Fixed Points

Hey, Vsauce! Michael here.

There is an art museum on the moon. Supposedly. We can't be sure until we go back and check. But as the story goes, in 1969, Fred Wall Tower from Bell Laboratories and sculptor Forrest Myers convinced an engineer working on the Apollo 12 lunar lander to hide an itsy-bitsy, roughly 2 by 1 centimeter ceramic tile within the gold blankets wrapped around parts of the spacecraft that would be left on the moon.

Etched onto that ship were artworks by famous artists. According to everyone involved, the plan worked. When the Apollo 12 team left the moon, the wafer was still there. Now, two days later, Myers told the New York Times what they had done and this image of the wafer was published. It contains a Rauschenberg straight line, a Oldenburg drawing of Mickey Mouse. Pretty cool!

The story is likely to be true. If confirmed, that would make this wafer the first and currently only art museum on the moon. But the thumb is covering something; in fact, it's covering Andy Warhol's submission. Why? Well, according to Warhol, all he did was innocently etch his initials on the chip. He stylized them like this: he drew a somewhat funny W for Warhol and then put a line right across here to make this bit look like an A. That's it; it's just his initials.

Here's an unobscured view of a replica that's on the moon right now. I, however, am on Earth right now in North America, and this is a map of North America, which is cool because if you are in the place, you have a map of… well, mathematically, there will always be some point on your map that is directly above the place in the real world it represents.

Always. It doesn't matter how you hold the map; it can be rotated, flipped, even twisted, folded, or crumbled. It is guaranteed by Brouwer's fixed point theorem. A fixed point is anything that goes nowhere after a transformation. Brouwer's theorem tells us that it is impossible to completely mix up a set of points if they are bounded without holes and transformed continuously, with no cutting or gluing.

Without bounds, every point can be mapped somewhere new; around holes, every point can be mapped somewhere new. And if you cut or glue, every point can be mapped somewhere new. But otherwise, mixing will always fail somewhere. A transformation is continuous if, as the distance between any two points reaches zero in the before state, it also approaches zero in the after state.

Each square of this checkerboard is filled with point-like pixels of a unique color. As we transform it, let's have the pixels still within the square they began in glow. Okay, as you can see, no matter how you resize or manipulate the checkerboard, it will always glow somewhere. You cannot get every pixel to be simultaneously outside of its original square unless you cut or move the whole shape outside of the space that used to fill.

Here's a question: do you think you can completely mix up coffee in a mug by stirring? Well, of course you cannot! Stirring coffee is a continuous transformation of the coffee, and everything stays within the same space. So, Brouwer's fixed-point theorem applies; no matter how well you try to stir, there will always be, after the liquid has settled, at least one point that you stirred right back to where it was before. A fixed point.

To be sure, coffee isn't made out of points, it's made out of molecules! But they're quite tiny and very numerous, so within a degree of error, it'll pretty much hold. Fixed points don't just frustrate your ability to fully mix things; they can also suck you toward them. They can be attractive, like the number nine.

Try this: think of a number with more than one digit and then add up its digits. Now subtract that sum from the original number to get a new number. If you do this over and over and over again until you're left with a single-digit number, you will always end up with nine. Every time! Also, notice that any number with two or more digits minus the sum of its digits becomes divisible by nine right away.

What's going on here? Well, there's nothing mystical about nine, misc in general. Instead, it's just a consequence of how we write numbers. Numbers can be written in all sorts of ways, but the most common base-ten positional notation makes the nine trick work. In this system, a number like twenty-five doesn't mean two and five; it means ten twos and one five—that's twenty-five!

So, subtracting out just the positional digits of a number removes one member from each group. The digit you have one of completely disappears. The digit you have ten of goes down to nine copies. The digit you have one hundred of goes down to ninety-nine copies, and so on. So, the whole thing becomes a multiple of nine.

Attractive fixed points also play a starring role in a famous method for calculating square roots. It's called the Babylonian method. The number you want to find the square root of is called the radicand, and its square root is some number that is equal to the radicand divided by itself. Okay, now step one of this method is guess.

If your guess is lower than the actual answer, the radicand divided by your guess will be larger than the answer, and vice-versa. The true answer will always be somewhere in between those two values. So, after you guess, take both values and find their average. It'll be a point in between. Now use this value as your guess and keep going. You will converge towards the true square root at a pretty nice speed.

The number of correct digits in your approximation will roughly double after every iteration. That's pretty cool! But let's talk about infinity, specifically Aleph numbers. They describe sizes of well-ordered infinities. The smallest is Aleph-null, equal to the quantity of integers that there are. But there are literally larger infinities than that.

In order of increasing size, they are Aleph 1, Aleph 2, Aleph 3, and so on. Each one is infinite but refers to a greater number of things than the infinity before. If you tried to pair up Aleph-null things with Aleph 1 things, you'd literally run out of Aleph-null things first, even though it's endless.

You can hear more about Aleph numbers in this video of mine, but here's the thing: there happens to be an Aleph fixed point. Notice that the subscript of each Aleph is equal to the number of infinities less than it—there are none less than Aleph-null, one less than Aleph 1, two less than Aleph 2, and so on. Each next Aleph number is monstrously bigger than the last, but only adds one to the growing list of Alephs.

Clearly, these rates are so different they'll never meet up, but they do. Take a look at this number: Aleph Aleph Aleph, Aleph all the way down and so on—an endless cascade of Alephs. How many infinities are smaller than this number? Well, look at its subscript! It's… well, it's an endless cascade of Alephs, just like the number itself. This is an Aleph fixed point, an infinity so large it is equal to the number of infinities smaller than itself.

That's cool! But probably my favorite fixed point-related thing is the force of Coulomb theorem. It states that at any given moment, there must be at least one pair of points on Earth's surface that are diametrically opposite one another, but nonetheless have the same temperature and atmospheric pressure. Diametrically opposite points on a sphere are called antipodes, and as I've discussed before, if you place a piece of bread on the ground somewhere on Earth and another one on that point's antipode, well, you've made yourself an Earth sandwich.

There are sites that help you locate such opposite points on land, but at this moment in history, you'll notice that most points on land are antipodal to water, which makes sense—the Earth's surface is mainly covered with water. But even though we know that, we don't always appreciate just how gigantic the Pacific Ocean is. Maybe because maps tend to cut right along it, dividing its power. But take a look at this: this is the Atlantic Ocean. All right, this is the Pacific Ocean—it's really more of a water hemisphere!

The Pacific Ocean is so large, in fact, it contains its own antipodes, meaning there are places in the Pacific Ocean where you could float and know that even if you dug a hole straight down through the center of the Earth and emerged on the other side, you would still be in the Pacific Ocean. Okay, anyway, back to Brouwer's Coulomb. To see how it works, let's imagine two thermometers on opposite sides of the Earth, A and B. The temperatures they record will probably be different, but if we swap their locations, always keeping them at all times on opposite sides of the planet, their temperature readings will just flip.

In order to swap, well, the readings are going to have to cross at least once. No matter how we swap these, always and typical thermometers, a crisscross will have to happen at least once somewhere. Moreover, these fixed points aren't just peppered around the globe willy-nilly; a continuous unbroken band of them must separate A's region from B's. Why? Well, because if there wasn't such a wall, that would mean there'd be a way to swap them without their readings having to ever meet, which we know can't happen.

Okay, next, let's pick a pair of antipodal points on this band and measure atmospheric pressure at both. If they're the same, hey, that's pretty good, our job is done! But if they're not, we can just swap the barometers along the band like with temperature. The pressures they measure swap, so somewhere they will have to have the same value. Even though weather is chaotic and always changing, and even though the other side of the world is very, very far away, there must always be at least two places on opposite ends of the Earth where the temperature and the pressure are the same.

This is true, by the way, for any two variables that vary continuously across Earth's surface. Four is often called a cosmic number. Why? Because try this: name any number in English—really, any at all. Positive, negative, rational, complex, uninteresting, surreal, infinite—it doesn't matter. Now count the number of letters in its name. This gives you a new number. Count the number of letters in its name and keep going. Eventually, every time, every single time, you will wind up at four, where you will be stuck looping forever.

Side note: if doing something over and over again doesn't produce a different result than just doing it once, the procedure is called identity—meaning same. Taking the absolute value of a number is an idempotent act. Doing it once or doing it a million times gives the same result. Pushing an elevator call button is identity too—pushing it again and again and again and again doesn't make it arrive faster or differently.

Anyway, we get stuck at four because it's the only number in the English language spelled with the same number of letters as the quantity it represents. About four years ago, Reddit user protocol seven showed that there are no other endless loops except for four, and that while negative fifteen and negative seventeen both contain their absolute value worth of letters, which makes them special, to be sure, they obviously aren't spelled with a negative number of letters. Four is four—junior. There's no escape! A path to it is forced!

Oh! Fork! It's even on my forearm! Well, I look forward to seeing you again soon, and as always, thanks for watching.

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I have some exciting news! If you are a parent with young children, or if you are a young child yourself, I have a video over on Sesame Studios that you should check out. Sesame Studios is from the creators of Sesame Street, and it's a great new channel full of really awesome videos for young people. So go check that channel out!

Subscribe! If your kids use a different account, subscribe through that account or whatever you have to do to get stuff from Sesame Studios—it's great! Kevin and Jake already have videos over there, so go watch mine, check that out, and as always, thanks for watching.

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