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• so I have to calipers or dividers, They're both the same.

• Set them up.

• You can see the jump from here.

• We have the jump from this one to this.

• And that's the same on this set from here to here.

• And when I line them up, you see that the second division sort of grows at the same rate.

• So now I've got this one over here.

• But turns up that and if you're following along carefully, you can now actually work out what the number is.

• But let's just stay with the dividers for a little bit.

• Now, if I sort of widen this out a little bit, that's lined up there on then this middle one is now lined up with the gap of the 1st 2 But this one isn't.

• I land up with these two, so I'm gonna widen it a little bit further so that I get out this one.

• Now this middle one is like that on this last one is the full thing.

• Are those two sets of calipers decide they are, sir, If I go back to they are exactly so we could try and do it with one pair, but it's hard to line something up against itself.

• So when I look at the different possible jumps from this one to here from here to here from here to these two from these two to these to and from these two to the full thing, it's the same ratio every time.

• Is that a specific ratio?

• Is it many things?

• Let's find out.

• So we sort of have from here to here.

• Let's call that one on.

• Then, from here to here, that's going to be X.

• This is our mysterious number.

• And then from here to here, that's going to be ex squared, because whatever proportion we took from here to here, the same proportion go similar to that.

• Now let's go from here to here, and that is going to be ex cubed.

• If you recall this leap hair fitted on to this here and now, we've got something which is going to give us an equation.

• We can say that X cubed is equal tow X plus, and this might be familiar to people who've seen the golden ratio, which is a very similar formula.

• But you've got next squared if X plus one is X cubed weaken, just multiply X and one.

• So this is going to be next to the four.

• And now we can look at what we've got him.

• I've got my X squared and here I've got my X cubed.

• When I combined them together, I have sort of x squared times x plus one.

• And by this same equation, we get X squared times X cube, which is X to the five across this whole thing.

• We get X to the fight.

• And so this value off X, this polynomial has three solutions, but two of them are complex numbers.

• They're clearly not what we want, but this is 1.3 to 5 and it is called the plastic ratio.

• Bit unfortunate as a name.

• Why is it called that?

• Well, it was named before plastic was name plastic on gets the old meaning of plastic, meaning a sort of flexible or multipurpose number on one of the first people to investigate it.

• Han Stallone, who was actually an architect and a monk.

• I think he named it the plastic ratio because he thought this was the rial ultimate system of beauty that you could use architecture and all that stuff and he even designed a building on the basis of it.

• And so he was really setting it up in opposition to the golden ratio.

• The real factors that there isn't a single number which can express all beauty.

• But this number the plastic ratio has many of interesting properties like the golden ratio on DSO.

• It is quite useful for architects to, you know, some nice ideas to use in design.

• So we've got these calipers.

• But maybe we could look at a couple of the other things thinking about the golden ratio.

• One of the things you can do is I can just take a square and then build another square.

• And then I can put a square onto them.

• Here, here, here.

• And I have 112 three, five Your classic golden spiral.

• What would happen if instead of using squares here, we used equilateral triangles.

• I start with a neck electoral triangle, put another one on there and get another one on the But now I've got a bigger one.

• It's going to get their bigger one again.

• One more over here.

• So now the side lengths of one one on one, and then I get This is gonna be too.

• This is the same size.

• So that's two.

• This is two on one.

• So that's three.

• And this is three and one that that's so that's four.

• This one here is going to be four on one.

• So that's five.

• So we get instead of the Fibonacci numbers 11235 We get what accord?

• The pad A van numbers, which are 111 22345 The next one is going to be two and five.

• So that's seven.

• Then we're going to get seven and two, so that will be nine similar to the Fibonacci.

• But you, instead of just taking the two previous ones, you skipped one.

• So I you know, I start, let's start with just 111 Fibonacci would be 11 I'd add these two, but I'm going to skip this and I'm gonna add these two together so I get to That's the sum of these two.

• Now, I'm going to skip that, too, and add the two ones.

• And that's to skip that on at these two.

• Three.

• Skip that, Matt.

• These 24 579 12 16 the effect of having the equilateral triangle is you get that skipping.

• If it because you're going round three sides, you miss out beside you just put in.

• Then you're connecting things up.

• How does that tie into the plastic ratio?

• It's, ah, similar rule back to the golden ratio.

• How did we find that the Fibonacci numbers end up being the ratio between one the next, with the way you look at it as I replace this drawing where everything is squares by one rectangle like hot a square off on Dhe.

• If this is five on, then one.

• So now I have a square.

• This is one and one.

• This is length one, and so this length is fi minus one, but because fi minus one is equal to 1/5.

• So this rectangle here is similar to the hole rectangle.

• As I build up my spiral, I get closer and closer to Fi with this.

• It's a slightly more complicated set up because I've got more sides there.

• But if I look at how the sides come together and I can say if what if we started with one instead of doing this addition?

• We said this was acts and then this is X squared.

• This term here is now going to be one plus x two.

• This is X Cube, but it's also one plus X.

• We again get X cubed is equal to one plus X, which is familiar from what we had for the plastic ratio.

• So I actually have a bit of a challenge for number file viewers.

• There's something that I came up with.

• It's just such a simple thing that I can't believe it's not out there somewhere on dso find find other references to this sort of idea.

• I guess we need to come up with a good Greek letter for the plastic ratio pie would be be good, but that's sort of taken to sigh for plastic.

• With the golden ratio, I cut off a square in this case, instead of cutting off the whole square, I'm gonna cut off a similar rectangle.

• So that gets me about Yeah.

• Now I'm gonna cut off a square, and I want this rectangle here to be the same as the hole.

• I think this is a similar rectangle.

• So if the long side is no one, this is going to be one of than this length is going to be a sigh minus one over.

• So we want this to be similar.

• So the length is going to be this thing divided by size.

• That's one minus one over Site square.

• This is a square.

• So this length is also signed minus one over.

• So now I can add these two together and I get one.

• So I have one equals Cy minus one.

• I have a sigh plus one minus one over.

• So squared.

• So now we've got one on both sides so we can cancel those.

• And then we're gonna multiply through my squared zero, huh?

• On this again gives us that same equation.

• So this box, if we have the property that we take off a similar rectangle, we take off a square and we're left with similar rectangle has the same proportion that we've been looking at again.

• We get the same equation coming out.

• You know that this idea of the golden ratio is one that's just such a classic.

• Anyone interested in mathematics and beauty would have at least seen that.

• But when you've got a very similar process, you take off a similar rectangle a square and you're left with a similar rectangle.

• I've found some small references and occasionally which people will will grow up something.

• But that doesn't seem to be any systematic study of these sort of proportion systems.

• And what, what, you can play around with them Gold ratios.

• Famous people will say it's the most irrational number.

• What's the plastic razors claim to fame?

• It turns up in some of the more exotic, sort of still ations of poly he drove.

• It turns up in this wonderful spiral, but I think actually, this this divider is the real plastic ratio claim to fame because we had the property.

• If you notice that I can choose any two of these pins on dhe, the gap between them will be the power of one number.

• There is a version of this you can do with the golden ratio, but that only has three pens, not not four.

• There's no way to do that with five pence.

• So if you want a set of dividers where the jump between any two pins is a power of the same number, then this is the best you can do, right, huh?

so I have to calipers or dividers, They're both the same.

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# The Plastic Ratio - Numberphile

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林宜悉 posted on 2020/04/05
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