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 a 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 and etched onto that ship or artworks by famous artists according to everyone involved the plan worked and 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 and 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 brower'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 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 can not stirring coffee is 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 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 100 of goes down to 99 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 it's 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 guests 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 and now use this value as your guests 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 a love - 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 2 less than a love 2 and so on each next Aleph number is monstrously bigger than the last but only adds one to the growing list of Olives clearly these rates are so different they'll never meet up but they do take a look at this number Aleph Aleph Aleph alfalfa a load rule and so on an endless cascade of Olives how many infinities are smaller than this number well look at its subscript it's well it's an endless cascade of a lafe's just like the number itself this is an Aleph fixed point and 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 points 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 at 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 ok anyway 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 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 one 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 bees 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 for is often called a cosmic number why well because try this name any number in English really any at all positive negative rational complex uninteresting surreal infinite 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 ident meaning same power taking the absolute value of a number is an ID impotent Act doing it once or doing it a million times gives the same result pushing an elevator call button is identity' new on spill come to you 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 and that while negative 15 and negative 17 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 for is for JUnit 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 [Music] 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 go watch mine check that out and as always thanks for watching [Music]

Channel: Vsauce

Views: 6,463,649

Rating: 4.9325085 out of 5

Keywords: fixed points, math, education, vsauce, michael stevens, topology, shapes, brouwer's fixed point theorem, borsuk-ulam, temperature, geometry, science, infinity, numbers, functions, learn

Id: csInNn6pfT4

Length: 16min 25sec (985 seconds)

Published: Wed Sep 28 2016

Reddit Comments

[u/SOberhoff]

I've always felt like fixed points are an underappreciated concept in mathematics. e^x is a fixed point of the derivative. Eigenvectors are sorta fixed points (for Markov matrices they are exactly). Newton iteration heads towards a fixed point. The golden mean is a fixed point of f(x) = 1 + 1/x. The normal distribution is a fixed point of the fourier transform. Any discussion of chaos theory usually starts with a section on fixed points. It goes on forever. Really, I think the concept deserves an entire book for itself.

[u/fenixfunkXMD5a]

What about the filthy Frank collabs Micheal?

[u/DR6]

At the end he claims that, if you start with a number, count the letters of its name in English, and apply the same process to the result iteratively, you always ends up with four. His argument is that four is the only fixed point of that operation. But he doesn't rule the possibility of loops or going to infinity, right? I am willing to believe the claim is in fact true because if you start with a huge number or a non natural number, the process quickly turns the number into a small natural number, but he doesn't mention this, or does he?

[u/banksyb00mb00m]

Well, in the coffee example several assumptions need to be mentioned, like you are only considering two time instants, the stirring is continuous and every point in the surface stays in the surface, etc.

Also, I was expecting some mention of the Hairy Ball Theorem.

[u/functor7]

vSauce just needs to go ahead and get a math degree already.

[u/oblivion5683]

Is he secretly working up to a video on the incompleteness theorems? First ordinals, now fixed points, I believe he had a video where he went into more axiomatic arithmetic some, all he'd need to do now would be to cover basic proof stuff in an interesting way (possibly a video on induction? Sounds like his style) then he could do one on the foundational crisis and russels paradox, and then he could easily make a long one where he sketched the whole proof out well enough for his audience to understand

[u/CornerSolution]

eye-DEM-poh-tent?? No, no, I don't think so. EYE-dem-POH-tent, you scoundrel.

[u/Maciek300]

At 7:21 isn't it a mistake? The number of right digits of the approximation will not double with each iteration. I'd say that for each iteration the approximation will get "half a digit more right". The example from wiki.

EDIT: I just calculated x_4 and x_5 from the wiki example and they were accurate to the 8th and 18th significant figures so I guess I was wrong and the number of right digits of the approximation is indeed doubling with each iteration of the babylonian method which I didn't believe some ancient technique would do.

[u/OriginalName667]

Is his statement about the coffee accurate? Wouldn't the stirring of the water molecules essentially "tear" them away from each other as they get mixed up, since water is a liquid, rather than a solid?