How to lie using visual proofs

Today I'd like to share with you three fake proofs in increasing order of subtlety,

and then discuss what each one of them has to tell us about math.

The first proof is for a formula for the surface area of a sphere,

and the way that it starts is to subdivide that sphere into vertical slices,

the way you might chop up an orange or paint a beach ball.

We then unravel all of those wedge slices from the northern hemisphere,

so that they poke up like this, and then symmetrically unravel all of those from the

southern hemisphere below, and now interlace those pieces to get a shape whose area we

want to figure out.

The base of this shape came from the circumference of the sphere,

it's an unraveled equator, so its length is 2 pi times the radius of the sphere,

and then the other side of this shape came from the height of one of these wedges,

which is a quarter of a walk around the sphere,

and so it has a length of pi halves times R.

The idea is that this is only an approximation,

the edges might not be perfectly straight, but if we think of the limit as we do finer

and finer slices of the sphere, this shape whose area we want to know gets closer to

being a perfect rectangle, one whose area will be pi halves R times 2 pi R,

or in other words pi squared times R squared.

The proof is elegant, it translates a hard problem into a situation that's easier to

understand, it has that element of surprise while still being intuitive, its only fault,

really, is that it's completely wrong, the true surface area of a sphere is 4 pi R

squared.

I originally saw this example thanks to Henry Reich, and to be fair,

it's not necessarily inconsistent with the 4 pi R squared formula,

just so long as pi is equal to 4.

For the next proof I'd like to show you a simple

argument for the fact that pi is equal to 4.

We start off with a circle, say with radius 1,

and we ask how can we figure out its circumference, after all,

pi is by definition the ratio of this circumference to the diameter of the circle.

We start off by drawing the square whose side lengths are all tangent to that circle.

It's not too hard to see that the perimeter of this square is 8.

Then, and some of you may have seen this before, it's a kind of classic argument,

the argument proceeds by producing a sequence of curves,

all of whom also have this perimeter of 8, but which more and more closely

approximate the circle.

But the full nuance of this example is not always emphasized.

First of all, just to make things crystal clear,

the way each of these iterations works is to fold in each of the corners of

the previous shape so that they just barely kiss the circle,

and you can take a moment to convince yourself that in each region where a

fold happened, the perimeter doesn't change.

For example, in the upper right here, instead of walking up and then left,

the new curve goes left and then up.

And something similar is true at all of the folds of all of the different iterations.

Wherever the previous iteration went direction A then direction B,

the new iteration goes direction B then direction A, but no length is lost or gained.

Some of you might say, well obviously this isn't going to give the true perimeter of the

circle, because no matter how many iterations you do, when you zoom in,

it remains jagged, it's not a smooth curve, you're taking these very inefficient steps

along the circle.

While that is true, and ultimately the reason things are wrong,

if you want to appreciate the lesson this example is teaching us,

the claim of the example is not that any one of these approximations equals the curve,

it's that the limit of all of the approximations equals our circle.

And to appreciate the lesson that this example teaches us,

it's worth taking a moment to be a little more mathematically