hey, vsauce. michael here.there's a famous way to seemingly create chocolate out ofnothing. maybe you've seen it before.this chocolate bar is 4 squares by 8 squares, but if you cut it like this and then like this and finally like this you can rearrange the pieces like so and wind up with the same 4 by 8 bar but with a leftover piece, apparentlycreated out of thin air. there's a popularanimation of this illusion
as well. i call it an illusion because it's just that. fake.in reality, the final bar is a bit smaller.it contains this much less chocolate. each squarealong the cut is shorter than it was in the original, but the cut makes it difficult to noticeright away. the animation is extra misleading, because it tries tocover up its deception. the lost height of each square issurreptitiously added in while the piece moves to makeit hard to notice.
i mean, come on, obviously you cannot cut upa chocolate bar and rearrange the pieces into more thanyou started with. or can you?one of the strangest theorems in modern mathematics is thebanach-tarski paradox. it proves that there is, in fact, a way totake an object and separate it into 5 different pieces. and then, with those five pieces, simply
rearrange them.no stretching required into two exact copies of the original item. same density, same size, same everything. seriously. to dive into the mind blow that it is and the way it fundamentallyquestions math and ourselves, we have to start by askinga few questions. first, what is infinity? a number?i mean, it's nowhere
on the number line, but we often say things like there's an infinite "number" of blah-blah-blah. and as far as we know, infinity could be real. the universe may be infinite in size and flat, extending out for ever and ever without end, beyond even the part we canobserve or ever hope to observe. that's exactly what infinity is.not a number per se, but rather a size.the size
of something that doesn't end.infinity is not the biggest number, instead, it is how many numbers there are. but there are differentsizes of infinity. the smallest type of infinity is countable infinity.the number of hours in forever. it's also the number of wholenumbers that there are, natural number, the numbers we use whencounting things, like 1, 2, 3, 4, 5, 6 and so on. sets like these are unending,
but they are countable. countablemeans that you can count them from one element to any other in a finite amount of time, even if that finiteamount of time is longer than you will live or the universe will exist for, it's still finite. uncountable infinity, on the other hand, is literally bigger.too big to even count. the number of real numbers that there are, not just whole numbers, but all numbers is
uncountably infinite.you literally cannot count even from 0 to 1 in a finite amount oftime by naming every real number in between.i mean, where do you even start?zero, okay. but what comes next? 0.000000... eventually, we would imagine a 1 going somewhere at the end, but there is no end. we could always add another 0.uncountability makes this set so much larger than the setof all whole numbers
that even between 0 and 1, there are more numbers than there are whole numbers on theentire endless number line. georg cantor's famous diagonal argument helps illustrate this.imagine listing every number between zero and one. since they areuncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats. each number regenerate can be paired with a whole number. if there's a one toone correspondence between the two, that is if we can match one whole numberto each real number
on our list, that would mean that countable and uncountable sets are the same size.but we can't do that, even though this list goes on for ever. forever isn't enough.watch this. if we go diagonally down our endlesslist of real numbers and take the first decimalof the first number and the second of the second number,the third of the third and so on and add one to each, subtracting one if it happens to be a nine, we cangenerate a new
real number that is obviously between 0 and 1, but since we've defined it to bedifferent from every number on our endless listand at least one place it's clearly not contained in the list. in other words, we've used up everysingle whole number, the entire infinity of them and yet wecan still come up with more real numbers.here's something else that is true but counter-intuitive.there are the same number of even numbers as there are even
and odd numbers. at first, that soundsridiculous. clearly, there are only half as many even numbers as all whole numbers,but that intuition is wrong. the set of all whole numbers is denser but every even number can be matched with awhole number. you will never run out of members eitherset, so this one to one correspondence shows that both sets are the same size. in other words, infinity divided by two is still infinity.
infinity plus one is also infinity. a good illustration of this is hilbert'sparadox up the grand hotel.imagine a hotel with a countably infinite number ofrooms. but now, imagine that there is a person bookedinto every single room. seemingly, it's fully booked, right?no. infinite sets go against common sense. you see, if a new guest shows up and wants a room, all the hotel has to do is move theguest in room number 1
to room number 2. and a guest in room 2 toroom 3 and 3 to 4 and 4 to 5 and so on. because the number of rooms is never ending we cannot run out of rooms.infinity -1 is also infinity again. if one guest leaves the hotel, we can shift every guest the other way.guest 2 goes to room 1, 3 to 2, 4 to 3 and so on, because we have an infinite amount of guests. that is anever ending supply of them.
no room will be left empty.as it turns out, you can subtract any finite number from infinity and still be left with infinity.it doesn't care. it's unending. banach-tarski hasn't leftour sights yet. all of this is related.we are now ready to move on to shapes.hilbert's hotel can be applied to a circle. points around thecircumference can be thought of as guests. if we remove one point from the circle that point is gone, right?infinity tells us
it doesn't matter.the circumference of a circle is irrational. it's the radius times 2pi. so, if we mark off points beginning fromthe whole, every radius length along thecircumference going clockwise we will never land on the same pointtwice, ever.we can count off each point we mark with a whole number.so this set is never-ending, but countable, just like guests androoms in hilbert's hotel. and like those guests,even though one has checked out,
we can just shift the rest.move them counterclockwise and every room will befilled point 1 moves to fill in the hole, point 2 fills in the place where point 1 used to be,3 fills in 2 and so on. since we have a unendingsupply of numbered points, no hole will be left unfilled. the missing point is forgotten.we apparently never needed it to be complete. there's one last needoconsequence of infinity we should discuss before tackling banach-tarski.ian stewart
famously proposed a brilliant dictionary. one that he called the hyperwebster.the hyperwebster lists every single possible word of any length formed from the 26 letters in theenglish alphabet. it begins with "a," followed by "aa," then "aaa," then "aaaa." and after an infinite number of those, "ab," then "aba," then "abaa", "abaaa," and so on until "z, "za,"
"zaa," et cetera, et cetera,until the final entry in infinite sequence of "z"s.such a dictionary would contain every single word.every single thought, definition, description, truth, lie, name, story.what happened to amelia earhart would be in that dictionary,as well as every single thing that didn't happened to amelia earhart. everything that could be said using our
alphabet.obviously, it would be huge, but the company publishing it mightrealize that they could take a shortcut. if they put all the wordsthat begin with a in a volume titled "a," they wouldn't have to print the initial "a."readers would know to just add the "a," because it's the "a" volume.by removing the initial "a," the publisher is left with every "a" word sans the first "a," which has surprisingly become every possible word.just one
of the 26 volumes has beendecomposed into the entire thing. it is now that we're ready toinvestigate this video's titular paradox.what if we turned an object, a 3d thing into a hyperwebster? could we decompose pieces of it into thewhole thing? yes.the first thing we need to do is give every single point on thesurface of the sphere one name and one name only. a good way todo this is to name them after how they can be reached by a given starting point.
if we move this starting point acrossthe surface of the sphere in steps that are just the right length,no matter how many times or in what direction we rotate, so longas we never backtrack, it will never wind up in thesame place twice. we only need to rotate in fourdirections to achieve this paradox. up, down, left and right around two perpendicular axes.we are going to need every single possible sequence that canbe made of any finite length out of just thesefour rotations.
that means we will need lef, right, up and down as well as left left, left up, left down, but of course not left right, because, well, that'sbacktracking. going left and then right means you're the same asyou were before you did anything, so no left rights, no right lefts and no updowns and no down ups. also notice that i'm writingthe rotations in order right to left, so the final rotation is the leftmost letter.that will be important later on.
anyway. a list of all possible sequencesof allowed rotations that are finite in lenght is, well, huge. countably infinite, in fact. but if we apply each one of them to astarting point in green here and then name the point weland on after the sequence that brought us there,we can name a countably infinite set of pointson the surface. let's look at how, say, these four stringson our list would work. right up left. okay, rotating the startingpoint this way takes
us here. let's colour code the pointbased on the final rotation in its string, in this case it's left and for that we will use purple.next up down down. that sequence takes us here.we name the point dd and color it blue, since we ended with a down rotation.rdr, that will be this point's name, takes us here.and for a final right rotation, let's use red.finally, for a sequence that end with up, let's colour code the point orange.
now, if we imagine completing thisprocess for every single sequence, we will have acountably infinite number of points named and color-coded.that's great, but not enough.there are an uncountably infinite number of points on a sphere's surface. but no worries, we can just pick a pointwe missed. any point and color it green, making it a new starting point and then run everysequence
from here.after doing this to an uncountably infinite number ofstarting point we will have indeed named and colored every single point onthe surface just once.with the exception of poles. every sequence has two poles ofrotation. locations on the sphere that come back toexactly where they started. for any sequence of right or left rotations, the polls are the north andsouth poles. the problem with poles like these isthat more than one sequence can lead us
to them. they can be named more than once and be colored in more than one color. for example, ifyou follow some other sequence to the north or south pole, any subsequent rights or lefts will be equally valid names. in order to dealwith this we're going to just count them out of the normal scheme and color them all yellow.every sequence has two, so there are a countably infinite amount
of them. now, with every point on thesphere given just one name and just one of six colors,we are ready to take the entire sphere apart. every point on the surfacecorresponds to a unique line of points below it all the way to the center point.and we will be dragging every point's line along with it.the lone center point we will set aside. okay, first we cut outand extract all the yellow poles, the green starting points, the orange up points, the blue down points
and the red and purple left and rightpoints. that's the entire sphere.with just these pieces you could build the wholething. but take a look at the left piece. it is defined by being a piece composed of every point, accessed via a sequence ending with a left rotation.if we rotate this piece right, that's the same as adding an "r" to every point's name.but left and then right is a backtrack, they cancel each otherout. and look what happens when you
reduce them away. the set becomes the same as a set of all points with namesthat end with l, but also u, d and every point reached with no rotation.that's the full set of starting points. we have turned less than a quarter ofthe sphere into nearly three-quarters just by rotating it. we added nothing. it's likethe hyperwebster. if we had the right piece and the poles of rotation and the centerpoint, well, we've got the entire sphere
again, but with stuff left over.to make a second copy, let's rotate the up piece down.the down ups cancel because, well,it's the same as going nowhere and we're left with a set of allstarting points, the entire up piece, the right piece and the leftpiece, but there's a problem here. we don't need this extra set of startingpoints. we still haven't used the original ones. no worries, let's juststart over. we can just move everything from the uppiece that turns into a starting point whenrotated down.
that means every point whose finalrotation is up. let's put them in the piece. of course, after rotatingpoints named uu will just turn into points named u,and that would give us a copy here and here.so, as it turns out, we need to move all points with any name that is just a string of us. we will put them in the down piece androtate the up piece down, which makes it congruent tothe up right and left pieces, add in the down piecealong with some up and the starting point piece and, well,we're almost done.
the poles of rotation and center are missing from this copy, but no worries.there's a countably infinite number of holes,where the poles of rotations used to be, which means there is some pole aroundwhich we can rotate this sphere such that every pole hole orbits around withouthitting another. well, this is just a bunch of circleswith one point missing. we fill them each like we did earlier.and we do the same for the centerpoint. imagine a circle that contains it insidethe sphere and just fill in from infinity and lookwhat we've done.
we have taken one sphere and turned itinto two identical spheres without adding anything. one plus one equals 1.that took a while to go through,but the implications are huge. and mathematicians, scientists andphilosophers are still debating them. could such a process happen in the realworld? i mean, it can happen mathematically andmath allows us to abstractly predict and describe a lot of things in the realworld with amazing accuracy, but does the banach-tarski paradox
take it too far?is it a place where math and physics separate?we still don't know. history is full of examples ofmathematical concepts developed in the abstract that we did not think would ever applyto the real world for years, decades, centuries,until eventually science caught up and realized they were totally applicable and useful. the banach-tarski paradox couldactually happen in our real-world, the only catch of course is that thefive pieces you cut your object into
aren't simple shapes.they must be infinitely complex and detailed. that's not possible to do inthe real world, where measurements can only get so small and there's only a finite amount of timeto do anything, but math says it's theoretically valid and some scientists think it may be physically valid too. there have been a number of paperspublished suggesting a link between by banach-tarski and the way tiny tiny sub-atomicparticles
can collide at high energies and turninto more particles than we began with. we are finite creatures. our lives are small and can only scientificallyconsider a small part of reality.what's common for us is just a sliver of what's available. we canonly see so much of the electromagnetic spectrum. we can only delve so deep intoextensions of space. common sense applies to that which wecan
access.but common sense is just that. common.if total sense is what we want, we should be prepared toaccept that we shouldn't call infinity weird or strange. the results we've arrived at byaccepting it are valid, true within the system we use tounderstand, measure, predict and order the universe. perhaps the system still needsperfecting, but at the end of day, history continues to show us that theuniverse isn't strange.
we are. and as always, thanks for watching. finally, as always, the description is fullof links to learn more. there are also a number of books linkeddown there that really helped me wrap my mind kinda around banach-tarski. first of all, leonard wapner's "the pea and the sun." this book is fantastic and it's full of lot of the preliminaries needed to understand the proof that comes later.he also talks a lot about the
ramifications of what banach-tarski and theirtheorem might mean for mathematics. also, if you wanna talk about math andwhether it's discovered or invented, whether it really truly will map onto the universe,yanofsky's "the outer limits of reason" is great. this is the favorite book of mine that i've readthis entire year. another good one is e. brian davies' "why beliefs matter." this is actuallycorn's favorite book,
as you might be able to see there.it's delicious and full of lots of great information about the limits of what wecan know and what science is and what mathematics is. if you love infinity and math, i cannotmore highly recommend matt parker's "things to make and do in the fourth dimension." he's hilarious and this book is very very great at explaining some prettyawesome things. so keep reading,and if you're looking for something to watch, i hope you've already watched kevinlieber's film on
field day. i already did a documentary about whittier, alaska over there. kevin's got a great short film aboutputting things out on the internet and having people react to them. there'sa rumor that jake roper might be doing something on field day soon. so check out mine, check out kevin's and subscribe to field day for upcoming jakeroper action, yeah? he's actually in this room right now, sayhi, jake. [jake:] hi. thanks for filming this, by the way. guys, i really appreciate who you all are.
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