2.3."},{"startTime":1739.41,"endTime":1742.801,"body":"So I thought, okay, it's"},{"startTime":1739.41,"endTime":1742.801,"body":"not true. (chuckles)"},{"startTime":1742.801,"endTime":1743.723,"body":"And this is, you know, I that's,"},{"startTime":1744.74,"endTime":1746.43,"body":"that's an N equals one example, but,"},{"startTime":1746.43,"endTime":1749.13,"body":"but there are so many measurements"},{"startTime":1746.43,"endTime":1749.13,"body":"you could take, right?"},{"startTime":1749.13,"endTime":1752.103,"body":"That particular, even if that"},{"startTime":1749.13,"endTime":1752.103,"body":"particular one is about 1.6,"},{"startTime":1753.308,"endTime":1756.71,"body":"what about, you know, the"},{"startTime":1753.308,"endTime":1756.71,"body":"ratio of the height of your ear"},{"startTime":1756.71,"endTime":1759.307,"body":"to the distance from your"},{"startTime":1756.71,"endTime":1759.307,"body":"chin to your eye or something?"},{"startTime":1759.307,"endTime":1761.33,"body":"You know, you could take so many hundreds"},{"startTime":1761.33,"endTime":1763.12,"body":"of different things,"},{"startTime":1761.33,"endTime":1763.12,"body":"and check them, and yes,"},{"startTime":1763.12,"endTime":1765.99,"body":"some of them are going to be about 1.618,"},{"startTime":1765.99,"endTime":1767.84,"body":"but some of them are going"},{"startTime":1765.99,"endTime":1767.84,"body":"to be completely different."},{"startTime":1767.84,"endTime":1770.15,"body":"So I think, you know, we got to be careful"},{"startTime":1770.15,"endTime":1771.96,"body":"about what is claimed here."},{"startTime":1771.96,"endTime":1773.04,"body":"But let's talk about spirals,"},{"startTime":1773.04,"endTime":1776.27,"body":"because you do get"},{"startTime":1773.04,"endTime":1776.27,"body":"lovely spirals in nature,"},{"startTime":1776.27,"endTime":1779.01,"body":"and sometimes people talk"},{"startTime":1776.27,"endTime":1779.01,"body":"about a Fibonacci spiral,"},{"startTime":1779.01,"endTime":1781.62,"body":"or a golden spiral, so what's that?"},{"startTime":1781.62,"endTime":1784.54,"body":"So here's how to make a Fibonacci spiral."},{"startTime":1784.54,"endTime":1787.95,"body":"You start off with the Fibonacci"},{"startTime":1784.54,"endTime":1787.95,"body":"sequence, 1, 1, 2, 3, 5, 8,"},{"startTime":1787.95,"endTime":1790.988,"body":"and so on, and you start,"},{"startTime":1787.95,"endTime":1790.988,"body":"I'm going to point to the,"},{"startTime":1790.988,"endTime":1793.069,"body":"you start here with the little red square."},{"startTime":1793.069,"endTime":1796.5,"body":"That has side one, so you"},{"startTime":1793.069,"endTime":1796.5,"body":"are taking a bunch of squares"},{"startTime":1796.5,"endTime":1800.21,"body":"whose side is, you know,"},{"startTime":1796.5,"endTime":1800.21,"body":"the next Fibonacci number."},{"startTime":1800.21,"endTime":1802.59,"body":"Put next to it, another"},{"startTime":1800.21,"endTime":1802.59,"body":"square of side 1, 1, 1,"},{"startTime":1802.59,"endTime":1805.0,"body":"and then two, so you add"},{"startTime":1802.59,"endTime":1805.0,"body":"a square of side two,"},{"startTime":1805.0,"endTime":1808.24,"body":"and you're kind of going"},{"startTime":1805.0,"endTime":1808.24,"body":"around 90 degrees each time,"},{"startTime":1808.24,"endTime":1809.99,"body":"and adding the next square"},{"startTime":1809.99,"endTime":1812.247,"body":"with a side the next Fibonacci number."},{"startTime":1812.247,"endTime":1814.05,"body":"And the reason it all fits together nicely"},{"startTime":1814.05,"endTime":1816.75,"body":"without gaps is because the sum"},{"startTime":1816.75,"endTime":1819.13,"body":"of the last two Fibonacci numbers equals"},{"startTime":1819.13,"endTime":1820.69,"body":"the next Fibonacci number."},{"startTime":1820.69,"endTime":1823.5,"body":"So if you want to, you know,"},{"startTime":1820.69,"endTime":1823.5,"body":"that square of side three,"},{"startTime":1823.5,"endTime":1825.96,"body":"that you can see there"},{"startTime":1823.5,"endTime":1825.96,"body":"goes next to a square"},{"startTime":1825.96,"endTime":1827.9,"body":"of side two and a square side one,"},{"startTime":1827.9,"endTime":1829.61,"body":"and two plus one is"},{"startTime":1827.9,"endTime":1829.61,"body":"three, so there's no gaps,"},{"startTime":1829.61,"endTime":1831.57,"body":"and it all fits together nicely."},{"startTime":1831.57,"endTime":1833.29,"body":"Where's the spiral coming in?"},{"startTime":1833.29,"endTime":1834.76,"body":"Well, in each of these squares,"},{"startTime":1834.76,"endTime":1837.312,"body":"you just put a quarter of a circle,"},{"startTime":1837.312,"endTime":1839.548,"body":"like this,"},{"startTime":1839.548,"endTime":1841.84,"body":"and that gives you something"},{"startTime":1839.548,"endTime":1841.84,"body":"that looks quite spirally."},{"startTime":1841.84,"endTime":1844.08,"body":"It's not exactly a spiral,"},{"startTime":1841.84,"endTime":1844.08,"body":"because it is just made"},{"startTime":1844.08,"endTime":1847.29,"body":"of a bunch of quarter"},{"startTime":1844.08,"endTime":1847.29,"body":"circles bolted together,"},{"startTime":1847.29,"endTime":1849.99,"body":"so the curvature is"},{"startTime":1847.29,"endTime":1849.99,"body":"changing kind of suddenly."},{"startTime":1849.99,"endTime":1852.149,"body":"It's not changing by much,"},{"startTime":1849.99,"endTime":1852.149,"body":"so it looks all right,"},{"startTime":1852.149,"endTime":1854.351,"body":"but it's not quite a spiral."},{"startTime":1854.351,"endTime":1856.811,"body":"Let me tell you what is a spiral."},{"startTime":1856.811,"endTime":1859.61,"body":"Here's two examples of spirals."},{"startTime":1859.61,"endTime":1861.571,"body":"There are different kinds of spiral."},{"startTime":1861.571,"endTime":1864.81,"body":"The one of the left is"},{"startTime":1861.571,"endTime":1864.81,"body":"called an Archimedean spiral."},{"startTime":1864.81,"endTime":1867.56,"body":"In that kind of spiral, so in each one,"},{"startTime":1867.56,"endTime":1869.79,"body":"you're kind of, you're"},{"startTime":1867.56,"endTime":1869.79,"body":"gradually increasing your angle,"},{"startTime":1869.79,"endTime":1871.09,"body":"and when the angle increases,"},{"startTime":1871.09,"endTime":1872.94,"body":"you get further away from the center."},{"startTime":1874.01,"endTime":1876.117,"body":"This one is called an Archimedean spiral,"},{"startTime":1876.117,"endTime":1880.42,"body":"and the distance from the"},{"startTime":1876.117,"endTime":1880.42,"body":"center is going up in multiples."},{"startTime":1880.42,"endTime":1883.237,"body":"So every time you do"},{"startTime":1880.42,"endTime":1883.237,"body":"another full revolution,"},{"startTime":1883.237,"endTime":1885.34,"body":"you kind of add another"},{"startTime":1883.237,"endTime":1885.34,"body":"multiple of whatever,"},{"startTime":1885.34,"endTime":1886.9,"body":"some constant you might have."},{"startTime":1886.9,"endTime":1890.13,"body":"So the one I've got here,"},{"startTime":1886.9,"endTime":1890.13,"body":"once you go one revolution"},{"startTime":1890.13,"endTime":1892.17,"body":"around your distance one from the center,"},{"startTime":1892.17,"endTime":1894.083,"body":"and two revolutions, two, distance two,"},{"startTime":1894.083,"endTime":1895.59,"body":"three revolutions, your distance three,"},{"startTime":1895.59,"endTime":1898.683,"body":"and so it's sort of adding"},{"startTime":1895.59,"endTime":1898.683,"body":"up like that each time"},{"startTime":1898.683,"endTime":1900.784,"body":"by some multiple of the angle."},{"startTime":1900.784,"endTime":1904.33,"body":"But this other kind of"},{"startTime":1900.784,"endTime":1904.33,"body":"spiral are on the right here,"},{"startTime":1904.33,"endTime":1905.63,"body":"it's more interesting, and it's more like"},{"startTime":1905.63,"endTime":1907.74,"body":"the kind of thing you see in nature,"},{"startTime":1907.74,"endTime":1911.13,"body":"because here, it's not about"},{"startTime":1907.74,"endTime":1911.13,"body":"multiples, it's about powers,"},{"startTime":1911.13,"endTime":1913.544,"body":"not quite powers of the"},{"startTime":1911.13,"endTime":1913.544,"body":"angle that I actually wrote."},{"startTime":1913.544,"endTime":1916.14,"body":"That's not quite what I"},{"startTime":1913.544,"endTime":1916.14,"body":"wanted to say, really,"},{"startTime":1916.14,"endTime":1919.6,"body":"but it's powers that this is changing by."},{"startTime":1919.6,"endTime":1921.57,"body":"So this time, every new"},{"startTime":1919.6,"endTime":1921.57,"body":"revolution you go by"},{"startTime":1921.57,"endTime":1923.83,"body":"in this particular example,"},{"startTime":1921.57,"endTime":1923.83,"body":"I've got what I call"},{"startTime":1923.83,"endTime":1925.965,"body":"a two spiral, the distance is doubling"},{"startTime":1925.965,"endTime":1927.76,"body":"after every revolution,"},{"startTime":1927.76,"endTime":1929.764,"body":"these are called logarithmic spirals."},{"startTime":1929.764,"endTime":1934.33,"body":"The reason is that the log, the logarithm"},{"startTime":1934.33,"endTime":1937.57,"body":"of the distance is dependent"},{"startTime":1934.33,"endTime":1937.57,"body":"on the angle, right?"},{"startTime":1937.57,"endTime":1939.87,"body":"So if you are, if you're"},{"startTime":1937.57,"endTime":1939.87,"body":"doing A level maths,"},{"startTime":1939.87,"endTime":1941.42,"body":"you might have done logarithms."},{"startTime":1942.686,"endTime":1945.18,"body":"And so this thing, what does it do for us?"},{"startTime":1945.18,"endTime":1947.29,"body":"You can see the distances"},{"startTime":1945.18,"endTime":1947.29,"body":"after every revolution"},{"startTime":1947.29,"endTime":1949.55,"body":"you're starting at, well,"},{"startTime":1947.29,"endTime":1949.55,"body":"at one point it's two,"},{"startTime":1949.55,"endTime":1951.49,"body":"then the next time round,"},{"startTime":1949.55,"endTime":1951.49,"body":"it's gone up to four,"},{"startTime":1951.49,"endTime":1954.17,"body":"it's doubled, and then it doubles"},{"startTime":1951.49,"endTime":1954.17,"body":"again to eight, and so on."},{"startTime":1954.17,"endTime":1957.35,"body":"Why is this more like"},{"startTime":1954.17,"endTime":1957.35,"body":"what happens in nature?"},{"startTime":1957.35,"endTime":1959.3,"body":"Well, it's because in a natural process,"},{"startTime":1959.3,"endTime":1962.27,"body":"when something is being created or growing"},{"startTime":1962.27,"endTime":1964.752,"body":"from a central point spiraling out,"},{"startTime":1964.752,"endTime":1968.71,"body":"the matter that's there"},{"startTime":1964.752,"endTime":1968.71,"body":"continues to expand and grow,"},{"startTime":1968.71,"endTime":1970.49,"body":"even after it's been produced."},{"startTime":1970.49,"endTime":1973.04,"body":"So this is true for something"},{"startTime":1970.49,"endTime":1973.04,"body":"like a spiral galaxy,"},{"startTime":1973.04,"endTime":1976.68,"body":"where the galaxy is, you"},{"startTime":1973.04,"endTime":1976.68,"body":"know, it's spinning very fast,"},{"startTime":1976.68,"endTime":1979.46,"body":"and as the matter out from the center,"},{"startTime":1979.46,"endTime":1982.24,"body":"it has still continued to"},{"startTime":1979.46,"endTime":1982.24,"body":"expand, but also in something"},{"startTime":1982.24,"endTime":1984.58,"body":"like a shell, it's"},{"startTime":1982.24,"endTime":1984.58,"body":"growing from the center,"},{"startTime":1984.58,"endTime":1986.695,"body":"but it doesn't suddenly stop growing,"},{"startTime":1986.695,"endTime":1987.528,"body":"and just sort of sit there,"},{"startTime":1987.528,"endTime":1989.0,"body":"like it would for an Archimedean spiral."},{"startTime":1989.0,"endTime":1990.98,"body":"So that is why you see, you do get"},{"startTime":1990.98,"endTime":1992.79,"body":"these kind of logarithmic spirals,"},{"startTime":1992.79,"endTime":1995.31,"body":"or spira mirabilis was"},{"startTime":1992.79,"endTime":1995.31,"body":"the name given to them"},{"startTime":1995.31,"endTime":1997.28,"body":"by the mathematician Bernoulli,"},{"startTime":1997.28,"endTime":1999.41,"body":"who did a lot of research on them."},{"startTime":1999.41,"endTime":2002.75,"body":"Sadly, on Bernoulli's gravestone,"},{"startTime":1999.41,"endTime":2002.75,"body":"I dunno if people know,"},{"startTime":2002.75,"endTime":2004.327,"body":"on Bernoulli's gravestone, they were like,"},{"startTime":2004.327,"endTime":2007.29,"body":"\"Oh, he did some stuff on"},{"startTime":2004.327,"endTime":2007.29,"body":"spirals, put a spiral on,\""},{"startTime":2007.29,"endTime":2009.698,"body":"and unfortunately, they"},{"startTime":2007.29,"endTime":2009.698,"body":"put an Archimedean spiral."},{"startTime":2009.698,"endTime":2011.87,"body":"(chuckles) They put the wrong"},{"startTime":2009.698,"endTime":2011.87,"body":"spiral on his gravestone."},{"startTime":2011.87,"endTime":2014.43,"body":"So, sorry, Bernoulli, that was done wrong."},{"startTime":2014.43,"endTime":2017.473,"body":"Anyway, these are beautiful spirals."},{"startTime":2017.473,"endTime":2019.903,"body":"They do look a bit like our golden spiral,"},{"startTime":2021.16,"endTime":2024.87,"body":"but what kind of logarithmic"},{"startTime":2021.16,"endTime":2024.87,"body":"spiral might be the best"},{"startTime":2024.87,"endTime":2027.8,"body":"approximation to this, or"},{"startTime":2024.87,"endTime":2027.8,"body":"what is this looking like?"},{"startTime":2027.8,"endTime":2030.38,"body":"So we can, there's a"},{"startTime":2027.8,"endTime":2030.38,"body":"bit of Fibonacci facts"},{"startTime":2030.38,"endTime":2031.213,"body":"I need to tell you."},{"startTime":2031.213,"endTime":2032.96,"body":"If you think which, if"},{"startTime":2031.213,"endTime":2032.96,"body":"we're trying to work out"},{"startTime":2032.96,"endTime":2037.275,"body":"how far this, this spiral-like"},{"startTime":2032.96,"endTime":2037.275,"body":"curve is from the center,"},{"startTime":2037.275,"endTime":2040.698,"body":"then what you're doing is at each,"},{"startTime":2040.698,"endTime":2042.66,"body":"after each kind of quarter turn,"},{"startTime":2042.66,"endTime":2047.37,"body":"you're traveling across a"},{"startTime":2042.66,"endTime":2047.37,"body":"square that's, you know,"},{"startTime":2047.37,"endTime":2049.15,"body":"the side of a Fibonacci number."},{"startTime":2049.15,"endTime":2050.53,"body":"So if you sort of keep"},{"startTime":2049.15,"endTime":2050.53,"body":"doing this, if you think"},{"startTime":2050.53,"endTime":2052.873,"body":"of the Pythagorean theorem,"},{"startTime":2050.53,"endTime":2052.873,"body":"the square of the distance is"},{"startTime":2052.873,"endTime":2055.52,"body":"the sum of the squares on the side, right?"},{"startTime":2055.52,"endTime":2058.97,"body":"So here, and you have"},{"startTime":2055.52,"endTime":2058.97,"body":"to slightly believe me,"},{"startTime":2058.97,"endTime":2060.4,"body":"'cause I'm going to say"},{"startTime":2058.97,"endTime":2060.4,"body":"something a bit vague,"},{"startTime":2060.4,"endTime":2064.51,"body":"but here, after kind of N"},{"startTime":2060.4,"endTime":2064.51,"body":"quarter turns, roughly speaking,"},{"startTime":2064.51,"endTime":2066.83,"body":"the square of the distance"},{"startTime":2064.51,"endTime":2066.83,"body":"you've traveled is the sum"},{"startTime":2066.83,"endTime":2070.65,"body":"of the first N squares"},{"startTime":2066.83,"endTime":2070.65,"body":"of Fibonacci numbers,"},{"startTime":2070.65,"endTime":2072.138,"body":"'cause those are the sides, right?"},{"startTime":2072.138,"endTime":2074.68,"body":"So you want to know what's"},{"startTime":2072.138,"endTime":2074.68,"body":"the sum of the squares"},{"startTime":2074.68,"endTime":2078.193,"body":"of the first N Fibonacci"},{"startTime":2074.68,"endTime":2078.193,"body":"numbers, and luckily for us,"},{"startTime":2078.193,"endTime":2081.41,"body":"that is known, and it's just the product"},{"startTime":2081.41,"endTime":2083.53,"body":"of the Nth and the N plus"},{"startTime":2081.41,"endTime":2083.53,"body":"first Fibonacci number."},{"startTime":2083.53,"endTime":2085.65,"body":"So that's not obvious at all,"},{"startTime":2083.53,"endTime":2085.65,"body":"and I'm not going to prove it"},{"startTime":2085.65,"endTime":2089.4,"body":"to you right now, but"},{"startTime":2085.65,"endTime":2089.4,"body":"this turns out to be true."},{"startTime":2089.4,"endTime":2092.475,"body":"So it's that number, actually,"},{"startTime":2089.4,"endTime":2092.475,"body":"the square root of that,"},{"startTime":2092.475,"endTime":2093.308,"body":"that we're interested in."},{"startTime":2093.308,"endTime":2095.77,"body":"That's the distance in the golden spiral"},{"startTime":2095.77,"endTime":2099.938,"body":"after N quarter turns, so"},{"startTime":2095.77,"endTime":2099.938,"body":"N of these sort of squares,"},{"startTime":2099.938,"endTime":2102.64,"body":"and actually, if we want"},{"startTime":2099.938,"endTime":2102.64,"body":"to account, you know,"},{"startTime":2102.64,"endTime":2105.66,"body":"with our logarithmic spiral,"},{"startTime":2102.64,"endTime":2105.66,"body":"it's how much you've multiplied"},{"startTime":2105.66,"endTime":2108.56,"body":"the distance by out that"},{"startTime":2105.66,"endTime":2108.56,"body":"you're interested in,"},{"startTime":2108.56,"endTime":2111.01,"body":"'cause you're multiplying"},{"startTime":2108.56,"endTime":2111.01,"body":"by that amount each time."},{"startTime":2111.01,"endTime":2112.5,"body":"So you want the ratio of this thing,"},{"startTime":2112.5,"endTime":2117.5,"body":"compared to the previous term,"},{"startTime":2112.5,"endTime":2117.5,"body":"or yes, and when you do that,"},{"startTime":2117.51,"endTime":2120.79,"body":"we remember that our"},{"startTime":2117.51,"endTime":2120.79,"body":"Fibonacci numbers, the ratio"},{"startTime":2120.79,"endTime":2123.298,"body":"of consecutive terms is the"},{"startTime":2120.79,"endTime":2123.298,"body":"golden ratio, approximately,"},{"startTime":2123.298,"endTime":2128.07,"body":"so you actually do get"},{"startTime":2123.298,"endTime":2128.07,"body":"phi coming out of this."},{"startTime":2128.07,"endTime":2131.54,"body":"So what this golden spiral"},{"startTime":2128.07,"endTime":2131.54,"body":"does, or approximately,"},{"startTime":2131.54,"endTime":2133.62,"body":"it approximately has a property that,"},{"startTime":2133.62,"endTime":2135.836,"body":"after every quarter turn,"},{"startTime":2133.62,"endTime":2135.836,"body":"you multiply the distance"},{"startTime":2135.836,"endTime":2138.69,"body":"from the center by phi."},{"startTime":2138.69,"endTime":2142.17,"body":"So after every whole turn,"},{"startTime":2138.69,"endTime":2142.17,"body":"you'd have done this four times,"},{"startTime":2142.17,"endTime":2145.05,"body":"so you'd have multiplied not"},{"startTime":2142.17,"endTime":2145.05,"body":"by two, like the example I had"},{"startTime":2145.05,"endTime":2147.94,"body":"before, but by phi to the power of four,"},{"startTime":2147.94,"endTime":2150.01,"body":"phi times phi, times phi, times phi."},{"startTime":2150.01,"endTime":2152.61,"body":"So actually what you're approximating is"},{"startTime":2152.61,"endTime":2155.65,"body":"a logarithmic phi to the four spiral."},{"startTime":2155.65,"endTime":2157.74,"body":"That's what this golden spiral does."},{"startTime":2157.74,"endTime":2159.18,"body":"Now, the question is, why would anything"},{"startTime":2159.18,"endTime":2160.97,"body":"in nature be interested in doing"},{"startTime":2160.97,"endTime":2163.76,"body":"a phi to the four spiral specifically?"},{"startTime":2163.76,"endTime":2165.75,"body":"And the answer is mostly they wouldn't."},{"startTime":2165.75,"endTime":2168.315,"body":"Mostly, the logarithmic spirals are there,"},{"startTime":2168.315,"endTime":2170.637,"body":"and there are loads of them to"},{"startTime":2168.315,"endTime":2170.637,"body":"be seen in natural phenomena,"},{"startTime":2170.637,"endTime":2173.621,"body":"in shells, for example, but if you measure"},{"startTime":2173.621,"endTime":2176.21,"body":"what number spiral it is,"},{"startTime":2176.21,"endTime":2179.1,"body":"it's usually not that"},{"startTime":2176.21,"endTime":2179.1,"body":"particular phi to the four one,"},{"startTime":2179.1,"endTime":2180.395,"body":"and why would it be?"},{"startTime":2180.395,"endTime":2181.93,"body":"There's a whole range, you know,"},{"startTime":2181.93,"endTime":2183.1,"body":"in nautilus shells and others."},{"startTime":2183.1,"endTime":2185.49,"body":"There's a range of"},{"startTime":2183.1,"endTime":2185.49,"body":"possibilities, and the ratio"},{"startTime":2185.49,"endTime":2189.612,"body":"that they are bound by,"},{"startTime":2185.49,"endTime":2189.612,"body":"you can find these things,"},{"startTime":2189.612,"endTime":2192.8,"body":"and you know that these are"},{"startTime":2189.612,"endTime":2192.8,"body":"interesting ratios in nature,"},{"startTime":2192.8,"endTime":2194.4,"body":"not usually the golden ratio."},{"startTime":2194.4,"endTime":2199.25,"body":"You do sometimes, for"},{"startTime":2194.4,"endTime":2199.25,"body":"real, see golden spirals,"},{"startTime":2199.25,"endTime":2201.0,"body":"or those, you know, the analogs of those"},{"startTime":2201.0,"endTime":2203.98,"body":"in algorithm spirals in"},{"startTime":2201.0,"endTime":2203.98,"body":"some special situations."},{"startTime":2203.98,"endTime":2207.731,"body":"For example, in sunflowers,"},{"startTime":2203.98,"endTime":2207.731,"body":"where the seeds are growing,"},{"startTime":2207.731,"endTime":2210.33,"body":"you do see them, and the"},{"startTime":2207.731,"endTime":2210.33,"body":"reason for that is actually"},{"startTime":2210.33,"endTime":2213.12,"body":"because phi is an irrational number,"},{"startTime":2213.12,"endTime":2215.85,"body":"it means after a whole"},{"startTime":2213.12,"endTime":2215.85,"body":"number of rotations,"},{"startTime":2215.85,"endTime":2218.11,"body":"you're very unlikely to"},{"startTime":2215.85,"endTime":2218.11,"body":"sort of be overlapping"},{"startTime":2218.11,"endTime":2219.39,"body":"something you've already had."},{"startTime":2219.39,"endTime":2222.19,"body":"So there's a good reason why you would,"},{"startTime":2222.19,"endTime":2224.43,"body":"it's often the case that you do get phi"},{"startTime":2224.43,"endTime":2226.38,"body":"in those kind of special situations."},{"startTime":2226.38,"endTime":2229.33,"body":"Most of the spirals in nature"},{"startTime":2226.38,"endTime":2229.33,"body":"are not golden spirals,"},{"startTime":2229.33,"endTime":2230.93,"body":"but just occasionally there are."},{"startTime":2231.89,"endTime":2234.303,"body":"There's something, some other"},{"startTime":2231.89,"endTime":2234.303,"body":"ratios you might see in nature"},{"startTime":2234.303,"endTime":2236.12,"body":"that I want to mention to you,"},{"startTime":2236.12,"endTime":2238.526,"body":"and it has to do with populations."},{"startTime":2238.526,"endTime":2241.65,"body":"So there's a few proportions"},{"startTime":2238.526,"endTime":2241.65,"body":"you could think about"},{"startTime":2241.65,"endTime":2243.58,"body":"to do with populations."},{"startTime":2243.58,"endTime":2245.604,"body":"The first is predator-prey."},{"startTime":2245.604,"endTime":2249.81,"body":"So we all know you have"},{"startTime":2245.604,"endTime":2249.81,"body":"fewer lions than zebras,"},{"startTime":2249.81,"endTime":2252.06,"body":"for obvious reasons, and"},{"startTime":2249.81,"endTime":2252.06,"body":"you know that there will be"},{"startTime":2252.06,"endTime":2254.18,"body":"a certain proportion"},{"startTime":2252.06,"endTime":2254.18,"body":"that seems to work well"},{"startTime":2254.18,"endTime":2255.013,"body":"in a steady state."},{"startTime":2255.013,"endTime":2256.6,"body":"You could find and study that."},{"startTime":2256.6,"endTime":2258.09,"body":"Or what about adults to offspring?"},{"startTime":2258.09,"endTime":2261.41,"body":"We know that, you know, a pair"},{"startTime":2258.09,"endTime":2261.41,"body":"of ducks, a duck and a drake,"},{"startTime":2261.41,"endTime":2263.969,"body":"I should say, will not"},{"startTime":2261.41,"endTime":2263.969,"body":"lay precisely two eggs"},{"startTime":2263.969,"endTime":2268.23,"body":"to reproduce the population."},{"startTime":2268.23,"endTime":2270.273,"body":"They will, there'll be more than two eggs,"},{"startTime":2270.273,"endTime":2272.04,"body":"'cause very sadly,"},{"startTime":2272.04,"endTime":2274.95,"body":"not all ducklings will"},{"startTime":2272.04,"endTime":2274.95,"body":"survive to adulthood."},{"startTime":2274.95,"endTime":2279.0,"body":"So there'll be, again,"},{"startTime":2274.95,"endTime":2279.0,"body":"there'll be a good amount."},{"startTime":2279.0,"endTime":2281.07,"body":"There'll be a proportion that you need"},{"startTime":2281.07,"endTime":2282.96,"body":"to reproduce more than yourself"},{"startTime":2282.96,"endTime":2285.21,"body":"in order to maintain a steady population."},{"startTime":2285.21,"endTime":2287.643,"body":"There's another statistic"},{"startTime":2285.21,"endTime":2287.643,"body":"about populations"},{"startTime":2287.643,"endTime":2289.947,"body":"that I want to talk"},{"startTime":2287.643,"endTime":2289.947,"body":"about, though, and this is"},{"startTime":2289.947,"endTime":2293.36,"body":"the proportion of males to"},{"startTime":2289.947,"endTime":2293.36,"body":"females in a population."},{"startTime":2293.36,"endTime":2295.81,"body":"So in humans, roughly speaking,"},{"startTime":2295.81,"endTime":2298.96,"body":"slightly more male babies"},{"startTime":2295.81,"endTime":2298.96,"body":"are born than female babies,"},{"startTime":2298.96,"endTime":2300.85,"body":"but by the time everyone"},{"startTime":2298.96,"endTime":2300.85,"body":"reaches adulthood,"},{"startTime":2300.85,"endTime":2304.61,"body":"it's approximately one to"},{"startTime":2300.85,"endTime":2304.61,"body":"one, 50%, male, 50% female."},{"startTime":2304.61,"endTime":2307.71,"body":"And kind of a justification for this,"},{"startTime":2307.71,"endTime":2310.84,"body":"why this might make sense,"},{"startTime":2307.71,"endTime":2310.84,"body":"is about efficiency."},{"startTime":2310.84,"endTime":2313.851,"body":"Because if you think about,"},{"startTime":2310.84,"endTime":2313.851,"body":"you know, all your ancestors,"},{"startTime":2313.851,"endTime":2317.14,"body":"you know, you've got"},{"startTime":2313.851,"endTime":2317.14,"body":"eight great grandparents,"},{"startTime":2317.14,"endTime":2318.93,"body":"four male, four female, right?"},{"startTime":2318.93,"endTime":2320.767,"body":"Half of your ancestors need to be male,"},{"startTime":2320.767,"endTime":2323.069,"body":"and half need to be female,"},{"startTime":2320.767,"endTime":2323.069,"body":"because to make a baby,"},{"startTime":2323.069,"endTime":2325.51,"body":"hopefully not startling anyone here,"},{"startTime":2325.51,"endTime":2328.06,"body":"you need one of each, right?"},{"startTime":2328.06,"endTime":2331.92,"body":"So it makes sense,"},{"startTime":2328.06,"endTime":2331.92,"body":"efficiency wise, to have"},{"startTime":2331.92,"endTime":2335.03,"body":"approximately half a population"},{"startTime":2331.92,"endTime":2335.03,"body":"male, and half female."},{"startTime":2335.03,"endTime":2338.486,"body":"But here's a pair of bees"},{"startTime":2335.03,"endTime":2338.486,"body":"going out for little stroll."},{"startTime":2338.486,"endTime":2339.8,"body":"Maybe they're courting."},{"startTime":2339.8,"endTime":2341.73,"body":"They're different, because of something"},{"startTime":2341.73,"endTime":2342.86,"body":"called haplodiploidy,"},{"startTime":2342.86,"endTime":2345.07,"body":"which I hope I'm pronouncing correctly."},{"startTime":2345.07,"endTime":2349.21,"body":"It's a curious phenomenon"},{"startTime":2345.07,"endTime":2349.21,"body":"with bees, where actually,"},{"startTime":2349.21,"endTime":2351.392,"body":"not all bees need to have two parents."},{"startTime":2351.392,"endTime":2355.31,"body":"Male bees hatch from unfertilized eggs,"},{"startTime":2355.31,"endTime":2357.503,"body":"so they only have a mother."},{"startTime":2358.52,"endTime":2360.93,"body":"Female bees hatch from fertilized eggs,"},{"startTime":2360.93,"endTime":2363.75,"body":"so female bees have a mother and a father."},{"startTime":2363.75,"endTime":2366.15,"body":"So have a think like, what does that imply"},{"startTime":2366.15,"endTime":2370.9,"body":"for the proportion of males"},{"startTime":2366.15,"endTime":2370.9,"body":"to females in a bee colony?"},{"startTime":2370.9,"endTime":2372.36,"body":"If the males only need a mother,"},{"startTime":2372.36,"endTime":2374.91,"body":"and the females need"},{"startTime":2372.36,"endTime":2374.91,"body":"a mother and a father,"},{"startTime":2374.91,"endTime":2376.86,"body":"is it sort of going to"},{"startTime":2374.91,"endTime":2376.86,"body":"be two thirds female?"},{"startTime":2376.86,"endTime":2377.76,"body":"What's it going to be?"},{"startTime":2377.76,"endTime":2379.467,"body":"So we can think about this,"},{"startTime":2379.467,"endTime":2382.05,"body":"and we can draw the bee family"},{"startTime":2379.467,"endTime":2382.05,"body":"tree, and we're kind of going"},{"startTime":2382.05,"endTime":2383.3,"body":"backwards through the generations, right?"},{"startTime":2383.3,"endTime":2384.64,"body":"We're looking at ancestors."},{"startTime":2384.64,"endTime":2385.86,"body":"So at the bottom, I've tried to,"},{"startTime":2385.86,"endTime":2388.92,"body":"I've got very gender stereotypes."},{"startTime":2388.92,"endTime":2392.48,"body":"They're blue squares for males,"},{"startTime":2388.92,"endTime":2392.48,"body":"and pink circle is female."},{"startTime":2392.48,"endTime":2394.92,"body":"I feel guilty now that"},{"startTime":2392.48,"endTime":2394.92,"body":"I've done that, but anyway,"},{"startTime":2394.92,"endTime":2396.562,"body":"it's just in case you"},{"startTime":2394.92,"endTime":2396.562,"body":"can't see the letters."},{"startTime":2396.562,"endTime":2399.7,"body":"So your male bee has a mother only, then,"},{"startTime":2399.7,"endTime":2401.624,"body":"so that's one generation back."},{"startTime":2401.624,"endTime":2403.47,"body":"Then you go back the next generation,"},{"startTime":2403.47,"endTime":2405.2,"body":"the female has a father and a mother,"},{"startTime":2405.2,"endTime":2409.728,"body":"oh, golly, (chuckles)"},{"startTime":2405.2,"endTime":2409.728,"body":"father begins with F."},{"startTime":2409.728,"endTime":2411.922,"body":"Right, mother begins with M. (chuckles)"},{"startTime":2411.922,"endTime":2414.45,"body":"A female parent and a male parent."},{"startTime":2414.45,"endTime":2417.408,"body":"And then going back again,"},{"startTime":2414.45,"endTime":2417.408,"body":"the female bee will have"},{"startTime":2417.408,"endTime":2420.57,"body":"two parents, and the male"},{"startTime":2417.408,"endTime":2420.57,"body":"bee will just have a mother."},{"startTime":2420.57,"endTime":2423.28,"body":"And going back, and going"},{"startTime":2420.57,"endTime":2423.28,"body":"back, you can keep going."},{"startTime":2423.28,"endTime":2425.14,"body":"But just have a look,"},{"startTime":2423.28,"endTime":2425.14,"body":"count up how many there are"},{"startTime":2425.14,"endTime":2430.14,"body":"on each generation back, 1, 1, 2, 3, 5, 8."},{"startTime":2432.52,"endTime":2434.96,"body":"That sounds like a familiar sequence."},{"startTime":2434.96,"endTime":2436.603,"body":"Why should this be the Fibonacci sequence?"},{"startTime":2436.603,"endTime":2438.3,"body":"What is going on?"},{"startTime":2438.3,"endTime":2440.7,"body":"Well, it's because there's"},{"startTime":2438.3,"endTime":2440.7,"body":"the same sort of pattern."},{"startTime":2440.7,"endTime":2444.81,"body":"It's the same sort of rule for"},{"startTime":2440.7,"endTime":2444.81,"body":"getting the numbers applies."},{"startTime":2444.81,"endTime":2447.16,"body":"So if we write something"},{"startTime":2444.81,"endTime":2447.16,"body":"like A N to be the number"},{"startTime":2447.16,"endTime":2451.07,"body":"of ancestors N generations"},{"startTime":2447.16,"endTime":2451.07,"body":"back, so five generations back,"},{"startTime":2451.07,"endTime":2453.201,"body":"which what I got at the top,"},{"startTime":2451.07,"endTime":2453.201,"body":"there are eight ancestors,"},{"startTime":2453.201,"endTime":2456.07,"body":"then we can work out the numbers of these."},{"startTime":2456.07,"endTime":2458.4,"body":"So that number will be"},{"startTime":2456.07,"endTime":2458.4,"body":"comprised of a certain number"},{"startTime":2458.4,"endTime":2460.539,"body":"of females, and a certain number of males."},{"startTime":2460.539,"endTime":2463.72,"body":"How many females are"},{"startTime":2460.539,"endTime":2463.72,"body":"there N generations back?"},{"startTime":2463.72,"endTime":2466.803,"body":"Well, everybody has a"},{"startTime":2463.72,"endTime":2466.803,"body":"female parent, right?"},{"startTime":2466.803,"endTime":2469.43,"body":"Everybody has a female parent."},{"startTime":2469.43,"endTime":2471.31,"body":"So the number of females"},{"startTime":2471.31,"endTime":2474.35,"body":"in N generations back is just"},{"startTime":2471.31,"endTime":2474.35,"body":"going to be the total number"},{"startTime":2474.35,"endTime":2476.813,"body":"that they were in minus"},{"startTime":2474.35,"endTime":2476.813,"body":"one generations back,"},{"startTime":2476.813,"endTime":2478.87,"body":"that A N minus one."},{"startTime":2478.87,"endTime":2480.2,"body":"What about males?"},{"startTime":2480.2,"endTime":2482.41,"body":"So how did you get to be"},{"startTime":2480.2,"endTime":2482.41,"body":"a male in that generation?"},{"startTime":2482.41,"endTime":2484.7,"body":"Well, you are the father of somebody,"},{"startTime":2484.7,"endTime":2487.31,"body":"but only females have fathers, right?"},{"startTime":2487.31,"endTime":2490.301,"body":"So if you're a male in"},{"startTime":2487.31,"endTime":2490.301,"body":"generation N, then that,"},{"startTime":2490.301,"endTime":2494.01,"body":"you are the father of a female bee"},{"startTime":2494.01,"endTime":2495.51,"body":"in the previous generation."},{"startTime":2495.51,"endTime":2497.97,"body":"So the number of males in"},{"startTime":2495.51,"endTime":2497.97,"body":"generation N is the number"},{"startTime":2497.97,"endTime":2500.11,"body":"of females in generation N minus one,"},{"startTime":2500.11,"endTime":2503.728,"body":"and that's the total number"},{"startTime":2500.11,"endTime":2503.728,"body":"in generation N minus two."},{"startTime":2503.728,"endTime":2505.11,"body":"So if we put that together,"},{"startTime":2505.11,"endTime":2507.31,"body":"we get exactly the same relationship."},{"startTime":2507.31,"endTime":2510.0,"body":"A N is A N minus one, plus A N minus two."},{"startTime":2510.0,"endTime":2512.603,"body":"That's exactly the Fibonacci relationship."},{"startTime":2513.55,"endTime":2514.86,"body":"So we started with one and one,"},{"startTime":2514.86,"endTime":2516.49,"body":"and then we can just work out,"},{"startTime":2516.49,"endTime":2519.46,"body":"and exactly the same rule gives"},{"startTime":2516.49,"endTime":2519.46,"body":"you the Fibonacci numbers."},{"startTime":2519.46,"endTime":2523.35,"body":"So we get the Fibonacci"},{"startTime":2519.46,"endTime":2523.35,"body":"numbers, which is amazing."},{"startTime":2523.35,"endTime":2526.46,"body":"And another amazing thing, if"},{"startTime":2523.35,"endTime":2526.46,"body":"I asked about the proportions,"},{"startTime":2526.46,"endTime":2528.08,"body":"so if you think, you"},{"startTime":2526.46,"endTime":2528.08,"body":"know, what proportions"},{"startTime":2528.08,"endTime":2532.05,"body":"of the ancestors are of each gender,"},{"startTime":2532.05,"endTime":2533.92,"body":"and that might tell us"},{"startTime":2532.05,"endTime":2533.92,"body":"what it is in the whole"},{"startTime":2533.92,"endTime":2536.099,"body":"of the society, you exactly get,"},{"startTime":2536.099,"endTime":2537.31,"body":"there are how many males in generation?"},{"startTime":2537.31,"endTime":2538.85,"body":"N it's A N minus two."},{"startTime":2538.85,"endTime":2540.19,"body":"How many females?"},{"startTime":2540.19,"endTime":2541.44,"body":"It's A N minus one."},{"startTime":2541.44,"endTime":2544.166,"body":"So the proportion, the"},{"startTime":2541.44,"endTime":2544.166,"body":"ratio of males to females is"},{"startTime":2544.166,"endTime":2548.24,"body":"the ratio of the N minus"},{"startTime":2544.166,"endTime":2548.24,"body":"second Fibonacci number"},{"startTime":2548.24,"endTime":2551.14,"body":"to the N minus first one, and"},{"startTime":2548.24,"endTime":2551.14,"body":"as N gets bigger and bigger,"},{"startTime":2551.14,"endTime":2554.48,"body":"this becomes the golden ratio, right?"},{"startTime":2554.48,"endTime":2556.53,"body":"So the proportion of"},{"startTime":2554.48,"endTime":2556.53,"body":"male bees to female bees"},{"startTime":2556.53,"endTime":2561.53,"body":"in a bee population is the"},{"startTime":2556.53,"endTime":2561.53,"body":"golden ratio, and that's lovely."},{"startTime":2561.95,"endTime":2563.53,"body":"That's lovely."},{"startTime":2563.53,"endTime":2565.57,"body":"So that's the place where, again,"},{"startTime":2565.57,"endTime":2567.12,"body":"we do see the golden ratio in nature,"},{"startTime":2567.12,"endTime":2570.353,"body":"and it's kind of an"},{"startTime":2567.12,"endTime":2570.353,"body":"unexpected joy to see that."},{"startTime":2571.457,"endTime":2573.51,"body":"I'll talk about some other things now"},{"startTime":2573.51,"endTime":2574.638,"body":"in the last 10 minutes,"},{"startTime":2574.638,"endTime":2578.41,"body":"and it's proportions that we"},{"startTime":2574.638,"endTime":2578.41,"body":"see around us in everyday life."},{"startTime":2578.41,"endTime":2581.598,"body":"So I said at the start"},{"startTime":2578.41,"endTime":2581.598,"body":"head of piece of A4 paper,"},{"startTime":2581.598,"endTime":2585.33,"body":"and I said, why it is A4 paper?"},{"startTime":2585.33,"endTime":2587.71,"body":"It's fairly standard"},{"startTime":2585.33,"endTime":2587.71,"body":"around most of the world,"},{"startTime":2587.71,"endTime":2590.74,"body":"these sort of A series of paper sizes, A4,"},{"startTime":2590.74,"endTime":2592.65,"body":"A3 is bigger, A5 is smaller, and so on,"},{"startTime":2592.65,"endTime":2596.5,"body":"why is it exactly 297 millimeters long?"},{"startTime":2596.5,"endTime":2597.817,"body":"And, you know, the first"},{"startTime":2596.5,"endTime":2597.817,"body":"guess at this might be,"},{"startTime":2597.817,"endTime":2599.82,"body":"\"Well, maybe it was"},{"startTime":2597.817,"endTime":2599.82,"body":"something in inches once,"},{"startTime":2599.82,"endTime":2600.78,"body":"and this is their conversion."},{"startTime":2600.78,"endTime":2601.66,"body":"That's often the case.\""},{"startTime":2601.66,"endTime":2604.11,"body":"It's nothing to do with"},{"startTime":2601.66,"endTime":2604.11,"body":"that, it's something else,"},{"startTime":2604.11,"endTime":2608.97,"body":"and it starts with this chap,"},{"startTime":2604.11,"endTime":2608.97,"body":"Georg Christoph Lichtenberg."},{"startTime":2608.97,"endTime":2613.52,"body":"He proposed that there's"},{"startTime":2608.97,"endTime":2613.52,"body":"lots of different sizes"},{"startTime":2613.52,"endTime":2615.481,"body":"of paper milling around, and you know,"},{"startTime":2615.481,"endTime":2617.23,"body":"how are we going to get,"},{"startTime":2617.23,"endTime":2619.98,"body":"what's the most efficient"},{"startTime":2617.23,"endTime":2619.98,"body":"series of paper sizes?"},{"startTime":2619.98,"endTime":2622.53,"body":"You know, we need different"},{"startTime":2619.98,"endTime":2622.53,"body":"sizes for different purposes."},{"startTime":2622.53,"endTime":2624.52,"body":"What's the most efficient"},{"startTime":2622.53,"endTime":2624.52,"body":"way of doing this?"},{"startTime":2624.52,"endTime":2628.24,"body":"Well, what's very good in terms"},{"startTime":2624.52,"endTime":2628.24,"body":"of efficiency is if you have"},{"startTime":2628.24,"endTime":2630.691,"body":"a great big piece of paper, a folio,"},{"startTime":2630.691,"endTime":2633.19,"body":"you ought to be able to"},{"startTime":2630.691,"endTime":2633.19,"body":"make the next size down"},{"startTime":2633.19,"endTime":2636.4,"body":"by just folding it in"},{"startTime":2633.19,"endTime":2636.4,"body":"half and cutting, right?"},{"startTime":2636.4,"endTime":2639.717,"body":"And then you can fold again,"},{"startTime":2636.4,"endTime":2639.717,"body":"you can get octavo, quarter,"},{"startTime":2639.717,"endTime":2642.95,"body":"and then you can fold again"},{"startTime":2639.717,"endTime":2642.95,"body":"and get octavo, and so on."},{"startTime":2642.95,"endTime":2646.2,"body":"So if you do that by folding,"},{"startTime":2642.95,"endTime":2646.2,"body":"and then you can cut,"},{"startTime":2646.2,"endTime":2647.63,"body":"you're not wasting any paper."},{"startTime":2647.63,"endTime":2649.54,"body":"So this is kind of a good system,"},{"startTime":2649.54,"endTime":2651.98,"body":"if these are your standard paper sizes."},{"startTime":2651.98,"endTime":2653.2,"body":"So that's what he suggested."},{"startTime":2653.2,"endTime":2656.359,"body":"So he said that what you want is the shape"},{"startTime":2656.359,"endTime":2659.01,"body":"of your paper should"},{"startTime":2656.359,"endTime":2659.01,"body":"be, have this property,"},{"startTime":2659.01,"endTime":2659.99,"body":"that if you cut it in half,"},{"startTime":2659.99,"endTime":2662.15,"body":"you get the same shape, but just smaller."},{"startTime":2662.15,"endTime":2663.3,"body":"It'll be half the area."},{"startTime":2664.318,"endTime":2666.24,"body":"So what do we need there?"},{"startTime":2666.24,"endTime":2668.29,"body":"I mean, we could solve"},{"startTime":2666.24,"endTime":2668.29,"body":"this now, mathematically."},{"startTime":2668.29,"endTime":2670.31,"body":"You've got your piece of"},{"startTime":2668.29,"endTime":2670.31,"body":"paper, you cut it in half,"},{"startTime":2670.31,"endTime":2671.76,"body":"and what you want is a rectangle that has"},{"startTime":2671.76,"endTime":2674.69,"body":"the same proportions, the same ratio."},{"startTime":2674.69,"endTime":2676.937,"body":"So if your original one"},{"startTime":2674.69,"endTime":2676.937,"body":"has side, let's say one,"},{"startTime":2676.937,"endTime":2678.68,"body":"and the long side is X,"},{"startTime":2678.68,"endTime":2681.01,"body":"then the one you create, now this time,"},{"startTime":2681.01,"endTime":2684.8,"body":"it's long side is one, and"},{"startTime":2681.01,"endTime":2684.8,"body":"it's short side is half X,"},{"startTime":2684.8,"endTime":2687.18,"body":"and you want those to"},{"startTime":2684.8,"endTime":2687.18,"body":"be in the same ratio."},{"startTime":2687.18,"endTime":2691.15,"body":"So you want the little"},{"startTime":2687.18,"endTime":2691.15,"body":"rectangle short side"},{"startTime":2691.15,"endTime":2694.35,"body":"over long side, half X"},{"startTime":2691.15,"endTime":2694.35,"body":"over one, should equal"},{"startTime":2694.35,"endTime":2697.03,"body":"short side over long side"},{"startTime":2694.35,"endTime":2697.03,"body":"of the big rectangle,"},{"startTime":2697.03,"endTime":2700.48,"body":"which is one over X, and"},{"startTime":2697.03,"endTime":2700.48,"body":"we can solve this equation."},{"startTime":2700.48,"endTime":2702.477,"body":"It's easier than the Fibonacci one."},{"startTime":2702.477,"endTime":2705.08,"body":"We just get X squared equals two."},{"startTime":2705.08,"endTime":2708.041,"body":"Now, of course, that has two solutions."},{"startTime":2708.041,"endTime":2712.31,"body":"I'm not saying that the"},{"startTime":2708.041,"endTime":2712.31,"body":"negative square root of two is"},{"startTime":2712.31,"endTime":2714.097,"body":"not a solution of this, but"},{"startTime":2712.31,"endTime":2714.097,"body":"we are dealing with pieces"},{"startTime":2714.097,"endTime":2717.59,"body":"of paper here, so it's length"},{"startTime":2714.097,"endTime":2717.59,"body":"can't be minus root two."},{"startTime":2717.59,"endTime":2721.357,"body":"So I feel okay in saying that"},{"startTime":2717.59,"endTime":2721.357,"body":"X must be plus root two here."},{"startTime":2721.357,"endTime":2723.27,"body":"So X is root two."},{"startTime":2723.27,"endTime":2725.63,"body":"So that's the relationship,"},{"startTime":2723.27,"endTime":2725.63,"body":"one to root two."},{"startTime":2725.63,"endTime":2727.3,"body":"That's what shape the paper should be."},{"startTime":2727.3,"endTime":2728.78,"body":"So for our A4 paper that's,"},{"startTime":2728.78,"endTime":2731.679,"body":"if we want it to be 210 millimeters wide,"},{"startTime":2731.679,"endTime":2734.91,"body":"then we know that the"},{"startTime":2731.679,"endTime":2734.91,"body":"lengths should be squared"},{"startTime":2734.91,"endTime":2738.77,"body":"to two times that, and that's 297, right?"},{"startTime":2738.77,"endTime":2740.28,"body":"Give or take."},{"startTime":2740.28,"endTime":2743.483,"body":"So that's why, that tells you if you know"},{"startTime":2743.483,"endTime":2745.37,"body":"it's 210 millimeters wide,"},{"startTime":2745.37,"endTime":2747.74,"body":"then it must be 297 millimeters long,"},{"startTime":2747.74,"endTime":2749.262,"body":"but that doesn't actually tell you"},{"startTime":2749.262,"endTime":2751.879,"body":"why those particular numbers, right?"},{"startTime":2751.879,"endTime":2756.12,"body":"The answer to that comes"},{"startTime":2751.879,"endTime":2756.12,"body":"from the French Revolution,"},{"startTime":2756.12,"endTime":2758.364,"body":"as I'm sure you were expecting me to say."},{"startTime":2758.364,"endTime":2761.121,"body":"There was a law passed."},{"startTime":2761.121,"endTime":2763.8,"body":"I love these French"},{"startTime":2761.121,"endTime":2763.8,"body":"Revolution calendar dates,"},{"startTime":2763.8,"endTime":2767.72,"body":"the 13th day of Brumaire"},{"startTime":2763.8,"endTime":2767.72,"body":"sur le timbre, I think"},{"startTime":2767.72,"endTime":2769.99,"body":"it's probably November"},{"startTime":2767.72,"endTime":2769.99,"body":"sometime, in the seventh year"},{"startTime":2769.99,"endTime":2774.097,"body":"of the Republic, and this was"},{"startTime":2769.99,"endTime":2774.097,"body":"a law, and it was about taxes"},{"startTime":2774.097,"endTime":2777.624,"body":"on various things, but they"},{"startTime":2774.097,"endTime":2777.624,"body":"specified some paper sizes,"},{"startTime":2777.624,"endTime":2782.46,"body":"a series of paper sizes"},{"startTime":2777.624,"endTime":2782.46,"body":"of the Lichtenberg kind,"},{"startTime":2782.46,"endTime":2785.34,"body":"with this exact, you"},{"startTime":2782.46,"endTime":2785.34,"body":"know, the exact shapes,"},{"startTime":2785.34,"endTime":2787.73,"body":"but they gave the sizes as"},{"startTime":2785.34,"endTime":2787.73,"body":"well, and they were interested"},{"startTime":2787.73,"endTime":2790.48,"body":"in them being fractions of a square meter,"},{"startTime":2790.48,"endTime":2793.02,"body":"this lovely new metric"},{"startTime":2790.48,"endTime":2793.02,"body":"thing that they'd invented."},{"startTime":2793.02,"endTime":2796.963,"body":"So the Moyen Papier is one"},{"startTime":2793.02,"endTime":2796.963,"body":"eighth of a square meter in area."},{"startTime":2797.84,"endTime":2799.28,"body":"And this is what we do now,"},{"startTime":2799.28,"endTime":2803.28,"body":"so our A paper series starts at A0."},{"startTime":2803.28,"endTime":2804.82,"body":"That's the biggest one you can get."},{"startTime":2804.82,"endTime":2809.52,"body":"It is, you know, by definition,"},{"startTime":2804.82,"endTime":2809.52,"body":"the paper of this size,"},{"startTime":2809.52,"endTime":2812.18,"body":"whose area is a square meter exactly."},{"startTime":2812.18,"endTime":2816.623,"body":"So that's A0, and then A1 is"},{"startTime":2812.18,"endTime":2816.623,"body":"half that, A2 is a quarter,"},{"startTime":2816.623,"endTime":2820.456,"body":"A3 is an eighth, A4, a"},{"startTime":2816.623,"endTime":2820.456,"body":"16th of a square meter."},{"startTime":2821.81,"endTime":2823.27,"body":"So once you know that,"},{"startTime":2823.27,"endTime":2825.9,"body":"you can then exactly work"},{"startTime":2823.27,"endTime":2825.9,"body":"out what it has to be."},{"startTime":2825.9,"endTime":2828.49,"body":"It's got to be a rectangle of that shape,"},{"startTime":2828.49,"endTime":2830.68,"body":"with the area 1/16th of a meter."},{"startTime":2830.68,"endTime":2832.37,"body":"So if it's with this A,"},{"startTime":2832.37,"endTime":2836.51,"body":"then the area will be A times"},{"startTime":2832.37,"endTime":2836.51,"body":"the square of two times A,"},{"startTime":2836.51,"endTime":2838.28,"body":"and that should equal a 16th of a meter."},{"startTime":2838.28,"endTime":2841.42,"body":"If you solve that, 210 millimeters."},{"startTime":2841.42,"endTime":2844.73,"body":"So that's why A4 paper,"},{"startTime":2841.42,"endTime":2844.73,"body":"not only, that's why"},{"startTime":2844.73,"endTime":2845.68,"body":"it's the shape it is,"},{"startTime":2845.68,"endTime":2847.46,"body":"but that's why it's"},{"startTime":2845.68,"endTime":2847.46,"body":"those exact dimensions."},{"startTime":2847.46,"endTime":2849.06,"body":"It's all about parts of meter."},{"startTime":2849.06,"endTime":2850.97,"body":"So we can thank the French Revolution"},{"startTime":2850.97,"endTime":2852.883,"body":"for the shape and size of our paper."},{"startTime":2853.8,"endTime":2856.88,"body":"While I was thinking about,"},{"startTime":2853.8,"endTime":2856.88,"body":"you know, that's kind of cool,"},{"startTime":2856.88,"endTime":2859.14,"body":"that's two dimensional,"},{"startTime":2859.14,"endTime":2860.74,"body":"could you do this in three dimensions?"},{"startTime":2860.74,"endTime":2862.93,"body":"Like, could you have a cuboid"},{"startTime":2860.74,"endTime":2862.93,"body":"that if you cut it in half,"},{"startTime":2862.93,"endTime":2864.69,"body":"you get two cuboids the same?"},{"startTime":2864.69,"endTime":2867.26,"body":"You can, you can solve that"},{"startTime":2864.69,"endTime":2867.26,"body":"equation, but you know,"},{"startTime":2867.26,"endTime":2869.299,"body":"let's think what cuboids"},{"startTime":2867.26,"endTime":2869.299,"body":"we use in everyday life."},{"startTime":2869.299,"endTime":2871.017,"body":"I mean, I thought of bricks."},{"startTime":2871.017,"endTime":2873.1,"body":"You don't fold bricks in half."},{"startTime":2873.1,"endTime":2875.034,"body":"So that isn't what you"},{"startTime":2873.1,"endTime":2875.034,"body":"want of a brick, right?"},{"startTime":2875.034,"endTime":2877.54,"body":"That's, it's not this that"},{"startTime":2875.034,"endTime":2877.54,"body":"we're interested in for bricks."},{"startTime":2877.54,"endTime":2880.07,"body":"You can solve that problem,"},{"startTime":2877.54,"endTime":2880.07,"body":"and it's kind of fun."},{"startTime":2880.07,"endTime":2882.29,"body":"I encourage you to do that"},{"startTime":2880.07,"endTime":2882.29,"body":"on the train the way home."},{"startTime":2882.29,"endTime":2884.76,"body":"But actually, if you think"},{"startTime":2882.29,"endTime":2884.76,"body":"about something like bricks,"},{"startTime":2884.76,"endTime":2887.776,"body":"you have a different, you want"},{"startTime":2884.76,"endTime":2887.776,"body":"them to do different things."},{"startTime":2887.776,"endTime":2890.633,"body":"So a brick has a height,"},{"startTime":2887.776,"endTime":2890.633,"body":"a length, and a width."},{"startTime":2890.633,"endTime":2892.6,"body":"What should those be?"},{"startTime":2892.6,"endTime":2895.06,"body":"Or at least what should"},{"startTime":2892.6,"endTime":2895.06,"body":"the proportions be?"},{"startTime":2895.06,"endTime":2898.82,"body":"Brick walls, we walk past"},{"startTime":2895.06,"endTime":2898.82,"body":"hundreds of them every day."},{"startTime":2898.82,"endTime":2899.98,"body":"As you'll know, if you've ever,"},{"startTime":2899.98,"endTime":2902.18,"body":"I have never built a"},{"startTime":2899.98,"endTime":2902.18,"body":"brick wall, I have to say,"},{"startTime":2902.18,"endTime":2904.7,"body":"I have built Lego walls, and"},{"startTime":2902.18,"endTime":2904.7,"body":"I know you have to alternate."},{"startTime":2904.7,"endTime":2907.85,"body":"You can't have all your"},{"startTime":2904.7,"endTime":2907.85,"body":"weak points lining up,"},{"startTime":2907.85,"endTime":2911.03,"body":"so you alternate them, and"},{"startTime":2907.85,"endTime":2911.03,"body":"often you will do this by having"},{"startTime":2911.03,"endTime":2913.79,"body":"a wall that's like, a full brick in depth,"},{"startTime":2913.79,"endTime":2915.78,"body":"so it's sort of two brick widths deep,"},{"startTime":2915.78,"endTime":2917.76,"body":"and that's the same as one brick length,"},{"startTime":2917.76,"endTime":2919.05,"body":"and that means some of your,"},{"startTime":2919.05,"endTime":2921.05,"body":"in some of your rows, or"},{"startTime":2919.05,"endTime":2921.05,"body":"courses, they're called,"},{"startTime":2921.05,"endTime":2923.17,"body":"the bricks can be facing"},{"startTime":2921.05,"endTime":2923.17,"body":"this way, and in others,"},{"startTime":2923.17,"endTime":2925.13,"body":"they can be just sort of length side on."},{"startTime":2925.13,"endTime":2928.73,"body":"So to do that, you need the"},{"startTime":2925.13,"endTime":2928.73,"body":"widths to be half the length,"},{"startTime":2928.73,"endTime":2930.61,"body":"or the length to be twice the width."},{"startTime":2930.61,"endTime":2931.873,"body":"You need that to do this."},{"startTime":2933.17,"endTime":2936.3,"body":"To give yourself a bit more"},{"startTime":2933.17,"endTime":2936.3,"body":"play to make different designs"},{"startTime":2936.3,"endTime":2938.7,"body":"of brick, there's another"},{"startTime":2936.3,"endTime":2938.7,"body":"thing that would be nice."},{"startTime":2938.7,"endTime":2941.0,"body":"There's a third dimension"},{"startTime":2938.7,"endTime":2941.0,"body":"here, the height,"},{"startTime":2941.0,"endTime":2942.7,"body":"and I dunno if it's a good enough picture."},{"startTime":2942.7,"endTime":2944.66,"body":"It's just a brick wall on my road."},{"startTime":2944.66,"endTime":2947.74,"body":"The top row, or course of this, has got"},{"startTime":2947.74,"endTime":2950.13,"body":"this slightly different"},{"startTime":2947.74,"endTime":2950.13,"body":"finish, just to sort of say,"},{"startTime":2950.13,"endTime":2952.461,"body":"this is top of the wall,"},{"startTime":2950.13,"endTime":2952.461,"body":"slightly ornamental."},{"startTime":2952.461,"endTime":2955.294,"body":"It's got bricks that are on their side,"},{"startTime":2955.294,"endTime":2957.75,"body":"and there are three of them that fit"},{"startTime":2957.75,"endTime":2960.113,"body":"into the length of one brick."},{"startTime":2961.14,"endTime":2964.19,"body":"So this particular kind"},{"startTime":2961.14,"endTime":2964.19,"body":"of course, to do that,"},{"startTime":2964.19,"endTime":2967.067,"body":"you need the length to be"},{"startTime":2964.19,"endTime":2967.067,"body":"three times the height."},{"startTime":2967.067,"endTime":2968.736,"body":"And so if you put those together,"},{"startTime":2968.736,"endTime":2971.08,"body":"you get this nice collection of ratios."},{"startTime":2971.08,"endTime":2974.46,"body":"Height to width to length"},{"startTime":2971.08,"endTime":2974.46,"body":"is two to three to six."},{"startTime":2974.46,"endTime":2976.78,"body":"And of course, you'll all be rushing home"},{"startTime":2976.78,"endTime":2977.8,"body":"to measure a brick."},{"startTime":2977.8,"endTime":2981.19,"body":"Now, I will say, you've got"},{"startTime":2977.8,"endTime":2981.19,"body":"to take off a little bit"},{"startTime":2981.19,"endTime":2982.39,"body":"to allow for mortar."},{"startTime":2982.39,"endTime":2984.66,"body":"So you, you take, you decide what size"},{"startTime":2984.66,"endTime":2986.11,"body":"your brick wants to be,"},{"startTime":2984.66,"endTime":2986.11,"body":"and it'll be, you know,"},{"startTime":2986.11,"endTime":2988.25,"body":"what you can hold in a hand, basically,"},{"startTime":2988.25,"endTime":2990.58,"body":"but then it won't quite"},{"startTime":2988.25,"endTime":2990.58,"body":"be two to three to six"},{"startTime":2990.58,"endTime":2992.41,"body":"in the finished brick, but when you add"},{"startTime":2992.41,"endTime":2993.95,"body":"in the 10 millimeters for mortar,"},{"startTime":2993.95,"endTime":2996.12,"body":"that is the ratios that"},{"startTime":2993.95,"endTime":2996.12,"body":"you're going to get."},{"startTime":2996.12,"endTime":2997.01,"body":"So that's bricks."},{"startTime":2997.01,"endTime":2998.93,"body":"That's something three dimensional."},{"startTime":2998.93,"endTime":3003.4,"body":"I want to finish in the"},{"startTime":2998.93,"endTime":3003.4,"body":"last five minutes, I guess,"},{"startTime":3003.4,"endTime":3006.121,"body":"by talking about everyone's"},{"startTime":3003.4,"endTime":3006.121,"body":"favorite thing, cake."},{"startTime":3006.121,"endTime":3009.39,"body":"And in particular, you know, in baking,"},{"startTime":3009.39,"endTime":3011.16,"body":"we've got circles here."},{"startTime":3011.16,"endTime":3013.01,"body":"We haven't really talked"},{"startTime":3011.16,"endTime":3013.01,"body":"about circles enough."},{"startTime":3013.01,"endTime":3015.967,"body":"We always talk about circles"},{"startTime":3013.01,"endTime":3015.967,"body":"in every mathematical talk,"},{"startTime":3015.967,"endTime":3018.19,"body":"and about cake."},{"startTime":3018.19,"endTime":3021.32,"body":"If you are baking, there is so"},{"startTime":3018.19,"endTime":3021.32,"body":"much mathematics in cooking."},{"startTime":3021.32,"endTime":3023.14,"body":"So much, it's wonderful."},{"startTime":3023.14,"endTime":3026.17,"body":"You've got volumes, and"},{"startTime":3023.14,"endTime":3026.17,"body":"you've got, you know,"},{"startTime":3026.17,"endTime":3028.14,"body":"if you're icing this thing, you've got,"},{"startTime":3028.14,"endTime":3030.87,"body":"think about surface areas,"},{"startTime":3028.14,"endTime":3030.87,"body":"how much icing do I make?"},{"startTime":3030.87,"endTime":3032.847,"body":"If you've got some"},{"startTime":3030.87,"endTime":3032.847,"body":"ornamental band or something,"},{"startTime":3032.847,"endTime":3035.74,"body":"you need to know about the"},{"startTime":3032.847,"endTime":3035.74,"body":"circumferences, or perimeters."},{"startTime":3035.74,"endTime":3037.15,"body":"There's a lot going on."},{"startTime":3037.15,"endTime":3037.983,"body":"The particular thing"},{"startTime":3037.983,"endTime":3041.13,"body":"I want to talk about is cake tins, right?"},{"startTime":3041.13,"endTime":3043.894,"body":"If you've ever baked, you will know that"},{"startTime":3043.894,"endTime":3047.37,"body":"there's kind of standard"},{"startTime":3043.894,"endTime":3047.37,"body":"sizes for cake tins."},{"startTime":3047.37,"endTime":3049.98,"body":"There are standard sizes,"},{"startTime":3047.37,"endTime":3049.98,"body":"and they're usually inches,"},{"startTime":3049.98,"endTime":3051.33,"body":"and there are standard shapes."},{"startTime":3051.33,"endTime":3054.53,"body":"You normally, cake tins"},{"startTime":3051.33,"endTime":3054.53,"body":"are either round or square."},{"startTime":3054.53,"endTime":3056.17,"body":"You can get other fancy-shaped ones,"},{"startTime":3056.17,"endTime":3057.42,"body":"but that's the main ones."},{"startTime":3058.56,"endTime":3062.86,"body":"And if you are working"},{"startTime":3058.56,"endTime":3062.86,"body":"out how much do I need"},{"startTime":3062.86,"endTime":3066.39,"body":"for my nine-inch-round cake tin?"},{"startTime":3066.39,"endTime":3068.937,"body":"Then lots of cookbooks will"},{"startTime":3066.39,"endTime":3068.937,"body":"have kind of tables that say,"},{"startTime":3068.937,"endTime":3071.0,"body":"\"You need this much for this,"},{"startTime":3068.937,"endTime":3071.0,"body":"and this much for that,\""},{"startTime":3071.0,"endTime":3073.609,"body":"and my cookbook that I have"},{"startTime":3071.0,"endTime":3073.609,"body":"at home, or one of them says,"},{"startTime":3073.609,"endTime":3077.371,"body":"\"Nine-inch-round tin"},{"startTime":3073.609,"endTime":3077.371,"body":"requires the same amount"},{"startTime":3077.371,"endTime":3080.899,"body":"of cake mixture as an"},{"startTime":3077.371,"endTime":3080.899,"body":"eight-inch-square tin.\""},{"startTime":3080.899,"endTime":3081.732,"body":"It's like, that's interesting."},{"startTime":3081.732,"endTime":3082.924,"body":"Does it exactly say, is it?"},{"startTime":3082.924,"endTime":3085.165,"body":"Because, of course, in there implicitly,"},{"startTime":3085.165,"endTime":3089.04,"body":"they're making an approximation"},{"startTime":3085.165,"endTime":3089.04,"body":"to pi, 'cause circles have"},{"startTime":3089.04,"endTime":3090.27,"body":"something to do with pi, right?"},{"startTime":3090.27,"endTime":3092.11,"body":"The area of a circle, pi R squared,"},{"startTime":3092.11,"endTime":3094.4,"body":"and squares, you know, don't."},{"startTime":3094.4,"endTime":3097.09,"body":"So they're making some"},{"startTime":3094.4,"endTime":3097.09,"body":"approximation to pi."},{"startTime":3097.09,"endTime":3102.09,"body":"Do I know any ancient,"},{"startTime":3097.09,"endTime":3102.09,"body":"wonderful, historical sources"},{"startTime":3102.72,"endTime":3104.3,"body":"of approximations to pi?"},{"startTime":3104.3,"endTime":3106.06,"body":"And I did know. (chuckles)"},{"startTime":3106.06,"endTime":3107.12,"body":"I did know one."},{"startTime":3107.12,"endTime":3110.65,"body":"This is the Rhind"},{"startTime":3107.12,"endTime":3110.65,"body":"Papyrus, Rhind with an H."},{"startTime":3110.65,"endTime":3111.83,"body":"It's not about cooking."},{"startTime":3111.83,"endTime":3115.04,"body":"It's got lots and lots of"},{"startTime":3111.83,"endTime":3115.04,"body":"mathematical problems on it."},{"startTime":3115.04,"endTime":3118.4,"body":"It's an ancient Egyptian"},{"startTime":3115.04,"endTime":3118.4,"body":"papyrus, and I mean,"},{"startTime":3118.4,"endTime":3120.35,"body":"you might be able to see"},{"startTime":3118.4,"endTime":3120.35,"body":"the sort of nice pictures"},{"startTime":3120.35,"endTime":3121.247,"body":"of triangles and things."},{"startTime":3121.247,"endTime":3124.5,"body":"But the problem I want to"},{"startTime":3121.247,"endTime":3124.5,"body":"mention to you is problem 50,"},{"startTime":3124.5,"endTime":3126.45,"body":"method of reckoning, a"},{"startTime":3124.5,"endTime":3126.45,"body":"circular piece of land,"},{"startTime":3126.45,"endTime":3129.3,"body":"diameter nine, what's its area?"},{"startTime":3129.3,"endTime":3131.17,"body":"And there's an algorithm given."},{"startTime":3131.17,"endTime":3135.603,"body":"Subtract one ninth of"},{"startTime":3131.17,"endTime":3135.603,"body":"it, and then square that,"},{"startTime":3135.603,"endTime":3137.257,"body":"and you get the area."},{"startTime":3137.257,"endTime":3139.56,"body":"And there's other problems"},{"startTime":3137.257,"endTime":3139.56,"body":"that do the same sort of thing,"},{"startTime":3139.56,"endTime":3143.76,"body":"so we can deduce that the"},{"startTime":3139.56,"endTime":3143.76,"body":"ancient Egyptian formula"},{"startTime":3143.76,"endTime":3147.68,"body":"for the area of a circle"},{"startTime":3143.76,"endTime":3147.68,"body":"of diameter D is find"},{"startTime":3147.68,"endTime":3150.82,"body":"eight ninths of the diameter,"},{"startTime":3147.68,"endTime":3150.82,"body":"and then square that."},{"startTime":3150.82,"endTime":3155.12,"body":"So that gives you 64, 80 firsts, 81ths,"},{"startTime":3155.12,"endTime":3158.29,"body":"64 over 81 times the diameter squared."},{"startTime":3158.29,"endTime":3161.39,"body":"Now we know what the area"},{"startTime":3158.29,"endTime":3161.39,"body":"of a circle is for real,"},{"startTime":3161.39,"endTime":3163.81,"body":"I mean, pi R squared,"},{"startTime":3161.39,"endTime":3163.81,"body":"but if you're working"},{"startTime":3163.81,"endTime":3165.59,"body":"with the diameter, pi"},{"startTime":3163.81,"endTime":3165.59,"body":"over four, D squared."},{"startTime":3165.59,"endTime":3170.25,"body":"So what the ancient Egyptian"},{"startTime":3165.59,"endTime":3170.25,"body":"scribe has done there is"},{"startTime":3170.25,"endTime":3171.93,"body":"they've got an approximation for pi."},{"startTime":3171.93,"endTime":3173.404,"body":"I mean, they're not thinking,"},{"startTime":3173.404,"endTime":3174.237,"body":"\"I've got an approximation for pi,\""},{"startTime":3174.237,"endTime":3177.689,"body":"but in effect, they are"},{"startTime":3174.237,"endTime":3177.689,"body":"using pi over four equals"},{"startTime":3177.689,"endTime":3182.689,"body":"64 over 81, or pi is 256"},{"startTime":3177.689,"endTime":3182.689,"body":"over 81, which is about 3.16."},{"startTime":3182.987,"endTime":3185.947,"body":"That's a pretty good approximation."},{"startTime":3185.947,"endTime":3188.07,"body":"So I thought, well, what does my cookbook,"},{"startTime":3188.07,"endTime":3189.627,"body":"what's it's approximation to pi?"},{"startTime":3189.627,"endTime":3192.12,"body":"So we've got the ancient Egyptian one."},{"startTime":3192.12,"endTime":3194.88,"body":"If I'm making a square"},{"startTime":3192.12,"endTime":3194.88,"body":"cake or a round cake,"},{"startTime":3194.88,"endTime":3198.86,"body":"my cookbook says nine inch"},{"startTime":3194.88,"endTime":3198.86,"body":"round equals eight inch square."},{"startTime":3198.86,"endTime":3201.2,"body":"So by the way, these tins are,"},{"startTime":3201.2,"endTime":3202.43,"body":"tend to be all the same height,"},{"startTime":3202.43,"endTime":3204.288,"body":"so we don't have to"},{"startTime":3202.43,"endTime":3204.288,"body":"worry about the height."},{"startTime":3204.288,"endTime":3205.69,"body":"That's just constant in there, H."},{"startTime":3205.69,"endTime":3208.36,"body":"So the area that you need,"},{"startTime":3205.69,"endTime":3208.36,"body":"or the volume will be"},{"startTime":3208.36,"endTime":3211.74,"body":"for a nine inch round tin if"},{"startTime":3208.36,"endTime":3211.74,"body":"that's the diameter would be"},{"startTime":3211.74,"endTime":3216.74,"body":"81 over four pi H, and the,"},{"startTime":3211.74,"endTime":3216.74,"body":"for the eight inch square tin,"},{"startTime":3216.76,"endTime":3218.968,"body":"it'll be 64 H whatever the height is."},{"startTime":3218.968,"endTime":3220.86,"body":"So you can see what the approximation is."},{"startTime":3220.86,"endTime":3224.843,"body":"81 over four pi equals"},{"startTime":3220.86,"endTime":3224.843,"body":"64, and low and behold,"},{"startTime":3224.843,"endTime":3227.64,"body":"wow, pi again."},{"startTime":3227.64,"endTime":3230.51,"body":"So, you know, pi in a"},{"startTime":3227.64,"endTime":3230.51,"body":"bit millennial later,"},{"startTime":3230.51,"endTime":3233.72,"body":"my little cookbook in my kitchen uses"},{"startTime":3233.72,"endTime":3235.93,"body":"exactly the same approximation to pi."},{"startTime":3235.93,"endTime":3237.54,"body":"So, you know, there's progress for you."},{"startTime":3237.54,"endTime":3239.53,"body":"So, but you know that it's,"},{"startTime":3239.53,"endTime":3240.95,"body":"so this is a good approximation."},{"startTime":3240.95,"endTime":3243.67,"body":"I just really like this fact."},{"startTime":3243.67,"endTime":3247.732,"body":"I will spend two more minutes,"},{"startTime":3243.67,"endTime":3247.732,"body":"'cause I can't resist telling"},{"startTime":3247.732,"endTime":3250.87,"body":"you about just one final proportion puzzle"},{"startTime":3250.87,"endTime":3252.31,"body":"which you can play with if you like."},{"startTime":3252.31,"endTime":3253.65,"body":"You know the old saying,"},{"startTime":3253.65,"endTime":3255.89,"body":"you can't fit a square"},{"startTime":3253.65,"endTime":3255.89,"body":"peg in a round hole."},{"startTime":3255.89,"endTime":3258.603,"body":"Well, you can, it's just there'll be gaps."},{"startTime":3258.603,"endTime":3260.54,"body":"So what I've got here, a square peg,"},{"startTime":3260.54,"endTime":3263.6,"body":"on the left, a square peg in"},{"startTime":3260.54,"endTime":3263.6,"body":"a round hole, on the right,"},{"startTime":3263.6,"endTime":3266.22,"body":"maybe this is better, a"},{"startTime":3263.6,"endTime":3266.22,"body":"round peg in a square hole."},{"startTime":3266.22,"endTime":3267.53,"body":"Right, what's better?"},{"startTime":3267.53,"endTime":3269.01,"body":"If you have to fit one"},{"startTime":3267.53,"endTime":3269.01,"body":"of them inside the other,"},{"startTime":3269.01,"endTime":3270.24,"body":"what's better?"},{"startTime":3270.24,"endTime":3272.832,"body":"And you can work out what's"},{"startTime":3270.24,"endTime":3272.832,"body":"better, 'cause you can work out"},{"startTime":3272.832,"endTime":3274.7,"body":"all the areas involved."},{"startTime":3274.7,"endTime":3277.97,"body":"So if the radius for this"},{"startTime":3274.7,"endTime":3277.97,"body":"circle is R, both of them,"},{"startTime":3277.97,"endTime":3279.55,"body":"then you get pi R squared."},{"startTime":3279.55,"endTime":3282.29,"body":"The big square there, it"},{"startTime":3279.55,"endTime":3282.29,"body":"has a diamond, so two R,"},{"startTime":3282.29,"endTime":3283.792,"body":"so it's area's four R squared."},{"startTime":3283.792,"endTime":3286.114,"body":"And what about the little square?"},{"startTime":3286.114,"endTime":3288.53,"body":"It's a tiny bit more"},{"startTime":3286.114,"endTime":3288.53,"body":"complicated, but actually look,"},{"startTime":3288.53,"endTime":3291.06,"body":"see how it fits just"},{"startTime":3288.53,"endTime":3291.06,"body":"exactly into the big square?"},{"startTime":3291.06,"endTime":3293.1,"body":"It takes up half that area."},{"startTime":3293.1,"endTime":3296.749,"body":"So the square peg has area two R squared,"},{"startTime":3296.749,"endTime":3298.99,"body":"and so you can work out what's better,"},{"startTime":3298.99,"endTime":3302.86,"body":"square peg in round hole,"},{"startTime":3298.99,"endTime":3302.86,"body":"you can work that out,"},{"startTime":3302.86,"endTime":3307.02,"body":"and you get that it takes"},{"startTime":3302.86,"endTime":3307.02,"body":"up about 64% of the area,"},{"startTime":3307.02,"endTime":3309.09,"body":"but the round peg in the square hole,"},{"startTime":3309.09,"endTime":3311.62,"body":"79%, that's much better."},{"startTime":3311.62,"endTime":3314.08,"body":"So it's better, categorically,"},{"startTime":3311.62,"endTime":3314.08,"body":"it's better to have"},{"startTime":3314.08,"endTime":3315.75,"body":"a round peg in a square hole"},{"startTime":3315.75,"endTime":3317.73,"body":"than a square peg in a round hole."},{"startTime":3317.73,"endTime":3319.551,"body":"That's very nice,"},{"startTime":3319.551,"endTime":3324.35,"body":"but this is peak"},{"startTime":3319.551,"endTime":3324.35,"body":"mathematician thing to do."},{"startTime":3324.35,"endTime":3326.29,"body":"In 1964, a mathematician"},{"startTime":3326.29,"endTime":3328.73,"body":"called David Singmaster"},{"startTime":3326.29,"endTime":3328.73,"body":"was thinking about this,"},{"startTime":3328.73,"endTime":3331.37,"body":"and he said, \"Well, that's all"},{"startTime":3328.73,"endTime":3331.37,"body":"very well in two dimensions,"},{"startTime":3331.37,"endTime":3333.04,"body":"but what happens in three dimensions?\""},{"startTime":3333.04,"endTime":3334.927,"body":"What if you get whatever, you"},{"startTime":3333.04,"endTime":3334.927,"body":"know, what's round and square"},{"startTime":3334.927,"endTime":3336.12,"body":"and three dimensions?"},{"startTime":3336.12,"endTime":3337.71,"body":"A sphere and a cube."},{"startTime":3337.71,"endTime":3340.08,"body":"What's the answer to this"},{"startTime":3337.71,"endTime":3340.08,"body":"question in three dimensions?"},{"startTime":3340.08,"endTime":3341.63,"body":"And you know, higher."},{"startTime":3341.63,"endTime":3344.187,"body":"What's better, square in"},{"startTime":3341.63,"endTime":3344.187,"body":"round, or round in square?"},{"startTime":3344.187,"endTime":3345.44,"body":"And you can work this out."},{"startTime":3345.44,"endTime":3347.08,"body":"We know the formula"},{"startTime":3345.44,"endTime":3347.08,"body":"for volume of a sphere,"},{"startTime":3347.08,"endTime":3349.29,"body":"and the volume of a cube."},{"startTime":3349.29,"endTime":3350.99,"body":"And it turns out that actually that, yeah,"},{"startTime":3350.99,"endTime":3354.526,"body":"the cube only fills"},{"startTime":3350.99,"endTime":3354.526,"body":"37% of sphere, rubbish."},{"startTime":3354.526,"endTime":3358.26,"body":"The sphere fills, well"},{"startTime":3354.526,"endTime":3358.26,"body":"still not very good,"},{"startTime":3358.26,"endTime":3359.093,"body":"but it fills more."},{"startTime":3359.093,"endTime":3361.77,"body":"It fills 52% of the cube"},{"startTime":3359.093,"endTime":3361.77,"body":"that it just sits inside."},{"startTime":3361.77,"endTime":3364.96,"body":"So again, round in square is"},{"startTime":3361.77,"endTime":3364.96,"body":"better in three dimensions."},{"startTime":3364.96,"endTime":3366.51,"body":"And you can check it in four dimensions,"},{"startTime":3366.51,"endTime":3368.157,"body":"and in five dimensions, round in square,"},{"startTime":3368.157,"endTime":3369.302,"body":"round in square is better."},{"startTime":3369.302,"endTime":3370.135,"body":"Round in square is better."},{"startTime":3370.135,"endTime":3371.81,"body":"So now we know round in square is better"},{"startTime":3371.81,"endTime":3373.856,"body":"in all dimensions, right?"},{"startTime":3373.856,"endTime":3377.952,"body":"No, (chuckles) it swaps"},{"startTime":3373.856,"endTime":3377.952,"body":"over nine dimensions."},{"startTime":3377.952,"endTime":3380.35,"body":"What? (chuckles)"},{"startTime":3380.35,"endTime":3381.974,"body":"This is really, I just"},{"startTime":3380.35,"endTime":3381.974,"body":"sort of wanted to leave you"},{"startTime":3381.974,"endTime":3384.63,"body":"with this thing,"},{"startTime":3381.974,"endTime":3384.63,"body":"actually, just a reminder."},{"startTime":3384.63,"endTime":3387.04,"body":"This is why we prove things in math,"},{"startTime":3387.04,"endTime":3389.52,"body":"because we can have a really"},{"startTime":3387.04,"endTime":3389.52,"body":"good idea of what's going on,"},{"startTime":3389.52,"endTime":3391.87,"body":"and then we can be completely confounded,"},{"startTime":3391.87,"endTime":3393.95,"body":"and that's why, you"},{"startTime":3391.87,"endTime":3393.95,"body":"know, math is so special,"},{"startTime":3393.95,"endTime":3396.42,"body":"because we don't just say,"},{"startTime":3393.95,"endTime":3396.42,"body":"\"Oh, that's probably right.\""},{"startTime":3396.42,"endTime":3398.83,"body":"We think about why it's right,"},{"startTime":3396.42,"endTime":3398.83,"body":"and we try and prove it."},{"startTime":3398.83,"endTime":3401.58,"body":"So I will finish there"},{"startTime":3398.83,"endTime":3401.58,"body":"with just a little plug"},{"startTime":3401.58,"endTime":3403.67,"body":"for my next lecture,"},{"startTime":3401.58,"endTime":3403.67,"body":"which is on the beauty"},{"startTime":3403.67,"endTime":3406.699,"body":"of geometrical curves, where"},{"startTime":3403.67,"endTime":3406.699,"body":"you can see this lovely thing"},{"startTime":3406.699,"endTime":3408.93,"body":"on March the 14th."},{"startTime":3408.93,"endTime":3410.495,"body":"All right, thank you very much."},{"startTime":3410.495,"endTime":3413.828,"body":"(participants clapping)"},{"startTime":3415.14,"endTime":3417.85,"body":"- Is the golden ratio considered to have"},{"startTime":3417.85,"endTime":3420.74,"body":"aesthetic properties across the world,"},{"startTime":3420.74,"endTime":3422.54,"body":"or is it just a European phenomenon?"},{"startTime":3423.47,"endTime":3427.157,"body":"- So I've, in the writings"},{"startTime":3423.47,"endTime":3427.157,"body":"I've seen about it,"},{"startTime":3427.157,"endTime":3431.56,"body":"it does seem to be quite a"},{"startTime":3427.157,"endTime":3431.56,"body":"Western idea, and as, you know,"},{"startTime":3431.56,"endTime":3435.07,"body":"as I said, actually, not"},{"startTime":3431.56,"endTime":3435.07,"body":"necessarily even a Western idea,"},{"startTime":3435.07,"endTime":3436.71,"body":"because this book"},{"startTime":3436.71,"endTime":3439.33,"body":"The Divine Proportion"},{"startTime":3436.71,"endTime":3439.33,"body":"actually didn't really make"},{"startTime":3439.33,"endTime":3441.283,"body":"claims for it, beyond, you know,"},{"startTime":3441.283,"endTime":3443.86,"body":"that it's lovely properties that are due,"},{"startTime":3443.86,"endTime":3446.02,"body":"to do with the mathematically"},{"startTime":3443.86,"endTime":3446.02,"body":"beautiful solids,"},{"startTime":3446.02,"endTime":3446.97,"body":"like the Platonic solids."},{"startTime":3446.97,"endTime":3449.63,"body":"So yes, it does have"},{"startTime":3446.97,"endTime":3449.63,"body":"associations with beauty,"},{"startTime":3449.63,"endTime":3451.89,"body":"but even in kind of the"},{"startTime":3449.63,"endTime":3451.89,"body":"ancient Western thought,"},{"startTime":3451.89,"endTime":3454.78,"body":"they didn't say it's,"},{"startTime":3451.89,"endTime":3454.78,"body":"these are the proportions"},{"startTime":3454.78,"endTime":3456.37,"body":"of the human face, or, you know,"},{"startTime":3456.37,"endTime":3458.62,"body":"Vitruvian man, not about gold proportions."},{"startTime":3458.62,"endTime":3461.28,"body":"It's about fitting into"},{"startTime":3458.62,"endTime":3461.28,"body":"squares and circles,"},{"startTime":3461.28,"endTime":3463.21,"body":"and if you look at the"},{"startTime":3461.28,"endTime":3463.21,"body":"actual ratios that are stated"},{"startTime":3463.21,"endTime":3465.16,"body":"by Vitruvius about the human body,"},{"startTime":3465.16,"endTime":3467.74,"body":"not one of them is the"},{"startTime":3465.16,"endTime":3467.74,"body":"golden ratio, so, you know,"},{"startTime":3467.74,"endTime":3470.29,"body":"maybe the answer is not"},{"startTime":3467.74,"endTime":3470.29,"body":"even Western. (chuckles)"},{"startTime":3470.29,"endTime":3472.56,"body":"- [Participant] Mentioning"},{"startTime":3470.29,"endTime":3472.56,"body":"the historical origins"},{"startTime":3472.56,"endTime":3477.09,"body":"of the measurements of"},{"startTime":3472.56,"endTime":3477.09,"body":"A4 paper, I had occasion,"},{"startTime":3477.09,"endTime":3479.42,"body":"I was reading a"},{"startTime":3477.09,"endTime":3479.42,"body":"mathematical economics text,"},{"startTime":3479.42,"endTime":3482.34,"body":"and I'd forgotten how to"},{"startTime":3479.42,"endTime":3482.34,"body":"solve a differential equation,"},{"startTime":3482.34,"endTime":3486.45,"body":"and I went back to my notes"},{"startTime":3482.34,"endTime":3486.45,"body":"as undergraduate 60 years ago,"},{"startTime":3486.45,"endTime":3488.682,"body":"and they're on foolscap paper."},{"startTime":3488.682,"endTime":3489.515,"body":"- Oh."},{"startTime":3489.515,"endTime":3490.531,"body":"- [Participant] Is there"},{"startTime":3489.515,"endTime":3490.531,"body":"any mathematical connection"},{"startTime":3490.531,"endTime":3493.953,"body":"between the dimensions of foolscap paper"},{"startTime":3493.953,"endTime":3496.94,"body":"in much the same way as"},{"startTime":3493.953,"endTime":3496.94,"body":"there is to A4 paper?"},{"startTime":3496.94,"endTime":3500.15,"body":"- I don't think, someone may correct me,"},{"startTime":3500.15,"endTime":3503.013,"body":"I don't think foolscap paper"},{"startTime":3500.15,"endTime":3503.013,"body":"has that same property."},{"startTime":3503.013,"endTime":3503.953,"body":"I mean, it's about,"},{"startTime":3503.953,"endTime":3506.82,"body":"it's just a little annoyingly"},{"startTime":3503.953,"endTime":3506.82,"body":"bit bigger than A4, right?"},{"startTime":3506.82,"endTime":3509.16,"body":"So if you have an A4 folder,"},{"startTime":3506.82,"endTime":3509.16,"body":"it won't quite fit in."},{"startTime":3509.16,"endTime":3511.038,"body":"I don't think, and in all those wonderful,"},{"startTime":3511.038,"endTime":3513.37,"body":"one does miss sometimes, I know, you know,"},{"startTime":3513.37,"endTime":3515.27,"body":"I love the A4 and all of that."},{"startTime":3515.27,"endTime":3517.92,"body":"I love it, I love the"},{"startTime":3515.27,"endTime":3517.92,"body":"mathematics of it, but, you know,"},{"startTime":3517.92,"endTime":3520.89,"body":"Double Elephant, and Imperial Eagle,"},{"startTime":3520.89,"endTime":3524.35,"body":"and all of these wonderful"},{"startTime":3520.89,"endTime":3524.35,"body":"old names for paper sizes,"},{"startTime":3524.35,"endTime":3525.6,"body":"you do miss them a bit, but yeah,"},{"startTime":3525.6,"endTime":3529.854,"body":"I don't think that foolscap"},{"startTime":3525.6,"endTime":3529.854,"body":"has that property, no."},{"startTime":3529.854,"endTime":3530.687,"body":"- [Participant] Thank you."},{"startTime":3530.687,"endTime":3531.73,"body":"- [Participant] Am I correcting saying"},{"startTime":3531.73,"endTime":3536.73,"body":"that one mile is 1.61 kilometers?"},{"startTime":3538.35,"endTime":3541.52,"body":"Is it actually the exact"},{"startTime":3538.35,"endTime":3541.52,"body":"value, the gold ratio,"},{"startTime":3541.52,"endTime":3544.236,"body":"or is the French Revolution something?"},{"startTime":3544.236,"endTime":3545.31,"body":"(instructor laughing)"},{"startTime":3545.31,"endTime":3547.237,"body":"- Oh, that would be, I"},{"startTime":3545.31,"endTime":3547.237,"body":"would love it if that-"},{"startTime":3547.237,"endTime":3549.6,"body":"It's, I believe it's a coincidence,"},{"startTime":3549.6,"endTime":3552.29,"body":"but it's a very useful coincidence,"},{"startTime":3549.6,"endTime":3552.29,"body":"because then you can use"},{"startTime":3552.29,"endTime":3554.92,"body":"the Fibonacci numbers to"},{"startTime":3552.29,"endTime":3554.92,"body":"convert miles to kilometers"},{"startTime":3554.92,"endTime":3556.3,"body":"in your head, right?"},{"startTime":3556.3,"endTime":3558.623,"body":"50 miles is 80 kilometers,"},{"startTime":3556.3,"endTime":3558.623,"body":"that's Fibonacci, right?"},{"startTime":3558.623,"endTime":3561.83,"body":"So it's a very useful"},{"startTime":3558.623,"endTime":3561.83,"body":"thing to be aware of,"},{"startTime":3561.83,"endTime":3565.09,"body":"and occasionally, I remember"},{"startTime":3561.83,"endTime":3565.09,"body":"a time when it feels"},{"startTime":3565.09,"endTime":3567.41,"body":"like it's impossible, but"},{"startTime":3565.09,"endTime":3567.41,"body":"I believe there was a time"},{"startTime":3567.41,"endTime":3569.21,"body":"when the exchange rate from the pound"},{"startTime":3569.21,"endTime":3572.87,"body":"to the dollar was also"},{"startTime":3569.21,"endTime":3572.87,"body":"one, (chuckles) like 1.6."},{"startTime":3572.87,"endTime":3575.51,"body":"Those days feel rather"},{"startTime":3572.87,"endTime":3575.51,"body":"distant now, but then again,"},{"startTime":3575.51,"endTime":3577.6,"body":"it's very easy to convert"},{"startTime":3575.51,"endTime":3577.6,"body":"backwards and forwards,"},{"startTime":3577.6,"endTime":3580.83,"body":"because of this one over phi"},{"startTime":3577.6,"endTime":3580.83,"body":"equals phi minus one thing,"},{"startTime":3580.83,"endTime":3582.879,"body":"so instead of having to"},{"startTime":3580.83,"endTime":3582.879,"body":"do one over something,"},{"startTime":3582.879,"endTime":3584.67,"body":"you can just take away one,"},{"startTime":3582.879,"endTime":3584.67,"body":"that's really, really useful."},{"startTime":3584.67,"endTime":3586.48,"body":"So I don't think it's on purpose,"},{"startTime":3586.48,"endTime":3588.6,"body":"but it's certainly a nice property"},{"startTime":3588.6,"endTime":3590.54,"body":"for kilometers and miles to have."},{"startTime":3590.54,"endTime":3593.2,"body":"- Last quick question,"},{"startTime":3590.54,"endTime":3593.2,"body":"which is an online one,"},{"startTime":3593.2,"endTime":3595.52,"body":"it's a Fibonacci one,"},{"startTime":3593.2,"endTime":3595.52,"body":"you'll be pleased to know."},{"startTime":3595.52,"endTime":3599.13,"body":"Given that the Fibonacci"},{"startTime":3595.52,"endTime":3599.13,"body":"sequence is divergent,"},{"startTime":3599.13,"endTime":3603.44,"body":"and its terms tend to"},{"startTime":3599.13,"endTime":3603.44,"body":"infinity, does it have a limit?"},{"startTime":3603.44,"endTime":3605.64,"body":"And if not, is there a set range"},{"startTime":3605.64,"endTime":3608.333,"body":"of values to consider as significant?"},{"startTime":3609.48,"endTime":3612.7,"body":"- So it itself, the"},{"startTime":3609.48,"endTime":3612.7,"body":"questioner is quite right,"},{"startTime":3612.7,"endTime":3614.15,"body":"that the terms just get"},{"startTime":3612.7,"endTime":3614.15,"body":"bigger and bigger and bigger,"},{"startTime":3614.15,"endTime":3615.91,"body":"so it does, it diverges."},{"startTime":3615.91,"endTime":3618.92,"body":"It goes off to infinity,"},{"startTime":3615.91,"endTime":3618.92,"body":"but the thing we can say is"},{"startTime":3618.92,"endTime":3620.68,"body":"when we take, you know, these ratios,"},{"startTime":3620.68,"endTime":3622.18,"body":"they are approaching a limit."},{"startTime":3622.18,"endTime":3624.277,"body":"So you know, it, as you"},{"startTime":3622.18,"endTime":3624.277,"body":"get further and further up,"},{"startTime":3624.277,"endTime":3627.91,"body":"each term will be about 1.61833, whatever,"},{"startTime":3627.91,"endTime":3629.84,"body":"times the previous one, but yeah,"},{"startTime":3629.84,"endTime":3632.256,"body":"the terms themselves"},{"startTime":3629.84,"endTime":3632.256,"body":"do go off to infinity."},{"startTime":3632.256,"endTime":3634.17,"body":"- Professor Hart, thank you very much"},{"startTime":3634.17,"endTime":3635.611,"body":"for another tremendous lecture."},{"startTime":3635.611,"endTime":3638.354,"body":"(participants clapping)"}]}