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  • I was actually e mailed by Cem number file viewers Young's and you and goggle.

  • Rajiv.

  • They've been doing a project about cutting your fingernails, and they sent it to me.

  • Andre was some pretty cool months in there, which I sort of developed.

  • It would be nice to talk about what we're talking about.

  • Here is what happens when you cook them with a pair of scissors.

  • You know, you might eyes just gonna freak you out, so I gotta cut.

  • That's a cut from the bottom, and you probably would trace it.

  • Cut to the top on.

  • Then you do another one like that and then on then I would cut across the top like that.

  • You can you carry Argo, right?

  • Did you get I smoothed things right?

  • I don't know where they went flying off to this door.

  • Picture of our fingernails will draw.

  • It's a semi circle.

  • Magic's got radius warm.

  • So the first cuts where we cut out a triangle, right?

  • This is a 45 degree angle.

  • So next cuts so you don't want a big pointy thing.

  • You know, it cuts across the top in such a way that each new side that we create, They're all the same length.

  • Do that again, We do another cuts the idea being that we now got four sides on that, these are all the same length.

  • And of course, if I do this enough times, it's going to start looking like a narc of a circle.

  • Okay, I'm not your goal.

  • That's where you want to endure pack, right?

  • Because you don't want, like, weird.

  • Yeah, you know what?

  • Stop the fingers.

  • Nails, Right?

  • You want nice, smooth, beautiful fingernails.

  • So you want to reach a circle?

  • So we do that by doing successive polygons.

  • Essentially.

  • So you could carry on with this.

  • Adam for Knighton human, you've got an infinite amount of time after 100 cuts.

  • Could definitely looked like a circle.

  • Okay, now the question is, how big is this circle?

  • Because we never caught at this point again after the first court.

  • Thes angles here are still going to be a 45 degrees, and that's very important.

  • That tells us a lot.

  • So I'm now going to draw a line there, which is a right angle to this new circle that I've traded here, and I'm going to do another one coming from this point on the circle again at right angles.

  • Now what does this tell me?

  • Because this is hitting the tangent to the Circle Triangle.

  • This is actually a radius of the new circle.

  • I can kind of complete the new circle.

  • The other thing, I noticed this point here.

  • Whilst it's also the center of the big circle, it's also the bottom of the top circle.

  • That's because of something that you know about circles is that if you take two points diametrically opposite and you connect them by any point on this a conference, they will meet in 90 degrees.

  • Do you know the picture now?

  • From this, we can very easily work out how much we've cut off our original circle had radius of one.

  • A little bit of Pythagoras tells you that this line here is route to from there to there is route to from there to there is one.

  • So from there to there is route to minus one.

  • So now we can compare the ratio between the size of our nail before to the sides of the nail Afterwards, they're sort of height before Call hates before ever hate Shafter hates before was one hate after his route to minus one.

  • And it's not too difficult to show that this is the same as route, too.

  • Flus one.

  • Well, big deal.

  • What's so good about by?

  • Cut my fingernails.

  • So the rank want out before was bigger than the one I have after by a factor of the silver issue.

  • That's what the silver rations there.

  • Okay, so what is the silver Asia you know about the golden ratio, right?

  • So golden ratio you think about with the Fibonacci sequence.

  • Okay, that's one way of understanding with the golden ratio comes from, so you start off with zero on one, and then you add these two together and you get one.

  • And then you have these two together, and you get to have these things together.

  • You get three pieces together, you get five carry on like that.

  • That's gives you the Fibonacci sequence.

  • The ratio between terms and this Fibonacci sequence tends towards the golden ratio.

  • The celebration is slightly different.

  • So you start off at the same 0.1 You double this and you add that gives us two double this.

  • And at that, that's not four plus one is five double this.

  • Not that.

  • That's 12.

  • Double this and add that That's gonna give me 29.

  • And I carry on like that with that pattern.

  • If I wanted to write down what that part is, it's like this city and is to pee and minus one close p and minus two.

  • So in other ways, I take the one before I double it.

  • And then I had the one before that You can see this ratio between consecutive ones will tend to the silver Asia.

  • So you do it the same way you should derive the golden ratio.

  • So I just take this expression here and I divide through by P N minus one.

  • So that gives me play and divided by P N minus.

  • One is two plus he and miners to over and minus one.

  • So at large, end this ratio because what I'm gonna call Delta S this is too.

  • And this is one over the ratio that's one over Delta.

  • And if you solve the equation, you get route to plus one.

  • Okay, so that's the silver issue.

  • So there is an entire family of these guys and they're called the metallic ratios.

  • This is the pale sequence.

  • Actually, there's some fun facts about this.

  • If the Pel number is prime, it's indexes prime.

  • Okay, look, it's position.

  • Yeah, exactly.

  • Its position is prime.

  • We'll call this P too.

  • Okay, so this is the second pound of count zero.

  • Not counting zero.

  • We'll call this one p three.

  • Okay.

  • So you can see that this is a prime, and this is prone.

  • This is prime.

  • And this is prime.

  • This one here is also a prime number and yet surprise.

  • Surprise.

  • It's the fifth pal number, which is also print.

  • So has this cool property that if they pal number is prime, it's indexes prime.

  • The other weird property of this sequence is it contains very few powers of anything.

  • So But by that, I mean, does it contain squares or cubes or fourth powers?

  • Well, very few, actually.

  • The only ones that contains in the sequence are zero one shows a power 169 is in there.

  • Okay.

  • Which is obviously 13 squared.

  • There are no others.

  • There are no fourth powers.

  • There are no other squares.

  • There are no other 26 powers not fit.

  • That's it.

  • this pal sequence gives you the silver A ship.

  • There are, of course, generalizations of this that give you the other metallic ratio.

  • So you've got the three been, actually sequence it.

  • 013 10 thirty, three hundred nine Got started with these two.

  • You take this one, multiply it by three.

  • And at this take this one.

  • Multiply it by three.

  • And at this take this one.

  • Multiply it by three and had this and you go on like that.

  • It is the Fibonacci sequence.

  • Multiply it by one and adding the one.

  • Yes, yes.

  • On all the other metallic ratios come from multiplying by a different number.

  • Thought it seemed arbitrary, but I hadn't thought of the Fibonacci sequence is fitting this, but it does.

  • Yeah, Absolutely.

  • Absolutely.

  • So in general, you can think about sort of end Banacci sequence, which would have the rule he and is an P N minus one close and minus two.

  • So you take one of the numbers, you multiply it by N.

  • You had the one before.

  • That gives you your new number.

  • So this would give you the bronze Rachel and equals four and equals five.

  • They don't have to really have names.

  • S O.

  • I think some people call it copper Nickel, but I think we're free to call him what we like, actually.

  • So the bronze ratio, which will call Delta Bi with limit of the ratio of these terms, That's three plus Route 13 over to okay, rights, which is roughly 3.30 Very.

  • Yeah, but they're not right, because they can think about what you get from the and form.

  • Okay, The 10th 1 The ratio that you would associate with that I'll call it Delta End.

  • And that is N plus the square root of n squared.

  • Plus four over to.

  • And you get that just by applying the same method as we did for the silver ratio here.

  • Everything you think about the golden ratio will generally have an analog here.

  • So you could talk about golden rectangles.

  • You can talk about silver rectangles, bronze rectangles.

  • You can talk about golden spirals.

  • You can talk about silver spirals, bronze spirals.

  • So you know.

  • So what were their properties?

  • Okay, so we probably need some fake.

  • They're actually really?

  • So the golden rectangle is the following sea.

  • Take a rectangle.

  • Okay.

  • On its sides have the ratio of the golden ratio.

  • So if this one's got length one, this one will have length.

  • The golden ratio, which is one plus Route five truly going, I already suspected, Not even right look at the state thinks we're really a golden rectangle.

  • What property would have if I removed the largest square?

  • Then what's left over would be another golden rectangle basically won over five minus one is actually equal to what's the silver version of this?

  • We draw a rectangle whose sides have ratio given by the silvery ship removed two large squares, both of side one on what we have left over is going to be another silver rectangle.

  • If you did it with the rectangle whose sides were in the bronze ratio, you'd have to remove three and then you'd get one left over, which was another runs rectangle in England.

  • Of course, we have a four paper.

  • Now you may know something about you for paper, the rip, the ratio between the length of the sides is route to This is one.

  • This would be rude to.

  • This is another kind of rectangle.

  • Actually, another kind of interesting one is not just a four.

  • This is sometimes also called a silver rectangle.

  • But it's not our silver rectangle.

  • So we're not gonna call it still director, because some people would call a route to the silver ratio.

  • Actually, we're gonna call it the Japanese race you because it is big in Japan, This ratio this with this route to and these what I'm gonna I'm gonna call the Japanese Rectangle.

  • Lots of Japanese architecture uses this particular ratio on these particular rectangles You associate Roman alters all these sorts of things, but we're interested in the real silver Asia.

  • So how would you create it from this guy?

  • You would just remove the largest square kind of a pro.

  • Phyllis feels like origami and talk about a Japanese rectangle.

  • So this would be the largest square here.

  • And if I remove it, I'm gonna do it quite crudely.

  • I would get this.

  • This is a silver rectangle.

  • This won't work in America.

  • Yeah, for paper is wrong.

  • In America, you have to beat England, finish the work.

  • Well, some other places that have a full faith of it definitely could.

  • This could work in England not necessary in America.

  • So this is still a rectangle.

  • Now, if I start removing two larger squares from this, I'll get with the silver rectangles ever decreasing.

  • So those are the rectangles that you can have.

  • It is again.

  • It's a whole family of them that you could talk about that more interesting than the rectangles, of course, And the spirals.

  • Okay, now the spirals of great because of the spirals of the things that everybody says appear everywhere now, people normally say, Oh, the golden ratio is everywhere, isn't it?

  • It's everywhere because they're talking there about the spirals, but it's not really the golden spiral.

  • It's appearing everywhere.

  • It's all These are the ones we start off with a square on.

  • We draw the circle that goes between there and there.

  • Now we create another square, which is a factor smaller by that racial were interested in.

  • So we decrease it by a factor of Delta.

  • Okay, what?

  • Delta's out of the golden ratio of the silver ratio, the brand ratio, whichever one you're interested in when we continue our circle.

  • But now we create another square.

  • But we dropped by the ratio again on we continue our curve around, and then we go on again, We're gonna have one over Delta.

  • Cute.

  • But then we go around like this and we create a spiral.

  • If this was the golden ratio, this would be a golden spiral.

  • If this was a silvery show, it's a silver spoil.

  • It was a bronze Bronze rations, a bronze spout.

  • These are all a family of spirals which have called logarithmic spirals.

  • So what is that?

  • A formula for these spirals of this particular one I can compare, You know, I can think of an angle, the angle that I've gone round.

  • So I'm going round like this, even changing angles.

  • I can ask, How does the distance from the centre change?

  • So the distance from the centre alcohol are Well, you can sort of see that if I if I were to go around this way, I gain a factor of delta every time I go around 1/4 turn.

  • So actually, it turns out that our will go like Delta two over Pi.

  • The tour of a pie, of course, is or pie over to is, of course, 1/4 turn in radiance.

  • Okay, so this is the formula for this.

  • This spiral.

  • It's pirate with y o u t o.

  • Because I need to scale it by the number of radiance.

  • Okay?

  • And this is part of a larger family.

  • The logarithmic spirals which looked like this.

  • Okay, so feet of times.

  • Hey, so what is P here?

  • He is called the pitch of the particular spiral that you're interested in, and that's very characteristic of it.

  • Imagine I'm coming along this spiral on.

  • I'm going along this direction, Call an Alfa and then I could draw a right angle there.

  • I'll call that he and that's called the pitch of the Spiral.

  • Now each spiral has its own pitch.

  • Okay, so we can write down what the pictures are for the different spirals that we're interested in.

  • So for the gold for the golden spiral, the pitch is actually you can show it's about 17 degrees.

  • You can wait out by comparing these two, formerly just comparing the two.

  • You put it in the golden ratio.

  • Here you can extract what the pictures.

  • It's where excited.

  • About 17 degrees with a silver.

  • It corresponds the pictures about 29 degrees For the bronze.

  • It's 37 degrees N equals four.