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?
A kinda think this recent video from Numberphile illustrates this concept a bit more intuitively (at least for me).