I want to show you the Golden Rectangle, I'm going to start here with a square
that's 2 on each side, so a square of side 2, I want to go to the midpoint
at this side and bisect it here, so that length is 1 and that length is 1.
Then I'm going to draw this diagonal in, let me get a ruler here,
draw this diagonal in that goes from this midpoint up to
that vertex of that square.
And then what I want to do is drop that down
that length
down
and extend this
side right here to meet that.
Now when I do that, then the
rectangle that I get is called the Golden Rectangle
There's my Golden Rectangle, and so what I want to do is find the ratio of the
length to width in the Golden Rectangle because in mathematics that's a very
special number.
So if I'd look in this little
right triangle right here, I see that this side is 1,
this side is 2, because that was my original square, and so by the Pythagorean Theorem
I know that this side is going to be √(1² + 2²),
√(1 + 4) = √(5).
So this length is √(5), that means that this length right here
is √(5)
and this little length is 1, so if I want the ratio of length
to width
in the Golden Rectangle, the length is
1 + √(5)
and the width is this width right here, which is 2.
So, this number right here [1 + √(5)]/2
is an irrational number, but it's also called the Golden Ratio.
A very special number in mathematics, if you work this out on a calculator, it is
approximately equal to
1.618
Of course, I can't write it as a decimal because it's an irrational
number.
So, the Golden Rectangle, when I find the ratio of the length of width in that,
I end up with what's called the Golden Ratio.
Now I want to show you a little postage stamp from Japan.
There are derivations on it as you can see, sort of vertical rather than horizontal,
the way we did it.
Very important derivation in mathematics, so important that they actually put it
on a postage stamp.
Now I want to walk over to the other board here and look at another Golden Rectangle.
Alright, this time I want to make a Golden Rectangle from a square with side 4,
rather than a square with side 2. So I do the same thing, I go to this midpoint
right here,
that means that each of these two lengths is 2
and then I draw in this diagonal right here, and then I want to drop it down
like this,
form this rectangle,
and that will be another Golden Rectangle because I've derided in
exactly the same way I did with my first one. What that means that it's similar
to my other Golden Rectangle, so the ratio of length to width should
come out to be that Golden Ratio.
So let's take a look and see.
This side right here is 2,
this side is going to be 4,
so this side will be the √(2² + 4²), so I have
2² = 4,
4² = 16, 4 + 16 = 20
√(20),
Now when I want the ratio of length to width,
that will be 2
plus this length right here is going to be the same as this line
so √(20)
all divided by the width, which is 4.
So just to go over that again: here's the width, 4,
here's the length right here which is 2 plus this length,
which is the same as this length, or √(20), so 2 + √(20), for that.
Well, this should come out to be the same as my other one, so let's see what I can
do here.
2 + √(20)
in simplified form, that's going to be √(4 x 5),
I can take that 4 out as 2
leave 5 inside the radical, that's all divided by 4.
Okay so 2 + 2√(5), 2√(5) being the same as √(20)
I'll factor a 2 from the numerator, 1 + √(5)
with the denominator, I'll write as 2(2), and let's see, did that go off the screen?
Yes it did, hang on for just a second. Okay, there I've moved it.
You can see it now.
2[1+√(5)] over (2)(2)
those 2s divide out, and I end up with
1+√(5)
all over 2.
Same number I got before, the
Golden Ratio so,
any Golden Rectangle, regardless of what size square you start with, will always
end up giving you the same ratio for length to width, and that ratio is
called the Golden Ratio. now I want to go back to the first Golden Rectangle
that we worked with, and look at it for a second.
One of the interesting things about the Golden Rectangle is that
when I construct it like this, this little rectangle that I get over here
also it turns out to be a Golden Rectangle. So the big one is a Golden Rectangle
but so is this one right here, and we can show that by finding the
ratio of length to width.
So with the little blue rectangle,
I have the ratio of the
length to width. Well that's the length in this rectangle, and this is the width.
Well, what's the width?
This length right here is √(5), that whole thing, that length is 1,
so this length right here on the inside
this must be √(5),
that's the whole length, -1.
So if I find the ratio of length to width
in my little blue rectangle right here I'm going to end up with
length of 2
divided by the width of √(5) - 1
Well, this should come out to be the same thing, 1+√(5)
over 2.
How can we show that?
We can show that by rationalizing the denominator of this expression.
So to do that, I'm going to multiply by √(5) + 1
in the denominator, and √(5) + 1 in the numerator.
In the numerator, I get
2[√(5) + 1]
and in the denominator, I get
√(5)² - 1², so 5 - 1
which is 4, let's see if I'm going to have enough room right here,
and that comes out to be 2[√(5) + 1]
all divided by 4, which is 2 x 2,
and that means that these 2s divide out, and I end up with
1+ √(5) over 2
for the ratio of length to width
in this little rectangle right here.
I want to show you a postage stamp, another posted step that has the golden
rectangle on it, and you can see this little rectangle and a bunch of other
ones also. So let's take a look at it.
Now when you look at you notice that there's a little spiral that comes off
of this, and in fact, there's a bunch of little golden rectangles that spiral on
down smaller and smaller, so in fact if this is a golden rectangle right here,
the little blue rectangle, that means that must have come from some square
itself.
Well, that means that this up here is also golden rectangle, and so it must
have come from
some square itself.
That's a golden rectangle, it came from some square,
so on, and so forth. So you can see that those golden rectangle spiral down
further and further as small as you want to get. So every golden rectangle also
contains an infinite number of other golden rectangles all with the same
ratio of linked to width.