Vsauce! Kevin here. Let’s split a pizza. I’ll eat one half and you eat the other.

I’m going to be using a laser guide and a protractor to make sure that it’s perfectly,

mathematically equal so that no one is complaining that one of us got more pizza than the other

one

Okay! Here we go.

There we go! Alright, let's just get this cutting board out of the way. Perfect! These

are two exactly equal halves. So grab your slices! Alright, I know that it doesn't look

like it but we can prove it mathematically, but before we do that let’s analyze how

pizzas are traditionally cut… and how they could be infinitely cut.

Pizza is usually a cut right down the center through its diameter, then cut again at a

90 degree angle perpendicular to the first cut. Two more cuts bisect each 90 to a 45

to make 8 equal slices. That feeds 4 people or 2 people equally… and maybe 3 if one

of them just isn’t as hungry. But what about cutting one pizza to feed 13 people? Or 31?

Slices won’t work. We need monohedral disk tiles.

By cutting the pizza into a pinwheel you can get twelve identically-shaped wedges. Monohedral

tiling means every tile is the same shape. In 2015, mathematicians Joel Haddley and Stephen

Worsley found you can create infinite families of monohedral disk tilings by subtiling forever.

The actual math proofs on this get rather complicated but basically, there’s no mathematical

end to the pattern of pizza wedges you could make. But obviously dividing up a real pizza

this way for little Billy’s birthday would be… impractical.

The pizza that I cut however is the same exact method pizzerias use everyday... I just moved

the center. Divisors Of A Circle was a problem proposed by L.J. Upton in 1967 and solved

in volume 41 of Mathematics Magazine challenging readers to show that alternating divisions

of a circle with 4 lines converging on concurrent point O add up to half of the circle. Michael

Goldberg solved Upton’s problem without using calculus.

In 1994, a follow-up piece in Mathematics Magazine explained how this problem demonstrates

a fair way to split up pizza, and the “Pizza Theorem” was born: If a pizza is divided

into eight slices by making cuts at 45 degree angles from any point in the pizza, then the

sums of the areas of alternate slices are equal. That’s why if we top alternating

slices with green peppers and yellow peppers it becomes clear we have two equal halves.

We can do the math but a ‘proof without words’ is considered by many to be the highest,

most elegant form of proof. And our proof is in the pizza.

We’ll label our pizza slices A through H… and then we’ll represent each slice as two

equal shapes. We’ll clone the four smaller slices, while the four larger ones will be

reduced to make up for that duplication. They’ll be weird shapes, but they’ll be equal. We’ll

label them with a capital letter and lowercase letter to differentiate which slice is yours

and which slice is mine. So here’s A. Here’s B.This is C. D. E. F. G. And finally H.

So the large slices, G, H, A and B are divided into two equal wedges and the small slices

F, E, D and C are duplicated within the pizza. Proving that we split our original pizza perfectly

for two people.

This will work regardless of the concurrent point on the disk or pizza. Here’s a wonderful

proof without words visualizer created by Christian Lawson-Perfect where you can move

the interior point and instantly see the different slices that fulfill the Pizza Theorem.

The most delicious theorem of all time.

What began as an obscure math problem dividing a circle in Mathematics Magazine, has continued

to be refined and reimagined 50 years later. Which is impressive no matter how you slice

it.

And as always, thanks for watching.

This pizza is three days old.

*skeleton chewing noises*