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# Rhapsody on the proof of pi = 4

Correction: when I mark where pi is on the graph, I meant pi/2!
Note: If this video were supposed to be teaching you, I'd probably have to make it boring and say that in one sense of limits, spoiler alert, you actually do approach a circle and a line, solving the apparent paradox by saying that the invariant of length does not hold over infinity. Luckily I am an artist, and this is a Rhapsody, and instead of "learning," you get to actually think, if you like. Created by Vi Hart.

Video transcript

Say you're me and
you're in math class, and you're supposed to
be graphing functions as if there were some deep
relationship between y and x that your teacher just
won't stop gossiping about. But like most gossip,
you really don't care about y's unhealthy
dependency on x. Really y, get a life. Luckily, your friend
passes you a note. You wait until the teacher
is facing the board, and sneakily open it up. And it's this proof
of pi equals 4 that you've seen like a
billion times on the internet. But because you're bored, and
at least it's not graphing, you'd take another look. It's like this. Start with a circle of diameter
one and circumference pi. Then draw a square around it. The length of a side
of the square is 1, so the perimeter of the
square is of course, 4. Now, you start
approaching a circle by zigzagging this parameter. This shape still has perimeter
4, and we do it again, and it's still 4, and again. And we approach a circle while
the perimeter never changes. So therefore, pi equals 4. Obviously, there's
something wrong here. The circumference
of a circle is pi, but pi is more like 3.14
numbers, and less like 4. So somehow what this
process approaches, is something that looks like
a circle, but is not a circle. I mean, what does
circle mean anyway? There's this loopy line
thing, And then there's this solid disk shape. And while they're
related and all, they're not actually the
same mathematical object. This troll proof is so much
fun because, as you repeat this zigzag move, all
the way to zig-finity, it approaches the shape of
a circle in the disk sense, and it approaches
the area of a circle, but it does not
approach a circle. It's all crinkled up. And you imagine that if you
stuck a straw in this thing and inflated it-- that
is, added as much area as you can while keeping
the perimeter as 4-- all the infinite wrinkles
would smooth out, and you'd get a circle
with perimeter 4 and diameter 4 over pi. In fact, you could inflate the
square to get the same thing. Probably has something
to do with how circles are the shape with the most area
possible given their perimeter, and why soap bubbles
like to be spheres, and raindrops are actually
pretty sphere-ish too. But anyway, you decide the best
way to respond to your friend is to try applying
the same fake proof process to something else. Maybe you can choose
another irrational number, like square root 2. In fact, square root
2 would work great, because it's also a
common geometry number. The ratio of the diagonal of
a square to its perimeter. I mean, like, if this
square has side length 1, then you've got
this right triangle. And a squared plus b
squared equals c squared. In fact, that's how
you decide to start your proof for your friend. Take a right isosceles triangle. Each leg has length 1. So the hypotenuse
is square root 2. Put this all in the
square of perimeter 4. Now we can approach a
triangle in the same way. And each time, the perimeter
of the whole shape is still 4. And thus, by the time
we get to a triangle, the length of the
hypotenuse must be 2. Thus, the square root of 2 is 2. So you pass that
over to you friend. Will he notice that
you're approaching the area of a triangle without
approaching a triangle? That a triangle has three sides,
but the shape you end up with is a polygon with
infinite sides? An infini-gon? But not a boring one, like how
a regular polygon approaches a circle as sides
go to infinity, because infini-gons
are more fun when there's actual
angles between sides. Zigzagging along
to make what you've decided to call a zig-fini-gon. You begin to wonder what other
zig-fini-gons you can make. Maybe if you
started with a star, and zigged all the points in. And did that again
and again to infinity. You get something that
looks like a pentagon, but it's actually
a zig-finite star with the same perimeter
as the original star. Or maybe you could have
a rule where at each step you zig down only part way. And then your zig-fini-gons
will have more texture to it. Maybe you could throw
some zags in there too. There's something
fractal-ly about it, except the perimeter
never changes. You could do that to
a triangle, or maybe make a square that turns
into a zig-fini-square. Uh oh, teacher's walking around. Better draw some axes, and
pretend to be doing math. So you turn the idea
sideways, and start at zero. Go to y equals 1 at x equals 1,
then back to 0 at x equals 2. The next iteration is like
folding this point down to 0. So the function zigzags from
0 to 1/2, to 0 to 1/2, to 0. The next step brings
the 1/2's to 0. And now the highest points
are at y equals 1/4. Each step brings the
highest points down to 0, and the new highest peaks
are only half as high. And each step keeps the total
length exactly the same. So what happens when you
do this to zig-finity? In one way, it approaches this
line, the x-axis, y equals 0. If there were any peaks,
they'd get folded down to 0. Therefore, there can't be any. Yet, at each step we
have twice as many peaks. So how can there be an infinite
number of peaks and no peaks? You might reconcile this by
saying that all infinite peaks are equally at zero,
since all peaks get brought to zero eventually. But if everything's
at y equals 0, you have just a line
segment of length 2. That doesn't make sense. The length of the zigzag
stays the same at each step, and at the beginning it's
like two hypotenuse of two right triangles,
so 2 square root 2. And 2 does not equal
2 square root 2. Another problem is that only
peaks ever get brought to 0, but not all numbers
can be peaks. Any fraction of
a power of 2 will be a peak at some
iteration, but a number like 1/3, or an
irrational number, will never be a peak
or a zig or a zag. So they must all be between
the zigs and the zags. But there can't be any
length between zigs and zags, or else that would
create a peak that would have been folded down. Somehow it has to be
infinitely zigzagged in a way where there's no
actual line segments of any length, but
only zigs and zags. Yet, there must be an
infinite number of numbers between each zig and zag to
fit all the irrationals in, and somehow all the
pieces of 0 length add up to be something
that does have length. You could imagine grabbing the
ends and stretching it out, accordion style, into a line
of length 2 square root 2. And then I suppose
you could crumple it all back down until the whole
length 2 square root 2 line is folded up into a single point.