"... i don't fucking care what happens when you turn a figure 8 into a fucking circle, i don't give a flying fuck about avoiding sharp bends, why are you avoiding me? ..."
If i got a nickel for every time someone mistook a hugsbee video for an educational video, i would have 2 nickels, which isn't a lot but it's surprising it happened twice. But hey! Now you are in the same level ad cnn
Every shape has an infinite number of points in it. Because of the Hilbert's Hotel paradox, if you take an infinitely small chunk out of a shape, there will always be another infinitely small chunk that replaces it, a chunk that replaces that chunk, a chunk that replaces that chunk... you get the idea. As a result, you can keep taking infinitely small chunks out of that shape and create an identical copy of that shape... or an infinite number of them.
I guess the word "infinity" means nothing because you can always divide it further and further. Real life objects dont work like that. You're trying to apply common, real sense to something which only makes sense as a concept in the minds of eggheads.
@thelibyanplzcomeback youre changing how you handle infinities halfway through, the perfect example of taking infinitely small pieces out infinite times is a tank of water make a hole in a tank of water, each infinitely small amount of time an infinitely small amount of water leaves, but if you integrate you realise it doesnt duplicate
@@Galinaceo0its very counter intuitive that you only need 6.28 metres added to the circumference to change the radius of the entire earth by a metre. i mean, it makes perfect sense when you think about it and enough exposure to it makes it much more intuitive, but at first hearing it sounds obviously false
Did you know that Google's translation of "Did you know that Banach-Tarski is an anagram of Banach-Tarski Banach-Tarski" into English is "Did you know that Banach-Tarski is an anagram of Banach-Tarski Banach-Tarski"?
7:58 usually when we talk about spaces like S^2 or R^3 we pronounce them as “S Two” or “R Three”, also the R should be blackboard bold since it denotes the real numbers
Just a comment regarding your explanation of the coin rotating around the other coin. Your explanation kind of gives the feeling, that the outer coin always has to double the amount of spins, when in reality it is the size ratio + 1 for the Rotation around itself. So for equaliy sized coins the size ratio is 1 and therefore the outer coin travels 1+1=2 times the circumfarance of the inner coin. However, if the outer ball has only a radius of 1/3, it spins 3+1 = 4 times around Insel and not 6 times.
@@TH-cam_username_not_found That just says that coin B traveled twice as far and therefore rolled twice as much. This logic would still apply if coin B had a radius of only 1/3 cm. In that case, if you rolled coin B across a flat surface, it would rotate by 1 turn after traveling (1/3)τ cm. Rolling it around coin A causes it to travel (4/3)τ cm, which is 4 times as far. Therefore, coin B rotates by 4 turns in this scenario.
3:48 also, the coin thing is so much more simple than it’s let on to be; they roll “against” each other at the same rate for the same distance, but since coin A doesn’t move, halfway through coin B’s journey the connection point is halfway around the coin, thus tada 🎉 it’s right side up because it’s connected under coin A rather than on top. Not all that math
😭😭😭😭😭yes cry some more😭😭😭😭for your understanding of what an actual paradox is, equals your General iq in well defined logic that is Just counterintuitive to incompenent idiots like you 😭😭😭😭😭
when I was young, I joked to friend that I could prove that 1 + 1 = 1 using the staircase limit. (I already knew that was silly and impossible, but still funny)
I don't know if you see this but I want you to know - keep making these types of videos of geometry - - - Also try to do a video on non euclieadian geometrical 3d planes
2:10 the easiest way to visualize this is to look at the top of the coins head orientation. At the top it is facing up, the it is facing up again at the bottom, then back to it originally at the top
On the coins one: Huh? It isn’t rotating twice. That’s just an illusion because at the halfway point the surface upon which it‘s rolling is fully inverted. If you unwrapped it to flat at that moment, the rolling coin would be upside-down. It’s rolling the same distance in both orientations.
I think the easiest way to mentally disprove the staircase paradox is to imagine that instead of the point off of the line being a right angle it's literally any other angle following the rest of the same rules. Under that restriction it's possible to make the limit of the length equal literally any value greater than ~1.4, which if you accept that then length has no meaning or it's a faulty measurement. It looked like one of the other proof diagrams you flashed up may have being along a similar thought process.
@@TheBalthassar The proof you reference appears to be a proof by contradiction: you begin with a set of assumptions, prove that those assumptions lead to an absurd result, and conclude that at least one of the assumptions must be incorrect. But there was already such a proof that was all but stated in the video, which I will write explicitly here: "Assume that the limit of the lengths of the staircases is equal to the length of the limit of the staircases. The limit of the lengths of the staircases is 2. The limit of the staircases is the diagonal line segment, whose length is √2. But 2 is not equal to √2, so this is a contradiction. Therefore, the limit of the lengths of the staircases cannot be equal to the length of the limit of the staircases." So, did you just not notice that one?
@@isavenewspapers8890 You appear to be under the mistaken impression that my point is disregarding that not building upon it. You're arguing with a straw man.
@@TheBalthassar Your claim was that your proof is the easiest way to do it. However, it involves a bit of extra work that can be entirely skipped to produce the same conclusion. I don't see how that's easier than the other way.
Hey, I know you. You're the one who was stirring up trouble in the comments of the video adaptation of The Tau Manifesto. Not sure if you're a troll or what, but it's funny running into you again. But yeah, a comprehensive explanation of the Banach-Tarski paradox doesn't work well with the 2-minute-per-section format. I feel like you'd need 10 minutes, bare minimum, even assuming the viewer already has some basic experience with set theory.
@@isavenewspapers8890 i was certainly not "stirring up trouble." tau evangelism is not mathematical or self-consistent, it's more like people preferring 432hz tuning to 440hz because the numbers are more "pure"
there are instances where tau is a more appropriate constant and instances where pi makes more sense. tau-absolutists who pretend to have forgotten about the existence of pi because they've reprogrammed their brains are closer in nature to cult leaders than mathematicians
The Dehn invariant arises as part of a problem involving finitely many straight cuts of a solid. Such restrictions do not apply in the case of Banach-Tarski.
I vividly remember watching the 'outside in' video so many times trying to imagine what happened. It reminds me that I wish I could select some faces of the original sphere to watch how the sphere everted patch-wise. Maybe someone build that as a 3D web app now, which wasn't possible at the time of course.
The circle one I know the other way around. You extend the circumference with one meter of string and the question is, if a mouse would fit under the rope. In this case I remember the radius to raise about 16cm which would allow some mice stacked under the rope.
@@vibbruh tau is equal to 2*pi. So any equation where you use pi, you could use tau instead. For example, the circumference of a circle is 2*pi*r, or you can say it's tau*r. Some people argue over which is better: pi or tau.
There's a great Vsauce video from years ago that goes over Banach-Tarski and the seeming paradoxes of infinity involved in some detail. I'd highly recommend it.
The first one isn’t just interesting because you only have to add 6.28 meters. It’s interesting because it’s entirely independent of the initial radius. So whether you’re talking about a 1m radius or Earth or the entire universe, you always have to add only 2pi meters to make the radius 1 meter bigger.
I understand that the math is correct but still cant wrap my head around the String girdling Earth. It's very counterintuitive that for the whole Earth it's just some meters.
If you dont find it intuitive think of a simpler shape like a square. If you strap a rope around it and do the same you wanted to for the earth you could do it like this: step1: cut the rope at every corner Step 2 place all 4 ropes 1 meter from the side of the square the rope was touching Step 3 notice the extra length needed. Its all at the corners, and this isnt alot at all, and it doesnt depend on how big the initial square is. Now expand this idea in your head to a circle.
It’s because the equation is just tau times the radius. It’s a linear relationship- adding 1 to the radius ALWAYS adds tau to the circumference. If you want the radius to affect the amount you add in the way you’re thinking, you would need an r^2 term
It's only a paradox when you apply it to something large. If you picture a very small object with a string around it, like a 1 cm sphere, it becomes much more apparent that increasing the radius by 1 meter can't be related to the diameter of the original object.
It’s because if you calculate the length of string to add, it’s independent of the original radius. And yes, it is indeed strange that adding 6.28 m to a rope around the universe has the same effect as adding 6.28 meters to a rope around a basketball.
A redefinition of the mathematical symbol π would be... unpleasant, to put it mildly. We'd have to destroy all the old records and write new ones, or else deal with ambiguity as to which number we're using when we write "π".
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Vsauce made a quite good video about it with a relatively intuitive explanation. I don't think you'll ever find a more intuitive explanation than theirs, at least til now.
this video is sponsored by adblock i totally understand why people use adblock i pay for youtube premium but still have to listen to sponsorship ads i really understand people now and weird youtube doesn't
I don't think the staircase works. If you always halve it, you'll never cover irrational points, which is almost every point on the diagonal. So the limit of the staircase is NOT the diagonal.
That's not how limits work. The sequence doesn't actually have to reach the point in question; it just has to approach it. For example, the function sin(x) / x never attains a value of 1, but its value approaches 1 as x approaches 0, so the limit of the function as x approaches 0 is 1.
No, that’s not what a paradox is. A paradox is something that cannot be proven or disproven like the grandfather paradox or the boots paradox. You will never find a contradiction in traditional versions of those two.
@@Bdcrock what I meant is that in attempt to solve a paradox often a contradiction arises. What the video contains are not paradoxes, the examples are just math problems that have an answer that is not intuitive at first.
@@titastotas1416 yes and you are correct the difference between what you’re saying and what I’m saying however is that all paradoxes have contradictions but that is just not true
@@Bdcrock listen up, I never stated that all paradoxes have contradictions, in fact I cant think of a paradox that has a contradiction in its formulation. What I meant and I think I have stated it clearly enough already is that in attempts to solve a paradox one will be faced by a contradiction and that is always true ,If you don't agree with that give me a paradox that does not result in a contradiction when an attempt to solve is made. We are not in disagreement, I agree with the definition of paradox you have provided previously. In fact I don't see what your issue is.
1) Counterintuitive facts can be referred to as paradoxes. This is a well-established usage of the term, and most people understand it. Stop fighting language. 2) How does the Banach-Tarski paradox not fall into the same category as the other four? To me, it just sounds like you're saying, "That's the only one I don't understand, therefore it can't be true."
the paradox part comes in when the coin rotates seemingly a different amount of times depending on where the focus is, this was not mentioned, also it obviously goes around 2 times idk how that could be unexpected
@@temmie5764 At least for me, my intuition says that since the two coins are touching, the revolving coin's edge will go exactly one circumference-distance. I know that's wrong, it's just that that's what my intuition says. Considering this situation is a common one to cite for unintuitive behavior (and an entire group of SAT questions creators got it wrong), obviously many people have intuition similar to what I described. Good on you for having a better intuition.
the thing is, it does only go around once, but it also goes around twice, it just depends on the observer, thats the paradoxical part that isnt mentioned
I don’t know if this will help but for the last paradox, you could just think of it if one ball has infinite points and he cut those infinite points in half both halves will still be infinite so they can both be reconstructedreconstructed into two separate balls
I can't really tell why you said "why" three times. Anyway, how is it a distraction? It's directly relevant to the math at hand, so I don't know what you're talking about.
I struggle to see how the staircase one is a paradox. It's obvious that a jagged line is longer than a straight line (no matter how small those jags are).
It's only called a paradox because to some people it's unintuitive that the two methods don't lead to the same answer. There's no logical contradiction in the situation, it's just a bit surprising, so it's a weaker kind of paradox.
@@benjaminhill6171 Coming from a background of philosophical logic, I've never liked the concept of "weaker forms of paradox", but I do accept that's a definition that is commonly used. My problem here is that I don't think this case even fits that weaker type of definition. Perhaps others think differently, but it isn't unintuitive to me. A jagged line between two points is always longer than a straight line between those same points, and the limit of a jagged line is still a jagged line.
@@DylanSargesson Ah, but that's where you actually *don't* understand. The limit of the sequence of jagged paths-well, they're technically called curves-is not a jagged curve itself; it is really, truly the actual diagonal line segment. This can be shown using the formal definition of a limit. If you're familiar with the definition of the limit L of a sequence of numbers, that states that for every choice of ε > 0, you can eventually get far enough in the sequence that no number in the sequence ever gets more than a distance of ε away from L ever again. We can do a similar thing with a the limit L of a sequence of curves, where whatever number you choose for ε > 0, I can eventually get to a part of the sequence where from here on out, the curves deviate from L by no more than a distance of ε. This is indeed the case for our staircase sequence.
@@isavenewspapers8890 However, the sequence of lengths of these curves converges to (and just always is) 2. The length of the approximating curve is 2 at every step. What I'm saying is that, ultimately, even though by your definition of convergence the jagged edge curve does converge to a diagonal, its length clearly doesn't converge to the length of the diagonal. I guess to me that's the real paradox. I hadn't thought of it in that way before, so that's interesting.
It's a paradox because it disproves that you can take lengths by bounding curves to be close to the original curve, which one might naively assume if you didn't see this paradox. All paradoxes are exactly that: something that disproves something one might naively assume (for example, Russel's paradox disproves that you are allowed to form sets using unrestricted comprehension, and so on).
These are indeed paradoxes-specifically veridical paradoxes, things that are true but sound false. However, they are not fallacies, as that implies that they are false.
Bro the first one isnt even a paradox i mean look how big earth is and then you lift it by one meter your animation is the only confusing part about it good try man but that video is an f.
Diagram not to scale, obviously. Did you want a to-scale version where the change wasn't even visible? That doesn't make much sense. And is that the only thing affecting your judgement of the video?
It doesn’t disprove it. Ordinarily transformations preserve volume, but the loophole that makes the theorem work is that this doesn’t hold if your pieces aren’t measurable by volume. So you can’t go from 1 to 2, but you CAN go from 1 to N/A to 2
@@elementgermanium Makes sense. But then it calls into question the validity of the proof. A point and a line are "breathless" things in the first place. If the proof works why does it depend on the axiom of choice? Shouldn't it work without it?
@@benjaminhill6171 True, but i mean that if such an axiom gives us something impossible, then its clearly not compatible with anything the deals with the real world. ie: its a bad axiom.
Let me know if there's a topic you'd like me to cover next. :)
9:38 WRONG VIDEO
lmao i was just about to comment that
"... i don't fucking care what happens when you turn a figure 8 into a fucking circle, i don't give a flying fuck about avoiding sharp bends, why are you avoiding me? ..."
REAL ONE APPEARS TO BE THIS: th-cam.com/video/OI-To1eUtuU/w-d-xo.html
LOL I had to check too I swear this happens to this guy all the time
i was about to say that lol
I like that the screenshot is from the Hugbees video
It had to have been an accident lmao
Sorry, which video?
@@VermillionPengu Huggbees parody "turning a sphere outside in"
He also kinda sounds like huggbees
I have a conspiracy theory that he is hugbees, they sound almost identical
If i got a nickel for every time someone mistook a hugsbee video for an educational video, i would have 2 nickels, which isn't a lot but it's surprising it happened twice.
But hey! Now you are in the same level ad cnn
Still watching Vsauce BanachTarski video from time to time, and still doesnt understand it fully
Every shape has an infinite number of points in it. Because of the Hilbert's Hotel paradox, if you take an infinitely small chunk out of a shape, there will always be another infinitely small chunk that replaces it, a chunk that replaces that chunk, a chunk that replaces that chunk... you get the idea. As a result, you can keep taking infinitely small chunks out of that shape and create an identical copy of that shape... or an infinite number of them.
Me too. I just love this vid
YoFeArIr
I guess the word "infinity" means nothing because you can always divide it further and further. Real life objects dont work like that. You're trying to apply common, real sense to something which only makes sense as a concept in the minds of eggheads.
@thelibyanplzcomeback youre changing how you handle infinities halfway through, the perfect example of taking infinitely small pieces out infinite times is a tank of water
make a hole in a tank of water, each infinitely small amount of time an infinitely small amount of water leaves, but if you integrate you realise it doesnt duplicate
9:39 Bro put the Huggbees version 😂 iykyk
I KNOW
@@Achieengo1209bro knows
How is the first one a paradox?
its a statement that sounds false but surprisingly is true. yes, thats a type of paradox
@@anaveragekiwi why does it sound false?
@@Galinaceo0Only because it's surprising, because most people expect it to be a lot more than 6m that you have to add.
@@Galinaceo0its very counter intuitive that you only need 6.28 metres added to the circumference to change the radius of the entire earth by a metre. i mean, it makes perfect sense when you think about it and enough exposure to it makes it much more intuitive, but at first hearing it sounds obviously false
Paradox doesn't just mean self-contradiction. It also means counterintuitive fact.
Did you know that Banach-Tarski is an anagram of Banach-Tarski Banach-Tarski
Did you know that Google's translation of "Did you know that Banach-Tarski is an anagram of Banach-Tarski Banach-Tarski" into English is "Did you know that Banach-Tarski is an anagram of Banach-Tarski Banach-Tarski"?
good one
7:58 usually when we talk about spaces like S^2 or R^3 we pronounce them as “S Two” or “R Three”, also the R should be blackboard bold since it denotes the real numbers
According to the Banach-Tarski paradox there shouldn't be such thing as people with one ball.
I have only 3😢
Just a comment regarding your explanation of the coin rotating around the other coin. Your explanation kind of gives the feeling, that the outer coin always has to double the amount of spins, when in reality it is the size ratio + 1 for the Rotation around itself. So for equaliy sized coins the size ratio is 1 and therefore the outer coin travels 1+1=2 times the circumfarance of the inner coin. However, if the outer ball has only a radius of 1/3, it spins 3+1 = 4 times around Insel and not 6 times.
What part of the explanation gives that feeling?
@@isavenewspapers88903:41
@@isavenewspapers8890Probably the part at 3:41
@@TH-cam_username_not_found That just says that coin B traveled twice as far and therefore rolled twice as much. This logic would still apply if coin B had a radius of only 1/3 cm. In that case, if you rolled coin B across a flat surface, it would rotate by 1 turn after traveling (1/3)τ cm. Rolling it around coin A causes it to travel (4/3)τ cm, which is 4 times as far. Therefore, coin B rotates by 4 turns in this scenario.
@@isavenewspapers8890 It feels like the distance will always double rather than it will always increase by 1 unit. Do you get what I mean?
3:48 also, the coin thing is so much more simple than it’s let on to be; they roll “against” each other at the same rate for the same distance, but since coin A doesn’t move, halfway through coin B’s journey the connection point is halfway around the coin, thus tada 🎉 it’s right side up because it’s connected under coin A rather than on top. Not all that math
The string girdling paradox confused most of my friends and needed to solve it mathematically 😭
😭😭😭😭😭yes cry some more😭😭😭😭for your understanding of what an actual paradox is, equals your General iq in well defined logic that is Just counterintuitive to incompenent idiots like you 😭😭😭😭😭
when I was young, I joked to friend that I could prove that 1 + 1 = 1 using the staircase limit. (I already knew that was silly and impossible, but still funny)
I don't know if you see this but I want you to know - keep making these types of videos of geometry - - -
Also try to do a video on non euclieadian geometrical 3d planes
Your videos are so good, thanks for putting the vector calculus. I was able to follow it
I love the idea that the function which, to an arc (basically any line), associate its length, is not continuous !
9:38 definitely the correct videos everyone should learn from huggbees :) (funnily i saw the original before the huggbees one)
2:10 the easiest way to visualize this is to look at the top of the coins head orientation. At the top it is facing up, the it is facing up again at the bottom, then back to it originally at the top
i have the power to take this video from 999-1000 likes and i’m abusing that
On the coins one: Huh? It isn’t rotating twice. That’s just an illusion because at the halfway point the surface upon which it‘s rolling is fully inverted. If you unwrapped it to flat at that moment, the rolling coin would be upside-down. It’s rolling the same distance in both orientations.
Upvoted for using tau.
that’s funny because I was like why are we using this just do pi*d lol
i think the first paradox feels weird at first because we see the visual and think about the area, instead of the circumference
I think the easiest way to mentally disprove the staircase paradox is to imagine that instead of the point off of the line being a right angle it's literally any other angle following the rest of the same rules. Under that restriction it's possible to make the limit of the length equal literally any value greater than ~1.4, which if you accept that then length has no meaning or it's a faulty measurement. It looked like one of the other proof diagrams you flashed up may have being along a similar thought process.
Why would "2 = √2" not be a problem to you, but then you draw the line at "length makes no sense"?
@@isavenewspapers8890 I don't know how you even came to that conclusion from what I said.
@@TheBalthassar The proof you reference appears to be a proof by contradiction: you begin with a set of assumptions, prove that those assumptions lead to an absurd result, and conclude that at least one of the assumptions must be incorrect. But there was already such a proof that was all but stated in the video, which I will write explicitly here:
"Assume that the limit of the lengths of the staircases is equal to the length of the limit of the staircases. The limit of the lengths of the staircases is 2. The limit of the staircases is the diagonal line segment, whose length is √2. But 2 is not equal to √2, so this is a contradiction. Therefore, the limit of the lengths of the staircases cannot be equal to the length of the limit of the staircases."
So, did you just not notice that one?
@@isavenewspapers8890 You appear to be under the mistaken impression that my point is disregarding that not building upon it. You're arguing with a straw man.
@@TheBalthassar Your claim was that your proof is the easiest way to do it. However, it involves a bit of extra work that can be entirely skipped to produce the same conclusion. I don't see how that's easier than the other way.
that has to be the least informative explanation of banach-tarski i've ever seen
Hey, I know you. You're the one who was stirring up trouble in the comments of the video adaptation of The Tau Manifesto. Not sure if you're a troll or what, but it's funny running into you again.
But yeah, a comprehensive explanation of the Banach-Tarski paradox doesn't work well with the 2-minute-per-section format. I feel like you'd need 10 minutes, bare minimum, even assuming the viewer already has some basic experience with set theory.
@@isavenewspapers8890 i was certainly not "stirring up trouble." tau evangelism is not mathematical or self-consistent, it's more like people preferring 432hz tuning to 440hz because the numbers are more "pure"
there are instances where tau is a more appropriate constant and instances where pi makes more sense. tau-absolutists who pretend to have forgotten about the existence of pi because they've reprogrammed their brains are closer in nature to cult leaders than mathematicians
Yt "deleted" the response lol
@@erner_wisal hes an idiot
"Given any two reasonable solids, either one can be chopped up and rearranged into the other."
But what if the two have different Dehn invariants?
The Dehn invariant arises as part of a problem involving finitely many straight cuts of a solid. Such restrictions do not apply in the case of Banach-Tarski.
@@isavenewspapers8890 Oh, that makes sense.
I vividly remember watching the 'outside in' video so many times trying to imagine what happened. It reminds me that I wish I could select some faces of the original sphere to watch how the sphere everted patch-wise. Maybe someone build that as a 3D web app now, which wasn't possible at the time of course.
In the first problem, it would be 2 meters bigger, because it is being lifted on both sides
Pardon my calculus but FUKKIN SUBSCRIBED
The circle one I know the other way around. You extend the circumference with one meter of string and the question is, if a mouse would fit under the rope. In this case I remember the radius to raise about 16cm which would allow some mice stacked under the rope.
Banach is pronunced banahh, not barrack
Well, that second one is also not how the narrator pronounced it.
@@isavenewspapers8890 he said it twice, first it was "banack", second was "barrack"
As a tau appreciater, thank you for using tau instead of pi in the first few examples :)
Newbie here, can u give some context? sorry for being dum
@@vibbruh tau is equal to 2*pi. So any equation where you use pi, you could use tau instead. For example, the circumference of a circle is 2*pi*r, or you can say it's tau*r. Some people argue over which is better: pi or tau.
Pi tastes better at least
I'm not a math person but I enjoyed this video!
There's a great Vsauce video from years ago that goes over Banach-Tarski and the seeming paradoxes of infinity involved in some detail. I'd highly recommend it.
8:20 oh no
Why tf did you use tau
WRONG OUTSIDE IN DONT WATCH THAT ONE
DO WATCH THAT ONE, IT'S HILARIOUS
Lovely profile picture
/genuine
PS. Please don't change it so that it makes me look bad. :)
I can't believe how amazingly good you are.
1:48
c a r
8:15 Hey look i found a material that can go through itself *accidentally folds it*
Why I always watch these at night
since the point on the circumference of the coin traces a cardioid shape, i wonder if it's related to the mandelbrot fractal in some way
The first one isn’t just interesting because you only have to add 6.28 meters. It’s interesting because it’s entirely independent of the initial radius.
So whether you’re talking about a 1m radius or Earth or the entire universe, you always have to add only 2pi meters to make the radius 1 meter bigger.
That's exactly what the video says at 1:38, so I don't know why you're just repeating it.
where is the paradox in the first one?
As soon as I heard tau I subbed lol
I understand that the math is correct but still cant wrap my head around the String girdling Earth. It's very counterintuitive that for the whole Earth it's just some meters.
If you dont find it intuitive think of a simpler shape like a square. If you strap a rope around it and do the same you wanted to for the earth you could do it like this: step1: cut the rope at every corner
Step 2 place all 4 ropes 1 meter from the side of the square the rope was touching
Step 3 notice the extra length needed. Its all at the corners, and this isnt alot at all, and it doesnt depend on how big the initial square is. Now expand this idea in your head to a circle.
It’s because the equation is just tau times the radius. It’s a linear relationship- adding 1 to the radius ALWAYS adds tau to the circumference.
If you want the radius to affect the amount you add in the way you’re thinking, you would need an r^2 term
It's only a paradox when you apply it to something large. If you picture a very small object with a string around it, like a 1 cm sphere, it becomes much more apparent that increasing the radius by 1 meter can't be related to the diameter of the original object.
It’s because if you calculate the length of string to add, it’s independent of the original radius. And yes, it is indeed strange that adding 6.28 m to a rope
around the universe has the same effect as adding 6.28 meters to a rope around a basketball.
I'm sad that they didn't flip tau and pi, since tau looks like half a pi, or rather pi looks like two taus next to each other.
A redefinition of the mathematical symbol π would be... unpleasant, to put it mildly. We'd have to destroy all the old records and write new ones, or else deal with ambiguity as to which number we're using when we write "π".
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Did you use the wrong sphere inversion video on purpose, you memester? XD
@@arcturuslight_ *
monster
how is the coin one a paradox?
I wish there was an intuitive way to understand the Banach-Tarski paradox.
That’s hard because it’s so counter-intuitive at its core.
Vsauce made a quite good video about it with a relatively intuitive explanation. I don't think you'll ever find a more intuitive explanation than theirs, at least til now.
this video is sponsored by adblock i totally understand why people use adblock i pay for youtube premium but still have to listen to sponsorship ads i really understand people now and weird youtube doesn't
I don't think the staircase works. If you always halve it, you'll never cover irrational points, which is almost every point on the diagonal. So the limit of the staircase is NOT the diagonal.
That's not how limits work. The sequence doesn't actually have to reach the point in question; it just has to approach it. For example, the function sin(x) / x never attains a value of 1, but its value approaches 1 as x approaches 0, so the limit of the function as x approaches 0 is 1.
Hmm 3:08 in, and none of this is paradoxical yet
I've watched outside in too many times 🙃
6.28... m 0:29
im here to leave a comment before you are famous
If there is no contradiction is it even a paradox?
No, that’s not what a paradox is. A paradox is something that cannot be proven or disproven like the grandfather paradox or the boots paradox. You will never find a contradiction in traditional versions of those two.
@@Bdcrock what I meant is that in attempt to solve a paradox often a contradiction arises. What the video contains are not paradoxes, the examples are just math problems that have an answer that is not intuitive at first.
@@titastotas1416 yes and you are correct the difference between what you’re saying and what I’m saying however is that all paradoxes have contradictions but that is just not true
@@Bdcrock listen up, I never stated that all paradoxes have contradictions, in fact I cant think of a paradox that has a contradiction in its formulation. What I meant and I think I have stated it clearly enough already is that in attempts to solve a paradox one will be faced by a contradiction and that is always true ,If you don't agree with that give me a paradox that does not result in a contradiction when an attempt to solve is made. We are not in disagreement, I agree with the definition of paradox you have provided previously. In fact I don't see what your issue is.
@@titastotas1416 oh i misunderstood i thought you meant the paridox itself is a contradiction
Paradox now means unintuitive i guess
Thats exactly what it means
8:30 EASY!!! you just have to kiss your sister
Only the last one was a paradox, the others were just ways that intuition doesn't akways line up with math.Especially the first one.
1) Counterintuitive facts can be referred to as paradoxes. This is a well-established usage of the term, and most people understand it. Stop fighting language.
2) How does the Banach-Tarski paradox not fall into the same category as the other four? To me, it just sounds like you're saying, "That's the only one I don't understand, therefore it can't be true."
90% counterintuitive logic, but was funi
The term "paradox" can refer to a counterintuitive fact.
You failed to mention the “paradox” part of the coin one at all
It's only called a paradox because the result is unexpected. There's no logical contradiction in the situation.
the paradox part comes in when the coin rotates seemingly a different amount of times depending on where the focus is, this was not mentioned, also it obviously goes around 2 times idk how that could be unexpected
@@temmie5764 At least for me, my intuition says that since the two coins are touching, the revolving coin's edge will go exactly one circumference-distance. I know that's wrong, it's just that that's what my intuition says. Considering this situation is a common one to cite for unintuitive behavior (and an entire group of SAT questions creators got it wrong), obviously many people have intuition similar to what I described. Good on you for having a better intuition.
the thing is, it does only go around once, but it also goes around twice, it just depends on the observer, thats the paradoxical part that isnt mentioned
I thought the video was a bit difficult to understand.
A lot of these comments are insufferable tho, good grief.
I think it was a foot
I was wrong
You thought what was a foot?
@@isavenewspapers8890 I honestly don't know
Wow chat I’m 1000 view
Bro TAU!? This gotta be bait 😂
No?
I don’t know if this will help but for the last paradox, you could just think of it if one ball has infinite points and he cut those infinite points in half both halves will still be infinite so they can both be reconstructedreconstructed into two separate balls
The fact I understood and already knew about all of these proves how way to nerdy I am💀💀💀
*way too nerdy
@@SweetRollTheif **squints** thats a typo
I mean. With a healthy dose of vsauce and… apparently huggbees? Im pretty sure ive become omnipotent
@@simpli_A ahhhhhhhhhhh AHHHHHHH *AHHHHHHHHHHHHH* OMNIPOTENTENCE HAS BEEN ACHIEVED
Can you simplify Banach tarsky
huggbees :)
klein bottle
A lot of these aren't paradoxes they're just basic math
Paradoxes have 3 types.
I love that you're making these beautiful concepts accessible to a general public!
a
Tau :(
Why?
Why?
Why distract from the already nice rope trick?
I can't really tell why you said "why" three times.
Anyway, how is it a distraction? It's directly relevant to the math at hand, so I don't know what you're talking about.
@@isavenewspapers8890 Relax. I said "why" pi times :)
@@bubblecast That makes no sense.
@@bubblecast Also, you didn't answer my question.
@@isavenewspapers8890 my bad, shouldn't have expected you to grok it
First from morocco😙😙
I struggle to see how the staircase one is a paradox. It's obvious that a jagged line is longer than a straight line (no matter how small those jags are).
It's only called a paradox because to some people it's unintuitive that the two methods don't lead to the same answer. There's no logical contradiction in the situation, it's just a bit surprising, so it's a weaker kind of paradox.
@@benjaminhill6171 Coming from a background of philosophical logic, I've never liked the concept of "weaker forms of paradox", but I do accept that's a definition that is commonly used.
My problem here is that I don't think this case even fits that weaker type of definition. Perhaps others think differently, but it isn't unintuitive to me. A jagged line between two points is always longer than a straight line between those same points, and the limit of a jagged line is still a jagged line.
@@DylanSargesson Ah, but that's where you actually *don't* understand. The limit of the sequence of jagged paths-well, they're technically called curves-is not a jagged curve itself; it is really, truly the actual diagonal line segment. This can be shown using the formal definition of a limit.
If you're familiar with the definition of the limit L of a sequence of numbers, that states that for every choice of ε > 0, you can eventually get far enough in the sequence that no number in the sequence ever gets more than a distance of ε away from L ever again. We can do a similar thing with a the limit L of a sequence of curves, where whatever number you choose for ε > 0, I can eventually get to a part of the sequence where from here on out, the curves deviate from L by no more than a distance of ε. This is indeed the case for our staircase sequence.
@@isavenewspapers8890 However, the sequence of lengths of these curves converges to (and just always is) 2. The length of the approximating curve is 2 at every step. What I'm saying is that, ultimately, even though by your definition of convergence the jagged edge curve does converge to a diagonal, its length clearly doesn't converge to the length of the diagonal. I guess to me that's the real paradox. I hadn't thought of it in that way before, so that's interesting.
It's a paradox because it disproves that you can take lengths by bounding curves to be close to the original curve, which one might naively assume if you didn't see this paradox. All paradoxes are exactly that: something that disproves something one might naively assume (for example, Russel's paradox disproves that you are allowed to form sets using unrestricted comprehension, and so on).
These aren’t paradoxes, they’re mathematical fallacies
These are indeed paradoxes-specifically veridical paradoxes, things that are true but sound false. However, they are not fallacies, as that implies that they are false.
First one is very stupid
Bro the first one isnt even a paradox i mean look how big earth is and then you lift it by one meter your animation is the only confusing part about it good try man but that video is an f.
Diagram not to scale, obviously. Did you want a to-scale version where the change wasn't even visible? That doesn't make much sense. And is that the only thing affecting your judgement of the video?
Why is the last one not called the "disproof of the axiom of choice"?
Because it doesn't disprove it?
It doesn’t disprove it. Ordinarily transformations preserve volume, but the loophole that makes the theorem work is that this doesn’t hold if your pieces aren’t measurable by volume.
So you can’t go from 1 to 2, but you CAN go from 1 to N/A to 2
Also, you simply can't disprove an axiom. 😂
@@elementgermanium Makes sense. But then it calls into question the validity of the proof. A point and a line are "breathless" things in the first place. If the proof works why does it depend on the axiom of choice? Shouldn't it work without it?
@@benjaminhill6171 True, but i mean that if such an axiom gives us something impossible, then its clearly not compatible with anything the deals with the real world.
ie: its a bad axiom.
How is the first one a paradox??