Fixed Points

Hello Vsauce Michael, here is an art museum on the moon. Supposedly we can't be sure until we go back and check it, but as the story goes, in 1969, Fred Wall Tower of Bell Laboratories and sculptor Forrest Myers convinced an engineer working on the Apollo 12 lunar lander to hide a tiny ceramic tile about 2 by 1 centimeters inside gold blankets wrapped around parts of the spaceship that would be left on the moon and etched on that ship or artwork by famous artists, according to everyone involved, the plan worked and when the Apollo 12 team left the moon, the wafer was still there two days later, Myers told the New York Times what they had done and this image of the wafer was published containing a Rauschenberg straight line, a drawing by Oldenburg's Mickey Mouse, very good and The story is probably 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 is covering Andy Warhol's presentation , why, according to Warhol, all he did was innocently carve his initials. on the chip he stylized them like this, he drew a kind of funny W for Warhol and then he put a line here to make this part look like an a, that's it, it's just his initials.

Here's a clear view of a replica that's on the moon. However, now I'm on Earth in North America and this is a map of North America, which is cool because if you're in the place where you have a map, mathematically there will always be some point on your map that is directly above the place in the real world that it represents always no matter how you hold the map it can be rotated, flipped, even twisted, folded or crumbled is guaranteed by Brower's

fixed

point theorem a

fixed

point is anything that doesn't go anywhere after a transformation, says Brouwer's theorem tells us that it is impossible to completely mix a set of

points

if they are bounded without holes and transformed continuously without cutting and pasting without limits each point can be mapped somewhere new around the holes each point can be mapped somewhere new and if you cut or paste each point can be assigned to some new place, but otherwise the blend will always fail somewhere a transformation is continuous if the distance between any two

points

reaches zero in the previous state and also approaches zero in the later state each square of this chessboard is filled with a dot like pixels of a single color as we transform them, let's keep the pixels still within the square they started in shining, okay, like You can see, no matter how you resize or manipulate the chessboard, it will always glow somewhere where you can't get all the pixels to be simultaneously outside of their original square unless you cut or move the entire shape out of the space it used to. fill here is a question do you think you can completely mix the coffee in a cup by stirring well?

Of course you can't stir the coffee it's a continuous transformation of the coffee and everything stays within the same space, so Brower'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 where you stirred it back to where it was before a fixed point to make sure that the coffee is not made of points, it is made of molecules, but they are quite small and very numerous, so which, within a certain degree of error, will pretty much keep points fixed, not just thwart your ability to completely mix things up.

They can also draw you towards them. They can be attractive like the number nine. Try this. Think of a number with more than one digit and then add its digits. Now subtract that sum from the original number to get a new number if you do this over and over again. over and over again until you're down to a single-digit number, you'll always end up with nine each time. Also note that any number with two or more digits minus the sum of its digits becomes divisible by nine immediately. What's going on here? Well, there is nothing mystical about miscellaneous nine in general, but it is just a consequence of how we write numbers.

Numbers can be written in many ways, but the most common base ten positional notation makes the nine trick work in this system. Not a number like twenty-five. It doesn't mean two and five, it means ten twos and one five, that's twenty-five, so subtracting just the positional digits from a number removes one member from each group, the digit you have one of disappears completely, the digit you have ten is reduced to nine copies, the digit of which you have 100 is reduced to 99 copies, and so on, until everything becomes a multiple of nine attractive fixed points that also play a starring role in a famous method for calculating square roots. , it is called the Babylonian method, the number you want. to find the square root of is called the radind and its square root is a number that is equal to the radind divided by itself.

Okay, now the first step of this method is to guess, if your guess is less than the actual answer, the radicand divided by your guests. will be greater than the answer and vice versa, the true answer will always be somewhere between those two values, so after guessing, take both values ​​and find their average, it will be a middle point and now use this value as your guests and continue, you will converge to the true square root at a pretty good rate, the number of correct digits in your approximation will roughly double after each iteration, that's pretty good, but let's talk about infinity, specifically the Aleph numbers, they describe sizes of well-ordered infinities, the smallest is aleph-null equal to the number of integers there are but there are literally infinite numbers larger than that in order of increasing size they are Aleph 1 a love - Aleph 3 and so on each one is infinite but refers to a larger number of things than infinite before, if you tried to pair aleph-null things with Aleph 1 things, you would 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 what's there. turns out to be a fixed point of Aleph, notice that the subscript of each Aleph is equal to the number of infinities less than it, there is nothing less than Aleph null one less than Aleph 1 2 less than one love 2 and so on every next number of Aleph is monstrously larger than the previous one, but it only adds one to the growing list of Olives. Clearly, these rates are so different that they will never match up, but do take a look at this number.

Aleph Aleph Aleph alfalfa a loading rule and so on an endless cascade of Olive Trees, how many infinities are smaller than this number? Well, look at its subscript, okay, it's an endless cascade of faith, like the number itself, this is a fixed point of Aleph and infinity is so large that it is equal to the number of infinities smaller than itself. Cool, but probably my favorite thing related to the fixed point is the strength of Coulomb's theorem, which states that at any given time there must be at least a pair of points on the Earth's surface that are diametrically opposite to each other but that, However, they have the same temperature and atmospheric pressure diametrically.

The opposite points on a sphere are called antipodes and as I have mentioned before, if you place a piece of bread on the ground somewhere on the earth and another on that point of the antipodes, well, you have made yourself a sandwich of earth, there are sites that help you. locate opposite points on the earth, but at this point in history you will notice that most points on the earth are antipodes to water, which makes sense, the surface of the Earth is mostly covered in water, but even though we know that it is not We always appreciate how Gigantic the Pacific Ocean is maybe because maps tend to divide its power, but take a look at this, this is the Atlantic Ocean, okay, this is the Pacific Ocean, it's actually more of a hemisphere of water , the Pacific Ocean is so big in fact.

It contains its own antipodes, meaning that there are places in the Pacific Ocean where you can float and know that even if you dug a hole in the center of the earth and emerged on the other side, you would still be in the Pacific Ocean, anyway. Let's go back to borsa 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 exchange their locations, always keeping them on opposite sides of the planet, their temperature readings will simply change so they can be exchanged well. readings will have to intersect at least once, no matter how we swap these thermometers always and typically, a crossover will have to occur at least one somewhere, furthermore, these fixed points are not scattered around the world, whether you like it or not, a band continuous and uninterrupted of them must separate the region A from the bees, why?

Well, because if there was no such wall, that would mean there would be a way to swap them without their readings having to match, which we know. can't happen, okay, now let's pick a couple of antipodal points in this band and measure the atmospheric pressure at both if they are equal, hey, that's pretty good, our job is done, but if they're not, we can just swap the barometers throughout the As with temperature, the pressures they measure interchange, so somewhere they will have to have the same value, even if the weather is chaotic and always changing and even if the other side of the world is very, very far away, there must always be at least two places at opposite ends of the earth where the temperature and pressure are the same, this is true, by the way, for any two variables that vary continuously across the Earth's surface, often It is called a cosmic number, why?

Well, because try with 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 continue eventually each time you will end up in 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 ident which means the same power taking the absolute value of a number is a powerless identification. once or doing it a million times gives the same result pressing the call button for an elevator is new identity in the spill come to you pressing it again and again and again and again does not make it arrive faster or differently from all Anyway we get stuck at four because it is the only number in the English language written 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 and that while negative 15 and negative 17 contain their absolute value of letters which makes them special to make sure that obviously are not written with a negative number of letters for it is for JUnit there is no escape a path to it it is forced Oh Fork, it's even on my forearm well, I hope to see you again soon and as always thanks for watching. I have exciting news if you are a parent of young children or a toddler yourself. I have a video about Sesame Studios that you should watch.

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