Welcome to another Mathologer video. The Golden Spiral over there is one of the

most iconic pictures of mathematics. The background of the picture is the special

spiral of squares and the golden spiral itself is made up of quarter circles

inscribed into these squares. Overall this quarter circle spiral is a very

close approximation of the true golden spiral which is a logarithmic spiral

that passes through these blue points here, the spiral here. Pretty good fit, hmm?

The golden spiral picture captures some of the amazing properties of one of

mathematics' superstars the golden ratio Phi. However the one feature that this

picture is most famous for is, sadly, just a mathematical urban myth pushed and

propagated by lots and lots,... and lots of wishful thinkers. These people,

I call them Phi-natics will assure you that the spiral that you see in Nautilus

shells are golden spirals, which is simply not true. Same thing for spirals

in spiral galaxies, cyclones and most other spirals found in the wild. What is

true is that just like our quarter circle spiral a lot of spirals we

observe in nature are approximately logarithmic spirals. However, there are

infinitely many different logarithmic spirals and most of the logarithmic

spirals found in nature are not even remotely golden. In fact, most of the

pictures that are supposed to prove the golden nature of naturally occurring

spirals are arrived at by roughly fitting a really thick golden spiral to some

suitably chosen and doctored picture. Having said that sometimes spiral

patterns that we observe in nature like, for example, those in flower heads do

have a connection to the golden ratio, However, in general, not even the spirals

in flower heads are golden and the connection is established in different

non-spiral ways. If you're interested I've linked to some articles that debunk

a vast portion of the golden-spiral-in-nature story. Phi-natics, sorry to

disappoint. What I'd like to do in the following is

to focus on some true and truly amazing features of this picture which even a

lot of mathematicians are not aware of. What will be important for us about this

picture is the curious spiral of squares at its core. In fact, as far as today's

story is concerned, the sole function of the golden spiral spiral is to highlight

this square spiral. It turns out that not only the golden ratio but in fact every

positive real number has an associated square spiral. For example, here's the

spiral of root 2. Hands up, who has seen a green golden spiral before? Anyway, these

square spirals which can be finite or infinite are very easy to construct and

provide a wealth of insight into the nature of numbers. For example, I'll show

you that if you look at root 2 s square spiral in just the right way it

magically morphs into a so called infinite descent proof of the

irrationality of root 2. In fact, I show you a simple

characterization of the irrational numbers in terms of their square spirals

and use this characterization to pin down and visualize the irrational nature

of many famous numbers. To finish off I'll show you how the squares spiral of a

number is really the geometric face of the so-called simple continued fraction

of that number. Those guys here. Anyway ready for some

really amazing and beautiful mathematics? Let's go. Okay to start with let me show

you how the square spiral of a number is constructed and why, if the resulting

spiral is infinite, the number has to be irrational.

I'll first focus on the number root 3 to construct the spiral. We start

with a root 3 rectangle like this one here. A root 3 rectangle is a

rectangle with sides A and B whose aspect ratio A over B is equal to root

3. Trivial but important observation: if you scale a root 3 rectangle, you

get another root 3 rectangle. Now here's the first square of the root

3 square spiral, here's the second square of the spiral, the third, the

fourth.The rule is that the next square is the largest square that fits into the

remaining green area, fitted in such a way that it continues the right turning

spiral. So next is this, then this and this and so on, pretty straightforward.

Right, let's quickly go back to the beginning and count the number of

squares of each size that we come across in this spiral. Okay,

first square again. There's only one square of this size. Next, also only one

square of this size. Next, two of those. Okay, then one, then two again. In

fact, from this point on things repeat so 121212, forever. Neat!

One way to convince ourselves that things really repeat is to show that this blue

rectangle here is also a root 3 rectangle just like the green one we

started with. This means that new squares fit into the blue rectangle in exactly

the same way as they do in the starting green rectangle, and so the pattern

repeats. Okay let's show that this blue rectangle really is a root 3

rectangle. Remember that we started with a root 3 rectangle and so the ratio

of the sides is root 3. Put the first square and so the dimensions of the

remaining green area are ... what? Well short side on top has lengths A minus B and the

long side obviously B. Put the next square in and calculate its

side lengths in exactly the same way. Now the third square and now let's check

that the aspect ratio of the blue rectangle is really root 3. This

aspect ratio is what? Well this. Now some straightforward algebra. Divide both the

numerator and denominator by B, that does not change the ratio. But remember A

over B is equal to root 3. The standard trick to

get rid of the root in the denominator is to multiply the bottom and the top by

a 2 plus root 3 like that. Just in case you have not seen this trick in action

let's highlight the denominator. The highlighted product is of the form U

minus V times U plus V which, of course, is equal to U squared minus V squared

which in this case is 2 squared minus root 3 squared and so you can see the

square root in the denominator vanish. Remember this clearing-the-denominator

of-roots trick. It really comes in handy very often in maths. Anyway

now just go on algebra autopilot and you'll see that the whole expression

simplifies to root 3. Wonderful! At this point we are ready to draw a couple of

pretty amazing conclusions. Let's start by using our square spiral to prove that

root 3 is irrational. This also ties in nicely with what I did in the last video.

Okay if root 3 was rational, that is, if root 3 could be written as a ratio of

positive integers A and B, then the rectangle with sides A and B would be a

root 3 rectangle. Now we just calculated the lengths of the sides of the first

couple of squares, right? Now since A and B are supposed to be integers, these

three side lengths B, A-B and 2B-A would have

to be integers as well. In fact, it's very easy to see that this continues. The side

lengths of all the infinitely many squares in our spiral must be some

integer multiple of A or B minus some other integer multiple of B or A, like

down there, integer times A minus integer times B. This implies that all the side

lengths of all the squares all the way down are positive integers. But, and

regulars have heard me say this a lot, this is impossible. Why, well the

infinitely many squares in our spiral shrink to a point and

therefore they must eventually have side lengths smaller than the smallest

possible positive integer 1. The only way to resolve this contradiction is to

conclude that the assumption we started with, namely that root 3 is a ratio

of positive integers is wrong. And so we conclude that root 3 is irrational. That is a

really, really pretty proof, don't you agree?

But it is much more than that. Why? Because all sorts of things we've just

said stay true beyond the special case of root 3. For example, it's really easy

to see that if we start with any rectangle with integer sides and if we

remove squares according to our recipe, then all those squares in the spiral

must also have integer sides. This means the same proof by contradiction shows

that any number with an infinite square spiral must be irrational. So, for example,

the golden ratio Phi is irrational because it's spiral is also infinite. Now

here's a really pretty way to picture what we've accomplished. The essence of

our proof by contradiction is called an infinite descent because our assumption

that a rational number has an infinite spiral implies the existence of an

impossible infinitely descending or decreasing sequence of positive integers.

Very nice but also notice that you can actually SEE the impossible infinite

descent in the spiral by interpreting the squares as steps of an ever

descending spiral staircase. There's our infinite spiral staircase and the

footsteps of someone going for the infinite descent. What's going to happen

when they reach the bottom? What do you think? Anyway, to round off this part of

the video, just remember that if we can show that a number has an infinite

square spiral, then we've also shown that this number is irrational. So what about

the spiral of a rational numbers. Well, obviously, it cannot have an infinite

spiral, that is, its spiral must end and after a finite number of steps. But

how does it end? Well let's have a look at an example. The aspect ratio of the

rectangular frame of this video is 1920 over 1080. That means that this rectangle

is a rectangle that corresponds to the rational number 1920 over 1080 and so as

you can see the square spiral of this number consists of only 7 squares. So the

spiral ends because when we place the 7s square the rectangle we started with is

completely covered, there is no space left for an eighth square. Here's an

interesting fact: the side lengths of the smallest square in this finite square

spiral is the greatest common divisor of the numbers 1920 and 1080. Puzzle for you:

Show that this is true in general. Second puzzle for those of you in the know.

Which super famous Greek mathematician is responsible for some closely related

mathematics? Okay so we can be sure that the square spiral

of a rational number is finite. How about going the other way? Is it also true that

every finite spiral comes from a rational number? Well let's see. Say I

give you a finite spiral like this one there. Here's how you can determine its

aspect ratio. First we scale things so that the smallest square has side

lengths 1. Then it's clear that the next larger square has side lengths 1 plus 1

plus 1 is 3. Then we can see that the largest square has side lengths 3 plus 3

plus 1 is equal to 7 and, finally, that the top side of our rectangle is of

length 3 plus 7 is equal to 10. And so our rectangle has aspect ratio 10 over 7

and of course we can do exactly the same for any finite spiral to show that it

corresponds to a rational number. Neat hmm? Okay, so that means that the rational

numbers are exactly the numbers with a finite spiral which then also implies

that the irrational numbers are exactly those numbers with an infinite

spiral. That's a pretty amazing characterization of rational and

irrational numbers, don't you think? Definitely made my day the first time I

read about this. Now, to actually use this characterization of irrational numbers

to prove that a particular number such as Phi is irrational we somehow have to

show that it's associated spiral is infinite. The way we were able to show

this for root 3 was by recognising that the square spiral repeats. In turn

this was possible because we were able to show that while building the spiral

we come across rectangles with the same aspect ratio. Now it's very easy to see

that this also happens for the golden ratio Phi. In fact, this repeating

property is part of the definition of the golden ratio, that is, a rectangle is

golden if when you cut off a square, like this, you end up with a scaled down

version of the original. So since things repeat after cutting off one square this

also means that Phi has the simplest possible square spiral, with every square

size occurring just once and the associated sequence of integers being

all 1s like that. Anyway, just remember things repeat for Phi. So next time

someone asks you why the golden ratio is irrational just point at the closest

Golden Spiral and say `infinite descent' in an ominous voice. Okay, as a final

repeating example here is root 2 and here's a nice little root 2 factoid that

I actually did not know myself until recently. All these pink rectangles are

root 2 rectangles, right? Of course an A4 piece of paper is basically a root 2

rectangle. What this means is that if you fold the paper in half you get a

scaled-down version of the original, that is, another root 2 rectangle? But did you

know that you also get another root 2 rectangle when you cut off two squares

like this? There, another root 2 rectangle. Very

cool. Maybe not earth-shatteringly cool but I enjoy little mats moments like

this almost as much as the really deep stuff? Okay at this point it's natural to

ask for which numbers this works. So which numbers have a repeating spiral.

Well the examples so far were Phi, root 2, root 3,

so all square rooty numbers. In fact, it turns out that the numbers with repeating

spiral are exactly the numbers of this type. And when I say of this type I mean

all positive irrational numbers that are roots of quadratic equations with

integer coefficients. These numbers are usually referred to as quadratic

irrationals. Now, the fact that a periodic spiral implies that we're

dealing with one of these rooty numbers is pretty easy and was first

shown by one of the usual suspects, Leonhard Euler. On the other hand, showing

that every quadratic irrational has a repeating square spiral is not super

hard but it's definitely a little bit fiddly. So let me just show you a sketch

of the easy direction: periodic spiral implies quadratic irrational. So let's

say X is a number with a repeating spiral. Then in this particular X

rectangle all side lengths of the resulting squares look like this. So

integer times X minus another integer OR integer minus another integer times X

This means that the aspect ratios of the rectangles that we come across during

spiral building are ratios of expressions like this. For example, we could

have something like that. Now we said the spiral repeats. What this means is that

two of these aspect ratios have to be the same.

But, obviously, after multiplying through with the denominators, any such equation

simplifies to a quadratic equation and so X, as a solution of this quadratic

equation, is a quadratic irrational. Easy peasy, lemon squeezy. Puzzle for you, what's the

solution to the equation over there and what do all the coefficients in this

equation have in common? Coincidence? I don't think so. Okay, now at the start of

this video I claimed that the square spiral of a number is really the same

thing as the simple continued fraction of the number. To explain this

correspondence let's have another look at the root 3 spiral. Okay here comes

the magic. To get the continued fraction you just take the sequence of numbers of

squares of each size at the top and do this ... so root 3 is 1 plus 1 divided by 1

plus 1 divided by 2 plus, and so on. Very cool, right? But how does this work.

Well, let me finish off this video by explaining. What I do is to run the

standard algorithm for generating the infinite fraction and our algorithm for

building the spiral side by side. This will make it clear why we are getting

the same sequence of green numbers. Okay root 3 is equal to 1.7320... and

so on let's rescale the short side of our root 3 rectangle to make it length 1. Then the

long side is equal to root 3, that is, 1.7320... and so on. Ok how many

squares of side lengths 1 can we fit? Well obviously just one, the integer part

of root 3. Next let's have a look at the rectangle that remains.

Let's rescale everything so that the short side of the green rectangle

becomes 1. The scale factor that does the trick is 1 over 0.7320... . Up on top we can

also do something, we can rewrite things like this. Now Mathologer regulars will be

familiar with this maneuver. Everybody else just think about it for a moment ... Ok,

all under control, great! So good, anyway this gets the continued fraction going

on top. now 1 over .7320 is 1.3360..., and so on

Now, again, from the start. How many squares of side lengths 1 fit into the

green? Well, obviously one, the integer part of 1.3360 ...

Focus on the remaining green rectangle

and rescale everything such that it's short side becomes 1. There we go.

Rewrite the top as before 1 over 1.3360 is 2.7320...

How many squares can we cut off the green. Two of course, and so on.

As you can see, the sequence of numbers that corresponds to the spiral

is exactly the sequence of numbers in the denominators of the infinite

fraction. And with this transition to simple continued fractions understood

you're ready for the Mathologer video dedicated to continued fractions and

some of the other amazing insights they offer into the nature of numbers. For

example, the amazing pattern in the continued fraction of the number e, the

continued fraction of pi, and the curious observation that the golden ratio, the

number was the simplest spiral and continued fraction, is the most

irrational number, etc. And that's it for today.

Except here is one more puzzle: apart from the golden spiral what else is

wrong with this picture here?