Everyone knows about the Fibonacci numbers
1, 1 and then I add those two together to get 2, then 3, 5, 8...
What would happen if instead I start with three numbers and now I'm going to add all three together.
So that gives me 3 and I'm now going to add these three together. That's gonna give me
5 and then these three together that's going to give me 9 and then I'm gonna get 9,
5 and
3;
17. And we're gonna get a different sequence of
numbers. Well, it's a bit like the Fibonacci, but it's got three instead of two. So it's called the Tribonacci numbers.
Let's call this Fn and so this is the Tribonacci Tn.
And you have that Fn plus 1
divided by Fn goes to the golden ratio.
So we can also do the same thing
with the Tribonacci numbers
but they go to what's called the Tribonacci constant. With the
Fibonacci numbers one of the ways we can sort of think about what, how they're going to end up is say, well
let's start this 1, x,
x squared,
and x squared was equal to the two previous ones together.
So here this is x squared is equal to x plus 1.
Here we're adding the three previous terms. So we get 1, x,
x squared,
this term is going to be the sum of all three.
So it's x cubed, and now we have x cubed is equal to x squared plus x
plus 1. And so that's a polynomial. Whilst I can reasonably well solve quadratics,
cubics have a far more messy to solve so I'm going to get a calculator to do that.
So this gives us 1.83929
and that's gonna... - (Brady: Tribonacci constant!) - Tribonacci constant.
(Brady: I'm assuming there's a fourbonacci constant and a fivebonacci?) - They're called the n-bonaccis and,
so here I've got an interesting shape. It's got a, a fractal boundary going around it.
Well, it's got a nice property to it.
So I've got this tile here and this is the same shape but a little bit smaller. And then I've got again the same
shape but smaller. The edges of these, if can find it, I have to get them the right way up and
there we go. So the edges are such so this shape, at different sizes, actually fits onto itself.
And that means that these edges might, must either be straight lines, which they're clearly not, or fractal.
It's not a fractal itself, but the boundary is a fractal. So now I will
take a third shape and put them there and
magically, I have my same shape.
These are three sizes of this one shape and when I put those three together, I
get the same shape. If this one scales up to
this one, this one scales up to
this one, and then this one
scales up to the whole thing, I
can take this one step further. So what I've done here is make a larger version of this object.
So this piece here becomes this tile here.
This piece here becomes this tile here. And then this tile here is replaced by
the whole piece here. It's the, the three tiles - if I rotate this to
there you see that this is just these three titles.
So now we could imagine doing this another time, and another time we can get larger and larger
regions. We could ask, well how many tiles am I going to have in them? Well, this pattern is made up of
one of the smallest tiles, one of the medium tiles, and one of the large tiles. This is made up of
one medium, one large, and then one
of the scale size beyond large, which itself is made up of:
one small, one medium, and one large. And when I go into the next step up, I'm gonna have one copy of this,
one copy of this, and then one additional large one.
So at each stage I'm going to take the three previous pieces,
put them together to get the next one up. And
so, what are we going to have? Well it's back over here.
I have my 1 1 1, I put those together so I get 3.
1, 1 and 3 gives me my 5. The next one up is going to be my 5,
my 3, and my 1 which is going to give me 9, and so on like that.
So this shape is called the Rauzy fractal and it was found by Gérard Rauzy. The interest is sort of studying how
matrices act on
three-dimensional space. And there's a lot of nice theory for how, for certain properties of how
matrices act on
two-dimensional space. Especially a matrix, which has one
expansion and one contraction. What that will do is you have these things called
eigenvectors and you pull down in one direction and you expand so that everything gets squashed down and pulled up.
There's a whole theory in that called Markov partitions,
which have - basically have nice properties as you get pulled down and smeared out,
you can turn that into a nice periodic product. And what rauzy asked,
well, what about for the the cubic case when we're dealing with three-dimensional space?
So the specific matrix you're studying here has as one of its
eigenvalues this number, the Tribonacci constant. And so because the Tribonacci constant is turning up in the
thing you're trying to study, that's how it turns out in these these tiles.
And the golden ratio and Fibonacci numbers are really important in the study of the two-dimensional Markov
partitions, and so when you go up to three dimensions, you have to deal with cubic things. It's quite natural,
it's sort of almost the ancient Greeks who, well, who felt that a cubic number really was a cube.
So they didn't, some people say they didn't even believe in fourth powers. When you're dealing with three dimensions
you naturally have cubic stuff. - (Brady: Do they exist in sunflowers and things like that?)
They don't exist in nature in the way that they, just like the Fibonacci
isn't quite as popular by - in nature as it's meant to be. Umm, the only place
beyond this that I know of them is an object called the Snub Cube, which is quite an attractive little polyhedron.
Just a reminder that there's now a Numberphile podcast. Have you had a listen yet?
This is what it looks like on your podcast apps, just search for Numberphile.
And if you haven't got into this whole podcasting thing but you still want to have a listen,
I am putting the episodes on the Numberphile2
YouTube channel. There are links on the screen and in the description.