The Pizza Theorem


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*

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