So I have two
calipers or dividers - they're both the same.
If I 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 that
turns up
there.
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
and
then this middle one is now lined up with the gap of the first two, but this one is
now lined up with these two.
So I'm going to widen it a little bit further so that I get out this one.
Now
this middle one is like that
and this last one is the full thing. - (Brady: Are those two sets of calipers the same?)
They are. So if I go back to, they are exactly
the same. 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 two, 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 1. And 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 x squared
because whatever proportion we took from here to here the same proportion goes from there to there. Now let's go from here
to here. And that is going to be x cubed.
If you recall this leap here
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 to x plus
1. And this might be familiar to people who've seen the golden ratio, which is a very similar formula,
but you've got an x squared here.
If x plus 1 is x cubed we can just multiply
x and 1
so this is going to be x to the 4. And now we can look at what we've got and here I've got my x
squared and
here I've got my x cubed. When I combine them together I
have sort of x squared times
x plus 1. And by this same equation we get x squared times x cubed
which is x to the 5. Across this whole thing we get x to the 5. And so this value of x,
this polynomial, has three solutions but two of them are complex numbers so they're clearly not what we want.
But this is one point three to five and
it is called the plastic ratio.
Which is a bit unfortunate as a name. - (Brady: Why it it called that?) - Well, it was named before
plastic was named plastic. And it's the old meaning of plastic, meaning a sort of flexible or multi-purpose number. And
one of the first people to
investigate it, Hans der Laan, who was actually an architect and a monk,
I think he named it the plastic ratio because he thought this was the real, ultimate
system of beauty that you could use for
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 fact is 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
and so it is quite useful for architects to you know, there's some nice ideas to use in design.
So we've got these calipers
but maybe we can 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 I build
another square and then I can put a square onto them here
and then a square here and a square here and I have 1, 1, 2, 3, 5:
your classic golden spiral. What would happen if instead of using squares here we used equilateral triangles?
I start with a equilateral triangle, put another one on there and get another one on there.
But now I've got a bigger one. It's gonna go there, bigger one again. One more over here.
So now the side lengths are 1,
1, and 1. And then I get this is going to be 2,
this is the same size so that's 2, this is 2 and 1,
so that's 3. And this is 3 and 1 so that's, so that's 4.
This one here
is going to be 4 and 1
so that's 5. So we get, instead of the Fibonacci numbers,
1, 1, 2, 3, 5; we get what are called the Padovan numbers which are 1, 1, 1, 2, 2,
3, 4, 5. The next one is going to be 2 and 5 so that's 7.
Then we're going to get 7 and 2 so that will be 9.
Similar to the Fibonacci, but you instead of just taking the two previous ones you skip one.
So I, you know I start,
let's start with just 1, 1, 1.
Fibonacci would be 1, 1; I'd add these two but I'm gonna skip this and I'm gonna add these two together.
So I get 2, that's the sum of these two. Now I'm going to skip that 2 and add the two ones and that's 2.
Skip that and add these two - 3. Skip that and add these two - 4. 5,
7, 9,
12,
16. The effect of having the equilateral
triangle is you get that skipping effect. Because you're going round three sides you miss out the side you just put in then you're
connecting things up. - (Brady: How does that tie into the plastic ratio?) - It's a similar rule back to the the golden ratio.
How did we find that the Fibonacci numbers end up being,
the ratio between one the next? Well the way you look at it is I replace this drawing where everything is squares
by one rectangle, and I cut a square off and if this is
5 and then 1, so now I have a square,
this is 1 and 1, this is length 1. And so this length is
5 minus 1.
But because
phi minus 1 is equal to 1 over phi,
so this rectangle here is similar to the whole
rectangle. As I build up my spiral I get closer and closer to phi. 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 - well if we started with 1, instead of
doing this addition we said this was x, and then this is x squared,
this term here is now going to be one plus x.
So this is x cubed
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, for Numberphile 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 and so 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. Pi would be be good
but that's sort of taken. - (Brady: Yeah) - Psi for plastic. With the golden ratio
I cut off a square. In this case instead of cutting off a whole square I'm going to cut off a
similar
rectangle. So that gets me about
here.
Now I'm going to cut off a square
and I want this rectangle here to be the same as the whole thing.
This is a similar rectangle, so if the long side is now 1, this is going to be 1 over psi.
Then this length is going to be psi minus 1 over psi.
And we want this to be similar
so the length is going to be this thing divided by psi. So that's 1
minus 1 over
psi squared.
This is a square, so this length is
also psi minus 1 over psi. So now I can add these two together and I get 1.
So I have 1
equals psi
minus 1 over psi,
plus
1 minus 1 over
psi squared.
So now we've got 1 on both sides so we can cancel those, and then we're going to multiply through by psi squareds.
0...
And 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 a similar
rectangle, has this 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 will, will, people will draw up something but there doesn't seem to be any
systematic study of these sort of proportion
systems and what what you can play around with them. - (Brady: The Golden ratio is famous. People will say it's the most irrational number!)
(What's the plastic ratio's claim to fame?) - It turns up in some of the more exotic, sort of stellations of polyhedra.
It turns up in this wonderful spiral, but I think actually this, this divider is the the real plastic ratio
claim to fame.
Because we had the property if you, you noticed that I can choose any two of these pins and
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 pins not,
not four. There's no way to do that with five pins.
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 the best you can do.