I will be the first to admit that math was not my favorite subject in school, in fact
it made me HELLA anxious.
But I’ve definitely warmed to it, especially ever since I heard about the Ham Sandwich
theorem--what, a girl loves to eat!--oh, and also the fact that it might help us better
understand our universe, but whatever.
Let’s take a step back and start with pancakes.
A bit of a departure from the universe and ham sandwiches, but this smorgasboard will
eventually all make sense, I promise.
The pancake theorem states that if I have a pancake on a plate, there’s at least one
cut that I can make to divide that pancake in half.
Simple.
If I add another pancake on the plate, no matter where it is there’s going to be at
least one cut that I can make through both of them that will allow me to divide both
pancakes in half.
Again, simple enough, right?
But, that’s because the pancake theorem technically only deals in 2-dimensional space,
like these pancakes that we’re talking about are perfectly flat, as if they were pancakes
that we’ve drawn on a piece of paper.
The Ham Sandwich theorem takes the concept and moves it into 3-dimensions, the world
we actually live in.
Ok, I’m about to drop a math formula on you here, but I want you to stay calm and
stay with me, ok?
Here we go *brings up piece of paper and read off it*: the ham sandwich theorem states that
given n measurable objects in n-dimensional Euclidean space, it is possible to divide
all of the objects in half (with respect to their volume) with a single (n-1)-dimensional
hyperplane . Yeah, I know.
I read that sentence when I was first researching this video and was just like, ‘No. there’s
no way.”
BUT it turns out this convoluted statement actually expresses something remarkably simple
and useful, so I want you to come on an imaginary adventure with me.
Say we have a ham sandwich.
That’s our n-measurable objects, our 3 measurable things: 2 slices of bread and 1 slice of ham.
(We’re being stingy with the ham, ok?)
In 3-dimensional space, it’s possible to use a single cut to divide the sandwich in
half, no matter where the 3 items are.
Now, this doesn’t JUST mean that a sandwich AS A WHOLE can be halved.
If we’re defining the sandwich as our 3 items (bread, bread, and ham), even if we
dropped our sandwich on the floor and the three items were all over the place, we could
draw one line that would divide the entire volume of the sandwich in two.
But it ALSO means, and this is where it gets real cool, that each individual item of the
in the sandwich, no matter where they are in space, can be divided perfectly in two.
This part is easier to picture if we get rid of gravity and picture the pieces of our ham
sandwich floating in the air.
Doesn’t matter where, they could be right next to each other or on opposite ends of
the universe and there is still at least one magic plane that could bisect all of the ingredients
perfectly, with one cut.
Ok, so this is cool and made us do some fun mental gymnastics, but I’m never going to
drop my ham sandwich on the floor and then whip out my calculator like, “lemme find
the magic line that divides all these in two”.
So exactly how is this important or useful?
Well, a paper that came out a few years ago wanted to explore exactly that, and came to
some cool conclusions, stating that “At any given instant of time, there is one planet,
one moon and one asteroid in our solar system with a single plane touching all three that
exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal
mass of the solar system.”
*mind blown gesture and noise* All of this is to say, theoretical mathematics
like this helps us understand the very nature of the way the universe behaves.
We can use things like the ham sandwich theorem, along with our understanding of physics and
the natural sciences, to help us make computer simulations of everything from atomic phenomena
to the movement of celestial bodies.
Man, the world is crazy and beautiful.
And who knew that talking about ham sandwiches could turn this former math-phobe into, dare
I say, a math enthusiast.
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And just FYI, the original creator of the ham sandwich theorem was named Hugo Steinhaus,
and worked out this problem while hiding from the Nazis in a Polish farmhouse during WWII.
BADASS.
I’m Maren, thanks for watching Seeker.