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*