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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?