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Let's build bridges.

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[SOUND]

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Normally I'd build a bridge say across a
river.

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So suppose that this blue area here is a
river.

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And what would it mean to build a bridge
across this river?

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Well I'd probably put pylon on either
side, on either of the two banks.

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And then I'd build the bridge across those
those two pylons.

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Here's a diagram.

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Alright I've got the earth, I've got the

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river, here are the two banks of the
river.

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And then I can build a bridge that goes
from

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one bank to the other, but it's attached
to both banks.

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But I want to think about a one-sided
bridge.

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Instead of being connected to both banks,
I just

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want to be connected to one of the banks,
sure.

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So instead of building a bridge that's
attached to both banks,

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I want to build a bridge that's just
attached to one bank.

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And then I want to know,

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how long can I make this bridge before it
collapses?

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Well, I mean look.

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If, if you allow me to build a bridge from

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some super strong metal like unobtanium,
well no problem right?

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I mean, I'll just build a really, really
long bridge, there's

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going to be no limit to how long that
bridge could be.

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And that's not really what I mean, right?
What I'm really going to ask you to do.

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Is to build the bridge out of these
blocks.

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All right, so you're allowed

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just to stock blocks on top of each other.

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And I want the thing to be stable, I don't
want it to fall over.

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And I want to know, if you start stocking
blocks,

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how long of an overhang can you get
alright?

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What's the maximum possible overhang that
you can achieve if I give you n blocks.

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I could try to get started just with some
of these foam blocks.

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I could try to build a

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stack of these blocks and see how far I
can get these blocks to overhang.

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Before they fall over.
But why did my block tower just fall over.

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Well the issue has to do with center of
mass.

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A block tower is going to fall over if the
center of mass isn't supported.

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So what I mean by that, that the center of
mass of the first block isn't

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supported by the second block, then that
first block will fall off.

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If the center of mass of just the first
two

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blocks together, which is maybe over here
somewhere isn't supported.

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If it isn't above the third block, then
those two blocks are going to fall.

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My tower's going to collapse.

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So that's the issue, I want to build a

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really tall tower of blocks with a really
long overhang.

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And yet I want it to be stable.

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Of course the easiest way to make it
stable is just to make my

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block tower perfectly vertical, but then I
don't have any over hang at all.

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So these two forces are really working
against each other.

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All right, my desire to have a really long
over hang should

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be playing against my desire to have my
block tower not collapse.

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Well let me, let me propose a specific
configuration.

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Here's the configuration I'm proposing.

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So imagine that all these blocks are
exactly the same.

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They're all one block long and I've
staggered them like this.

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So this first block is offset by half a
block width, from the second block.

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The second block is a quarter of a block
width pushed in.

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The next block is a sixth pushed in.

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The next block is an eighth, this distance
here is an eighth of a block.

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The next block here is a tenth of a block.

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If I put another block under there, I'd
offset it by twelfth of a block.

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The next block will be by fourteenth of a

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block, then sixteenth of a block and so
on.

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I need to check that that configuration is
stable.

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So the center of mass of the top block is
smack in the middle of the top block.

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And since

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I pushed the top block over half a block
from the second block, that puts the

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center mass of the top block right above

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the edge of the second block so it's
stable.

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What of the second block?

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Well the center of mass of the second
block is right

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in the middle of the second block, but
that's not relevant.

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I mean yes the second block isn't tipping
over but what I really need to know.

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Is what's the combined center of mass of

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the first block and the second block
together?

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So figure that out, I just need to
remember that I pushed the second

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block over a quarter of a block from the
edge of the third block.

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And then I could compute the center of

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mass of the first and the second block
together.

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And I find out that in that case the
center of mass is right there.

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It puts it right above the third block

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which means the first two blocks together
are stable.

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The first block is stable, the first two
blocks are stable, but that's just the

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top two blocks.
I need to know this general.

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So I need to compute the center of mass,
the top

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n blocks relative to the right edge of the
next block.

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And to do this I'll start by averaging
some center of masses.

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And I want to average the center of masses
of the top end blocks.

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I don't really need to know the y
coordinate of the center of masses.

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So I'm just going to add together

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the x coordinates of the center of masses.
Where I put the origin

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at the right-hand edge of the block right
under the stack.

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Alright so I'm going to add together these
x coordinates.

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And then once I've added together these x
coordinates

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to average them, I need to divide by n.

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Okay, so just got to figure out where this
blocks really are in in space.

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So first of all, where is block number
one?

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Where is the top block relative to the

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next block under this collection of the
top

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end blocks.

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Well that blocks center of mass is right
here at

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one half plus a half, plus a fourth, plus
dot dot dot plus one over two n, right.

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The, the center of the mass is just a
block by itself is at

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one half, and then this records how far
over I've pushed the top block.

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Relative to the next block in the stack
after the top n

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blocks, okay.
What about block number two?

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Well that looks very similar right, again
it's one half, because that's

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where the center of mass is just in the
block by itself.

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But the second block doesn't get pushed
over half, it gets pushed over a fourth.

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And then a sixth and so on until it's 1
over 2n.

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So a fourth plus a sixth plus until I get
to 1 over 2n.

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That's how far over I pushed it relative
to the next block.

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And a half then moves me over to the
middle of block number two.

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Then block number three has a similar
looking formula.

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It's a half plus now a sixth plus an
eighth until I get to 1 over 2n.

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And finally I get to the nth block, which

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is right above the block that I'm
measuring everything from.

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So its center of mass is at a half plus
just how

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much I pushed over the nth block which is
1 over 2n.

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So now I gotta look at this and see

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if there's anything I can say about this
complicated sum.

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Well, I've got a half, I've got a half,
I've got a half, I've got a half.

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Every single one of these n terms has a
half, so that gives me n over 2.

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I've got n halves all together.
I've also got a one half here and no extra

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one halves, so I can add just this one
half coming from right here.

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I've got a quarter here and a quarter here
and then no more quarters.

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So I've got two quarters, and I can add
those.

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How many sixths do I have?

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Well, I've got a sixth in here inside the
dot

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dot dot, I've got sixth here, I've got
sixth here.

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The next term doesn't have any sixths, so
I've got 3 6ths altogether.

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And then

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it was going to keep on going right, I
could count how many eighths

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I have, I could count how many tenths I
have and so on.

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And eventually I'll notice that I've got a
1 over 2n, a 1 over

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2n, a 1 over 2n and a 1 over 2n, I've got
n 2ns.

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And

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that's it.
Alright, that's all the terms in a sum.

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So then I'm dividing this whole thing by
n.

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Well this is a half, this is also a half,
this is a half and this is a half.

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Here I've got n halves.

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So instead of writing a half plus 2 4ths
plus

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3 6ths and everything, I could just write
n halves.

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So I've got n over 2 plus n over 2 over n
or altogether right, that's just

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n over n.
That's just one.

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So relative to the right hand edge of the
next

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block, the center of mass of the top n
blocks.

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Is right on the left hand edge of the next
block, which means it's stable.

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So it's stable, but now what kind

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of overhang can I get with that
configuration?

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So this case, I just used six blocks, but
the total

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overhang is easy to calculate, right, I
want to count what the total overhang.

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It's the distance from the left edge of
the

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top block to the left edge of the bottom
block.

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And it's just a half plus a forth plus a
sixth plus a eighth

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plus tenth, it's the total amount that I
shifted all the blocks over by.

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Now if I build the same kind of
configuration, but instead of 6 blocks,

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I built it with some large number, call it
big N number of blocks.

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Then this total overhang

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is the sum little n from 1 to big N minus
1, over 1 over 2 n.

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Cause this is the total amount that I'd be
shifting all the locks over by.

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That looks like the harmonic series, and
indeed the harmonic series diverges.

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Well, that means the sum of 1 over 2 to

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the n, n goes from 1 to infinity, also
diverges.

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But that means that by choosing n big
enough, I can make this overhang

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as large as I desire.
Well, here it is in, in the real world.

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I've built one of these harmonic towers,
and you can

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see that I've got a pretty significant
amount of overhang here.

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I'll rotate it a little bit, and I've got
this river here.

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Anyway, I could make the

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overhang as long as I'd like, as long as
I'm willing to use more slabs.

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[NOISE]