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## The golden ratio

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# The golden ratio

## Video transcript

What I want to explore
in this video is, given some length
of string or a line or some line segment
right here, b, can I set up an a, so
that the ratio of a to b is equal to the ratio
of the sum of these two to the longer side? So it's equal to the
ratio of a plus b to a. So I want to sit and think
about this a little bit. I want to see is can I
construct some a that's on this ratio,
this perfect ratio that I'm somehow
referring to right here, so that the ratio of the
longer side to the shorter side is equal to the ratio of the
whole thing to the longer side. And let's just assume that we
can find a ratio like that. And we'll call it phi. We'll use the Greek letter
phi for that ratio over there. So let's see what we can learn
about this special ratio phi. Well if phi is equal to a over
b, which is equal to a plus b over a, we know that
a plus b over a is the same thing as a
over a plus b over a. a over a is just 1. And b over a is just the
inverse of this statement right over here. So b over a-- this thing
right here over here is phi-- so b over a is
going to be 1 over phi. This is going to be 1 over phi. So this is interesting. We've now set up a
number, which we're going to call this special
ratio, phi is equal to 1 plus 1 over phi. Well just that is kind of a
neat statement right over there. First of all, you
could, if you subtract 1 from both sides of this,
you get phi minus 1 is equal to its inverse. That seems to be a pretty
neat property of any number that if I just
subtract 1 from it, I get its
multiplicative inverse. And so that, already that
seems kind of intriguing. But then even this
statement over here is kind of interesting because
we've defined phi in terms of 1 plus 1 over phi. So we can actually
think of it this way. We could say that phi is
equal to 1 plus 1 over phi. But instead of writing
phi, we're saying, wait, phi is just 1
plus 1 over-- instead of saying phi-- I could say,
well, that's just 1 plus 1 over and I could just write phi again
or I could just keep on going. I could just keep on
going like this forever. I could say that's
1 over 1 plus 1 over and just keep on going on
and on and on, forever. And this is a recursive
definition of a function, or a recursive
definition of a variable, where it's defined
in terms of itself. But even this seems like
a pretty neat property. But we want to get a
little bit further into it. We actually eventually want
to figure out what phi is. What is the value of phi, this
weird number, this weird ratio that we're beginning to explore? So let's see if we can turn
it into a quadratic equation that we can solve using
fairly traditional methods. And the easiest
way to do that is to multiply both sides
of this equation by phi. And then you get
phi squared-- let me write a little bit
different-- phi squared is equal to phi plus 1. phi squared is
equal to phi plus 1. And then, actually, I'm going
to take a little bit of a side here. But even this is
interesting, because then if we take the square root
of both sides of this, you get-- let me scroll
down a little bit-- you get phi is equal to the
square root of-- and I'll just switch the order here-- the
square root of 1 plus phi. So once again, we can set up
another recursive definition. phi is equal to the
square root of 1 plus phi. And I could write phi
there, but hey, phi is equal to the
square root of 1 plus and I could write phi
there, but hey, phi is just equal to the square root
of 1 plus, the square root of 1 plus. And we could just keep going
on and on like this forever. So even this is neat. The same number that can
be expressed this way, the same number where if
I just subtract 1 from it, you get its inverse. It can also be
expressed in these kind of recursive square roots
underneath each other. So this is already starting
to get very, very, very intriguing, but let's
get back to business. Let's actually solve for this
magic number, this magic ratio that we started thinking about. And really from a
very simple idea, that the ratio of the longer
side to the shorter side is equal to the ratio of the sum
of the two to the longer side. So let's just solve this
as a traditional quadratic. Let's get everything
on the left-hand side. So we're going to subtract
phi plus 1 from both sides. And we get phi squared minus
phi minus 1 is equal to 0. And we can solve
for phi now using the quadratic formula, which
we've proven in other videos. You can prove using
completing the square. But the quadratic formula
you say, negative b. Negative b is the coefficient
on this term right here. So let me just
write it down, a is equal to 1, that's the
coefficient on this term. b is equal to negative 1, that's
the coefficient on this term. c is equal to negative 1,
that's the coefficient, or it's really the constant
term right over there. So the solutions to
this, phi-- and we're actually only going to care
about the positive solution because we're thinking
about a positive-- when we go to our
original problem here, we're assuming that these
are both positive distances, so we care about a positive
value right over here. We get phi is equal to-- do
it in orange-- negative b. Well negative negative
1 is 1 plus or minus the square root of b squared. b squared is going
to be 1 minus 4ac. a is 1, c is negative 1. So negative 4 times
negative 1 is positive 4. So 1 plus 4, all
of that over 2a. So a is 1, so all
of that over 2. So phi is equal to 1. And once again, we only care
about the positive solution here. This is going to be
the square root of 5. If you have 1 minus
the square root of 5, you're going to get a
negative in the numerator. So we only care about
the positive solution. 1 plus the square
root of 5 over 2. So this seems like a
pretty interesting number. Let's actually take
a calculator out and see if we can get the first
few places of this magic number phi. So let me get my calculator out. Let's just actually evaluate it. And you might recognize
that square root of 5 is an irrational number. And so this whole thing is going
to be an irrational number, but I'll prove that in
another video, which means it never repeats. It goes on and on
and on forever. But let's actually evaluate it. So it's 1 plus the square
root of 5 divided by 2. So it says 1.6180339. So let me put that aside. Let me write it down. And this is where
it starts to get really interesting
and mysterious. So this number right
over here is 1.618033988 and it just keeps on going
on and on and on, keeps on never terminating,
never repeating. So that by itself,
it's this cool number. It's this ratio that has all
of these neat properties, which are pretty crazy anyway
that you express it. But what's really neat is if
we revisit this thing right over here. Because what is 1
over phi going to be? So 1 over phi, which we
sometimes denote with a capital phi. We already know 1 over
phi is just phi minus 1. So we actually can
do this in our heads. 1 over this is just
going to be 0.618033988. I don't know. There's just something
wacky about that, that the inverse of
the number is really just the decimals left over
after you get rid of the 1. That, by itself, is
kind of a crazy idea. But it gets even crazier
because this number is showing up everywhere. And as you might imagine
from the title of this video, this phi right over here, this
is called the golden ratio. This is the golden ratio
and it shows up everywhere. It shows up in art. It shows up in music. It shows up in nature. And just to get an idea of
where it shows up in nature, it shows up in very pure ideas. So if I were to just
draw a perfect star, if I were to just draw a
regular star like this. Let me just draw it like this. I'll draw it right over here. So this is just a regular star. All the lengths are equal. So I want to draw it a
little bit better than that. So if I just draw a star
like this right over here or sometimes this would
be called a pentagram, some amazing things
start to happen here. The ratio of this pink
side to this blue length right over here, that's
the golden ratio. The ratio of this
magenta to this pink is the golden ratio, as
it should, by definition. Now the ratio of the
magenta to this orange is also the golden ratio. It just keeps on showing up
in a ton of different ways when you look at a
pentagram like this. If you look at something
like a pentagon, a regular pentagon
where all the angles are the same and all
the sides are the same, a regular pentagon. If you take any of the
diagonals of a regular pentagon, so right over here, if you take
this diagonal right over here, the ratio of this green side
to--and when I'm talking about the diagonals, ones
that actually aren't one of the edges-- the ratio of
any of the diagonals to any of the sides is once
again, this golden ratio. So it keeps showing
up on and on and on. And we can do interesting
things with the golden ratio. Let's say that we had
a rectangle, where the ratio of the width to the
height is the golden ratio. So let's try that out. So let's say that
this is its height. This is its width. And that the ratio--
So let's call this a. Let's call this b. And the ratio of a
to b is equal to phi. That 1.61 so on and so forth. Let me scroll down a little bit. So that is going
to be equal to phi. So that's something
interesting to do. Maybe that's a nice looking
rectangle of some sort. But let me put
out a square here. So let me separate this
into a b by b square. So this is a b by b
square right over here. And then-- actually let me do
it a little bit, let me draw it a little bit differently,
this rectangle actually isn't exactly the way I
would want to draw it-- so the ratio might look
a little bit like this. So the ratio of the
width to the length, or the width to the height, is
going to be the golden ratio. So a over b is going to
be that golden ratio. And let me separate out a
little b by b square over here. So this has width b as well. And so this distance right over
here is going to be a minus b. So now it is a b by
a minus b square. Actually, I should say,
we have a b by b square, right over here. This is b by b. And then we're left with a
b by a minus b rectangle. Now wouldn't it be cool if
this was also the golden ratio? And so let's try it out. Let's find the ratio
of b to a minus b. So the ratio of b to a minus b. Well, that's going to be equal
to 1 over the ratio of a minus b to b. I just took the reciprocal
of this right over here. And this is just going to
be equal to 1 over a over b. Let me write this,
a over b minus 1. I just rewrote this right there. And that's just going to
be equal to 1 over phi. The ratio of a to b, we said,
by definition was phi minus 1. But what is phi minus 1? Well phi minus 1 is 1 over phi. It's this cool number. So it's equal to 1 over 1 over
1 over phi, which is once again, just equal to phi. So once again, the ratio
of this smaller rectangle, of its height to its
width, is once again this golden ratio, this
number that keeps showing up. And then we could do
the same thing again. We could separate this into an
a minus b by a minus b square. Just like that. And then we'll have
another golden rectangle, sometimes it's called,
right over there. And then we could separate
that into a square and another golden rectangle. Then we could separate
that into a square and then another golden rectangle. Then another golden rectangle. Actually let me do it like this. This would be better. So let me separate. Let me do the square up here. So this is an a minus
b by a minus b square and then we have another golden
rectangle right over here. I could put a square
right in there. Then we'll have another
golden rectangle. Then we could put another
square right over there, you have another
golden rectangle. I think you see
where this is going. Another square, another
golden rectangle, which by itself starts to create
a cool design that we can keep kind of circling
in and in and in. And then, if we actually
draw an arc here, something kind of cool happens. If we have an arc that
traces these things out, we have something,
a pattern that you might have seen
many times before. And that pattern does not
look too different than what you might see in something
like a nautilus shell. And it shows up all over
the place in nature. And that makes sense
because just the way cells construct themselves. It kind of makes sense to be
the same at different scales and the ratio from
one scale to the next is maybe the same as
the constituent ratios. This right here,
and it shows up all over a lot of Leonardo
DaVinci's paintings. He never explicitly
stated it, but there's a lot of interesting
ratios in them. But Salvador Dali, this painting
right here, The Sacrament of the Last Supper,
he explicitly used the golden ratio. So the actual ratio of
the width to the height is the golden ratio, so
this is a golden rectangle. And also there's
all sorts of ratios and I'll invite
you to explore it. The ratio of the different
parts of the tables to where it sits
in the painting. The golden ratio shows up
a ton in this painting. And then he does have
the pentagons over here. And we know that the ratio of
the diagonal of the pentagon to the sides of the pentagon
are also the golden ratio. And so he just thought it
was a really cool thing. There's all sorts
of neat things. That if you find where these
two guys are bowing down are, if you draw that line right over
here, this is the golden ratio. The ratio of this
length right over here to that length over there. Once again, the golden ratio. It just keeps showing
up in this painting. So it's a really,
really, really cool thing and I really encourage
you to explore this further because it's
kind of exciting.