Prática de Shadowing: The Banach–Tarski Paradox - Aprenda a falar inglês com vídeo
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Hey Vsauce, Michael here.
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There's a famous way to seemingly create chocolate out of nothing.
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Maybe you've seen it before.
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This chocolate bar is four squares by eight squares.
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But if you cut it like this, and then like this, and finally like this,
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you can rearrange the pieces like so and wind up with the same four by eight bar,
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but with a leftover piece apparently created out of thin air.
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There's a popular animation of this illusion as well.
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I call it an illusion because it's just that, fake.
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In reality, the final bar is a bit smaller.
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It contains this much less chocolate.
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Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away.
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The animation is extra misleading because it tries to cover up its deception.
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The lost height of each Each square is surreptitiously added in while the piece moves to make it hard to notice.
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I mean, come on.
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Obviously, you cannot cut up a chocolate bar and rearrange the pieces into more than you started with.
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- Okay.
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Or can you?
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One of the strangest theorems in modern mathematics is the Banach -Tarski paradox.
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It proves that there is in fact a way to take an object and separate it into five different pieces. of the original item.
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Same density, same size, same everything.
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Seriously, to dive into the mind blow that it is and the way it fundamentally questions math and ourselves,
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we have to start by asking a few questions.
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First, what is infinity?
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A number?
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I mean, it's nowhere on the number line, but we often say things like, "There's an infinite number of blah blah blah.
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And as far as we know, infinity could be real.
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The universe may be infinite in size and flat, extending out forever and ever without end,
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beyond even the part we can't observe or ever hope to observe.
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That's exactly what infinity is.
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Not a number per se, but rather a size.
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the size of something that doesn't end.
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Infinity is not the biggest number.
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Instead, it is how many numbers there are.
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But there are different sizes of infinity.
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The smallest type of infinity is countable infinity, the number of hours in forever.
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It's also the number of whole numbers that there are, natural numbers, the numbers we use when counting things like 1,
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2, 3, 4, 5, 6, and so on.
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Sets like these are unending, but they are countable.
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Countable means that you can count them from one element to any other in a finite amount of time.
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Even if that finite amount of time is longer than you will live, or the universe will exist for, it's still finite.
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Uncountable infinity, on the other hand, is literally bigger.
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Too big to even count.
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The number of real numbers that there are, not just whole numbers, but all numbers, is uncountably infinite.
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You literally cannot count even from zero to one in a finite amount of time by naming every real number in between.
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I mean, where do you even start?
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Zero?
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Okay, but What comes next?
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0 .000000.
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Eventually, we would imagine a 1 going somewhere at the end, but there is no end.
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We could always add another 0.
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Uncountability makes this set so much larger than the set of all whole numbers
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that even between 0 and 1 there are more numbers than there are whole numbers on the entire endless number line.
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Derokantor's famous diagonal argument helps illustrate this.
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Imagine listing every number between 0 and 1.
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Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats.
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Bye.
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Each number we generate can be paired with a whole number.
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If there's a one -to -one correspondence between the two, that is if we can match one whole number to each real number on our list,
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that would mean that countable and uncountable sets are the same size.
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But we can't do that.
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Even though this list goes on forever, Forever isn't enough.
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Watch this.
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If we go diagonally down our endless list of real numbers and take the first decimal of the first number, and the second of the second number,
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the third of the third, and so on, and add one to each, subtracting one if it happens to be a nine,
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we can generate a new real number that is obviously between 0 and 1,
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but since we've defined it to be different from every number on our endless list in at least one place,
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It's clearly not contained in the list.
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In other words, we've used up every single whole number, the entire infinity of them,
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and yet we can still come up with more real numbers.
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Here's something else that is true but counterintuitive.
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There are the same number of even numbers as there are even and odd numbers.
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At first, that sounds ridiculous.
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Clearly there are only half as many even numbers as all whole numbers.
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But that intuition is wrong.
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The set of all whole numbers is denser, but every even number can be matched with a whole number.
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You will never run out of members of either set, so this one -to -one correspondence shows that both sets are the same size.
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In other words, infinity divided by two is still infinity.
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Infinity plus one is also infinity. of the Grand Hotel.
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Imagine a hotel with a countably infinite number of rooms.
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But now, imagine that there is a person booked into every single room.
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Seemingly, it's fully booked.
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Right?
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No. Infinite sets go against common sense.
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You see, if a new guest shows up and wants a room, all the hotel has to do is move the guest in room number one to room number two.
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And the guest in room two to room three and three to four and four to five and so on.
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Because the number of rooms is never ending, we cannot run out of rooms.
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Infinity minus one is also infinity again.
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If one guest leaves the hotel, we can shift every guest the other way.
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Guest two goes to room one, three to two, four to three, and so on.
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Because we have an infinite amount of guests, that is a never -ending supply of them, no room will be left empty As it turns out,
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you can subtract any finite number from infinity and still be left with infinity.
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It doesn't care.
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It's unending.
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Banoktarski hasn't left our sights yet.
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All of this is related.
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We are now ready to move on to Shapes.
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Hilbert's Hotel can be applied to a circle.
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Points around the circumference can be thought of as guests.
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If we remove one point from the circle, That point is gone.
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Right?
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Infinity tells us it doesn't matter.
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The circumference of a circle is irrational.
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It's the radius times 2 pi.
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So if we mark off points beginning from the hole, every radius length along the circumference going clockwise,
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we will never land on the same point twice.
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Ever.
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has checked out, we can just shift the rest, move them counterclockwise, and every room will be filled.
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Point 1 moves to fill in the hole.
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Point 2 fills in the place where point 1 used to be.
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3 fills in 2 and so on.
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Since we have an unending supply of numbered points, No hole will be left unfilled.
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The missing point is forgotten.
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we apparently never needed it to be complete.
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There's one last neato consequence of infinity we should discuss before tackling Bonnachtarski.
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Ian Stewart famously proposed a brilliant dictionary, one that he called the Hyper Webster.
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The Hyper Webster lists every single possible word of any length formed from the 26 letters in the English alphabet.
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It begins with A followed by AA, then AAA, then AAAA, and after an infinite number of those,
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AB then ABA, then ABAA, ABAAA, and so on until Z.
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ZA, ZAA, etc, etc, until the final entry, an infinite sequence of Zs.
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Such a dictionary would contain every single word.
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Nay, every single thought, definition, description, truth, lie, name, story, what happened to Amelia Earhart would be in that dictionary.
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as well as every single thing that didn't happened to Amelia Earhart.
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everything that could be said using our alphabet.
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Obviously, it would be huge.
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But the company publishing it might realize that they could take a shortcut.
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If they put all the words that begin with "A" in a volume titled "A", they wouldn't have to print the initial "A".
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Readers would know to just add the A because it's the A volume.
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By removing the initial A, the publisher is left with every A word Sans the first A.
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which has surprisingly become every possible word.
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Just one of the 26 volumes has been decomposed into the entire thing.
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It is now.
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that we're ready to investigate this video's titular paradox.
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What if we turned an object, a 3D thing, into a Hyper Webster?
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Could we decompose pieces of it into the whole thing?
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Yes.
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The first thing we need to do is give every single point on the surface of the sphere, one name and one name only.
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A good way to do this is to name them after how they can be reached by a given starting point.
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If we move this starting point across the surface of the sphere in steps that are just the right length,
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no matter how many times or in what direction we rotate, So long as we never backtrack, it will never wind up in the same place twice.
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We only need to rotate in four directions to achieve this paradox: up, down, left and right, around two perpendicular axes.
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We are going to need every single possible sequence
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that can be made of any finite length out of just these four rotations.
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That means we will need left, right, up and down, as well as left left,
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left up, left down, but of course not left right because, well, that's backtracking.
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Going left and then right means you're the same as you were before you did anything.
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So, no left rights, no right lefts, and no up downs, and no down ups.
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Also, notice that I'm writing the rotations in order right to left.
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So the final rotation is the leftmost letter.
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That will be important later on.
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Anyway, a list of all possible sequences of allowed rotations that are finite in length, is Well, huge.
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countably infinite, in fact.
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But if we apply each one of them to a starting point, in green here, and then name the point we land on after the sequence that brought us there,
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we can name a countably infinite set of points on the surface.
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Let's look at how, say, these four strings on our list would work.
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right up left.
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Okay, rotating the starting point this way takes us here.
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Let's color code the point based on the final rotation in its string.
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In this case, it's left and and for that we will use purple.
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Next up, "Down, down." that sequence takes us here.
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We name the point DD and color it blue since we ended with a down rotation.
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RDR, that will be this point's name, takes us here, and for a final right rotation, let's use red.
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Finally, for a sequence that ends with "up", let's color code the point.
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Orange.
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Now, if we imagine completing this process for every single sequence, we will have a countably infinite number of points named and color -coded.
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That's great!
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But not enough there are an uncountably infinite number of points on a spheres.
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surface.
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But no worries, we can just pick a point we missed.
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any point and color it green, making it a new starting point, and then run every sequence from here.
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After doing this, to an uncountably infinite number of starting points, we will have indeed named and colored every single point on the surface.
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just once.
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with the exception of Poles.
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Every sequence has two poles of rotation, locations on the sphere that come back to exactly where they started.
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for any sequence of right, or left rotations, the poles are the north and south poles.
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The problem with poles like these is
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that more than one sequence can lead us to them they can be named more than once
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and be colored in more than one color.
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For example, if you follow some other sequence to the North or South Pole, any subsequent rights or lefts will be equally valid names.
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In order to deal with this, we're going to just count them out of the normal scheme and color them all yellow.
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Every sequence has two, so there are a countably infinite amount of them.
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Now, with every point on the sphere given just one name, in just one of six colors we are ready to take the entire sphere apart.
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Every point on the surface corresponds to a unique line of points below it,
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all the way to the center point and we will be dragging every points line along with it.
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The lone center point we will set aside.
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Okay, first we cut out and extract all the yellow poles.
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the green starting points, the orange up points the blue down points, and the red and purple left and right points.
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That's the entire sphere.
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With just these pieces, you could build the whole thing.
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But take a look at the left piece.
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It is defined by being a piece composed of every point accessed via a sequence ending with a left rotation.
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If we rotate this piece right, that's the same as adding an R to every point's name.
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But left and then right is a backtrack.
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They cancel each other out.
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And look what happens when we reduce them away.
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The set becomes the same as a set of all points with names that end with L,
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but also U, D, and every point reached with no rotation.
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That's the full set of starting points.
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We have turned less than a quarter of the sphere into nearly three quarters.
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Just by rotating it, we added nothing.
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It's like the Hyper Webster.
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If we add the right piece and the poles of rotation in the center point, well, we've got the entire sphere again, but with stuff left over.
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To make a second copy, let's rotate the up piece down.
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The down -ups cancel because, well, it's the same as going nowhere, and we're left with a set of all starting points,
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the entire up piece, the right piece and the left piece But there's a problem here.
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We don't need this extra set of starting points.
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We still haven't used the original ones.
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No worries.
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Let's just start over we can just move everything from the up piece that turns into a starting point when rotated down.
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That means every point whose final rotation is up.
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Let's put them in the down piece.
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Of course, after rotating, points named UU will just turn into points named U.
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And that would give us a copy here and here.
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So, as it turns out, we need to move all points with any name that is just a string of U's.
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We will put them in the down piece and rotate the up piece down, which makes it congruent to the up, right, and left pieces.
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Add in the down piece along with some up and the starting point piece and, well, we're almost done.
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Rotation and center are missing from this copy, but no worries.
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There's a countably infinite number of holes where the poles of rotation used to be.
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Which means there is some pole around which we can rotate this sphere such that every pole hole orbits around
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without hitting another.
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Well, this is just a bunch of circles with one point missing.
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We fill them each like we did earlier.
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And we do the same for the center point.
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Imagine a circle that contains it inside the sphere and just fill in from infinity.
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And look what we've done.
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We have taken one sphere, and turned it into two identical spheres without adding Anything.
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1 plus 1 equals 1.
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Thank you.
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That took a while to go through, but the implications are huge and mathematicians, scientists, and philosophers are still debating them.
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Could such a process happen in the real world I mean, it can happen mathematically, and math allows us to abstractly predict
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and describe a lot of things in the real world with amazing accuracy.
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But does the Banach -Tarski paradox take it too far?
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Is it a place where math and physics Separate.
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We still don't know.
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History is full of examples of mathematical concepts developed in the abstract that we
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did not think would ever apply to the real world for years,
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decades, centuries, until Eventually, science caught up and realized they were totally applicable and useful,
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The Bonnachtarski paradox could actually happen in our real world.
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The only catch, of course, is that the five pieces you cut your object into aren't simple shapes,
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they must be infinitely complex and detailed That's not possible to do in the real world, where measurements can only get so small,
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and there's only a finite amount of time to do anything.
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but math says it's theoretically valid and some scientists think it may be physically valid 2.
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There have been a number of papers published suggesting a link between Banach Tarski,
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and the way tiny, tiny subatomic particles can collide.
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high energies and turn into more particles than we began with.
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We are finite creatures.
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Our lives are small, and can only scientifically consider a small part of reality,
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What's common for us is just a sliver of what's available.
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We can only see so much of the electromagnetic spectrum.
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We can only delve so deep into extensions of space.
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Common sense applies to that which we can access.
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But common sense is just that.
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"common." If total sense is what we want, we should be prepared to accept that we shouldn't call infinity weird or strange.
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The results we've arrived at by accepting it are valid, true within the system we use to understand, measure, predict, and order the universe.
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Perhaps the system still needs perfecting.
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But at the end of the day, history continues to show us that the universe isn't strange.
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We are.
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And as always, thanks for watching there that really helped me wrap my mind kind of, around Banat Tarski.
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First of all, Leonard Wapner's The Pea and The Sun.
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This book is fantastic and it's full of a lot of the preliminaries needed to understand the proof that comes later.
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He also talks a lot about the ramifications of what Bonnachtarski and their theorem might mean for mathematics.
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Also, if you want to talk about math and whether it's discovered or invented, whether it really truly will map onto the universe, .
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Yanofsky's...
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The Outer Limits of Reason is great.
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This is the favorite book of mine that I've read this entire year.
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Another good one is E.
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Brian Davies.
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Why Beliefs Matter.
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This is actually Korn's favorite book as you might be able to see there.
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It's delicious and full of lots of great information about the limits of what we can know, and what science is, and what mathematics is.
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If you love infinity and math, I cannot more highly recommend Matt Parker's Things to Make and Do in the Fourth Dimension.
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He is hilarious and this book is very very great at explaining some pretty awesome things.
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So, Keep reading and if you're looking for something to watch, I hope you've already watched Kevin Lieber's film on Field Day.
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I already did a documentary about Whittier, Alaska over there.
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Kevin's got a great short film about putting things out on the internet and having people react to them.
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There's a rumor that Jake Roper might be doing something on Field Day soon, so check out mine, check out Kevin's,
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and subscribe to Field Day for upcoming Jake Roper action, yeah?
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He's actually in this room right now.
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Say hi, Jake.
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Hi.
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Thanks for filming this by the way.
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Guys, I really appreciate
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who you all are, and as always, thanks for watching.
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Sobre Esta Aula
Nesta aula, você terá a oportunidade de praticar o seu inglês através de um conteúdo inovador que explora o Paradoxo de Banach-Tarski. Este tema, peculiar no campo da matemática, questiona conceitos de infinitude e formas geométricas. Ao trabalhar com este conteúdo, você irá desenvolver sua compreensão auditiva e suas habilidades de fala, aprendendo a se expressar sobre tópicos complexos e, ao mesmo tempo, aprimorando sua pronúncia e fluência no idioma. A prática inclui shadow speech, onde você imita a fala de um falante nativo, ajudando a melhorar a pronúncia em inglês.
Vocabulário e Frases Chave
- Paradoxo de Banach-Tarski - Um teorema que explora a possibilidade de duplicar objetos através de separação e rearranjo.
- Infinidade - Um conceito matemático que descreve algo sem fim.
- Geometria - O estudo das formas e suas propriedades.
- Contável - Um tipo de infinidade que pode ser listado, mesmo que seja infinito.
- Pontos na Circunferência - Referência a elementos em um círculo, usados para ilustrar conceitos de infinito.
- Amostragem de Sequência - O processo de gerar e classificar pontos em uma superfície.
Dicas de Prática
Ao se engajar com o vídeo, use a técnica de shadowing em inglês para maximizar seu aprendizado. Repita as frases assim que elas forem ditas, tentando imitar a entonação e o ritmo do falante. A velocidade do vídeo pode parecer rápida no início, mas isso é normal. Aqui estão algumas dicas:
- Comece assistindo o vídeo uma vez sem repetir, apenas para entender o conteúdo geral.
- Na segunda execução, pause após cada frase e repita em voz alta, focando em imitar a pronúncia.
- Se necessário, utilize trechos do vídeo para praticar partes que você achou mais desafiadoras.
- Não se preocupe em ser perfeito; a prática constante levará à melhoria. O importante é se familiarizar com os sons e a estrutura da fala.
- Experimente gravar sua voz enquanto faz shadow speak e, em seguida, ouça a gravação para autoavaliação.
Com dedicação e prática, você conseguirá melhorar a pronúncia em inglês e se sentirá mais confiante ao discutir temas complexos.
Gramática neste vídeo
As estruturas que o falante mais usa, com as palavras exatas do vídeo:
| Estrutura | No vídeo |
|---|---|
| “Used to” used to + verbo — um hábito ou estado do passado que não é mais verdade | used to be |
| Present perfect have/has + particípio passado — uma ação passada que ainda importa agora | you've seen · we've defined · hasn't left |
| Voz passiva be + particípio passado — o foco está no que acontece, não em quem faz | can't be listed · can be paired · can be matched |
O que é a Técnica de Shadowing?
Shadowing é uma técnica de aprendizado de idiomas com base científica, originalmente desenvolvida para o treinamento de intérpretes profissionais. O método é simples, mas poderoso: você ouve áudio em inglês nativo e repete imediatamente em voz alta — como uma sombra seguindo o falante com 1-2 segundos de atraso. Pesquisas mostram melhora significativa na precisão da pronúncia, entonação, ritmo, sons conectados, compreensão auditiva e fluência na fala.