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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.