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This right over here
is a self-portrait

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that Rembrandt made
in 1640, and what's

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really interesting about it
is like other great artists

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like Leonardo da Vinci
and Salvador Dali

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and many, many, many
others, Rembrandt really

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cared about something
called the golden ratio.

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And I've done whole
videos about it.

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And it's this fascinating,
fascinating, fascinating number

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that's usually denoted
by the Greek letter phi.

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And if you were
to expand it out,

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it's an irrational
number, 1.61803,

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and it just goes on
and on and on forever,

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but there's some very neat
mathematical properties of phi,

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or the golden ratio.

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If you start with phi, and
if you were to add to that,

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or actually let's
start it this way.

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If you were to start with 1
and add to that 1 over phi.

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Let me write my phi
a little bit better.

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You add to that 1 over
phi, that gives you phi.

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So that's kind of a neat thing.

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Now, if you were to multiply
both sides of this equation

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by phi, you get that,
if you start with phi,

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and then if you add 1,
you get phi squared.

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So it's a number you add
1, you get its square.

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These are all really,
really neat things.

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It can even be written
as a continued fraction.

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Phi could be rewritten as 1 plus
1 over 1 plus 1 over 1 plus 1

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over, and we just go like that
forever and ever and ever.

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That also gives you phi.

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So hopefully this gives
you a little appreciation

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that this is a
really cool number.

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And not only is it
cool mathematically,

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but it shows up
throughout nature,

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and it's something that artists
have cared about because they

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believe that it helps
define human beauty.

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And we see that Rembrandt
really cared about it

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in this painting.

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And how can we tell?

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Well that's what we're going
to analyze a little bit

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through this exercise
in this video.

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We can construct a triangle.

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Obviously these triangles aren't
part of his original painting.

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We superimposed these.

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But if you were to put a
base of a triangle right

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where his arm is
resting, and then

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if you were to have the
two sides of the triangle

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outline his arms and shoulders
and then meet at the tip right

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at the top of this arch, you
would construct triangle ABD

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just like we have here.

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And then if you were
to go to his eyes,

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and you could imagine
human eyes are

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what we naturally look at,
whether we're looking at a face

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or a painting of a face.

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If you look at his eyes, and if
you were to draw a line there

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that's parallel, well, that
really connects the eyes,

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and that's parallel to the
BD right over here-- so let's

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call that segment PR
right over there--

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we'll see that this ratio,
the ratio between this smaller

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triangle and this larger
triangle, involves phi.

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So this is what we
know, what we're

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being told about this painting,
and this is quite fascinating.

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The ratio between the length of
segment CD and BC is phi to 1.

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So you drop an altitude
from this larger triangle,

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this ratio, the ratio of CD, the
length of CD to BC, that's phi.

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So clearly, Rembrandt
probably thought about this.

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Even more, we know that
PR is parallel to BD.

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We've actually
constructed it that way.

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So that is going to be parallel
to that right over there.

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And so the next clue
is what tells us

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that Rembrandt really
thought about this.

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The ratio of AC to AQ.

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So AC is the altitude
of the larger triangle.

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The ratio of that
to AQ, which is

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the altitude of this top
triangle, that ratio is phi

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plus 1 to 1, or you could even
say that ratio is phi plus 1.

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So clearly, Rembrandt
thought a lot about this.

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Now using all that
information, let's

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just explore a little bit.

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Let's see if we can come
up with an expression that

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is the ratio of the
area of triangle ABD,

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so the area of the
larger triangle,

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to the area of triangle APR.

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So that's this smaller
triangle right up here.

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So we want to find the ratio of
the area of the larger triangle

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to the area of the
smaller triangle,

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and I want to see if we
can do it in terms of phi,

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if we can come up with some
expression here that only

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involves phi, or
constant numbers,

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or manipulating phi in some way.

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So I encourage you to pause the
video now and try to do that.

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So let's take it step by step.

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What is the area of a triangle?

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Well, the area of any triangle
is 1/2 times base times height.

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So the area of
triangle ABD we could

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write as 1/2 times our base.

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Our base is the
length of segment BD.

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So 1/2 times BD.

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And what's our height?

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Well that's the
length of segment AC.

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1/2 times BD-- Maybe
I'll do segment AC.

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Well, let me do it
in the same color--

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times the length of segment AC.

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Now what's the area?

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This is the area
of triangle ABD.

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1/2 base times height.

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Now what's the area
of triangle APR?

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Well, it's going to be 1/2 times
the length of our base, which

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is PR, segment PR,
the length of that,

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times the height, which is,
the height is segment AQ,

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so the length of segment
AQ, we could just

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write it like that, times
the length of segment AQ.

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So how can we simplify
this a little bit?

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Well, we could divide
the 1/2 by the 1/2.

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Those two cancel out.

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But what else do we know?

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Well, they gave us the
ratio between AC and AQ.

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The ratio of AC to AQ right
over here is phi plus 1 to 1.

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Or we could just say
this is equal to phi.

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Or we could say this is
just equal to phi plus 1.

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So let me rewrite this.

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Actually, let me
write it this way.

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This is going to
be equal to-- So we

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have the length of segment BD
over the length of segment PR,

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and then this part right
over here we can rewrite,

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this is equal to
phi plus 1 over 1.

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So I'll just write it that way.

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So times phi plus 1 over 1.

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So what's the ratio of BD to PR?

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So the ratio of the base of
the larger triangle to the base

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of the smaller triangle.

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So let's think about
it a little bit.

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What might jump out at you
is that the larger triangle

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and the smaller triangle, that
they are similar triangles.

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They both obviously
have angle A in common,

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and since PR is
parallel to BD, we

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know that this angle
corresponds to this angle.

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So these are going to
be congruent angles.

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And we know that this
angle corresponds

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to this angle right over here.

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So now we have three
correspondingly angles

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are congruent.

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This is congruent to itself,
which is in both triangles.

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This is congruent to this.

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This is congruent to that.

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You have three
congruent angles, you're

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dealing with two
similar triangles.

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And what's useful
about similar triangles

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are the ratio between
corresponding parts.

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Corresponding lengths of
the corresponding parts

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of the similar triangles
are going to be the same.

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And they gave us
one of those ratios.

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They gave us the ratio of the
altitude of the larger triangle

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to the altitude of
the smaller triangle.

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AC to AQ is phi plus 1 to phi.

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But since this is true
for one corresponding part

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of the similar
triangles, this is

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true for any corresponding
parts of the similar triangle,

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that the ratio is going
to be phi plus 1 to 1.

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So the ratio of BD, the ratio of
the base of the larger triangle

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to the base of the
smaller one, that's

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also going to be
phi plus 1 over 1.

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Let me just write it this way.

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This could also be rewritten
as phi plus 1 over 1.

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So what does this simplify to?

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Well, we have phi plus 1 over
1 times phi plus 1 over 1.

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Well, we could just divide by 1.

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You're not changing the value.

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This is just going
to be equal to,

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and we deserve a drum roll now.

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This is equal to
phi plus 1 squared.

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So that was pretty neat.

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And I encourage you to even
think about this because we

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already saw that phi plus
1 is equal to phi squared,

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and there's all sorts of
weird, interesting ways

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you could continue
to analyze this.