The Painter's Paradox - These Weird Objects Will Blow Your Mind
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- čas přidán 2. 05. 2024
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6:02 What is the length of this line?
About 3 and a half things
10 and a half
10 2/3 apples square
1 hour is not longer than 2 yards
The length is such that light takes x amount of time to cross from one end to the other :)
"Imagine a cow that isn't perfectly spherical" Physicists: What is this? biology?
* AHHAHAHAGA
Well, definitely not topology.
This was never said in this video. You probably commented under the wrong one?
@@PeterDercsar but your comment isn't said in the video either
@@Censeo nice one
So if it takes forever for a single note to leave Gabriel’s Horn, should we conclude that Judgment Day will never come?
How would anyone (other than Gabriel) know that the Horn had been blown?
Does the infinite surface area of Gabriel horn implies the horn's infinite length?
@@MarcelinoDeseo I mean, the integration to infinity kinda does..
@@stephenrichards5860 More importantly, how does Gabriel even hold it?
@@vigilantcosmicpenguin8721 since the horn and Gabriel are improbable, why do you care
That comparison of units (time vs. length) was a really effective and clear example- a great 'aha!' moment when it was applied back to the original problem.
I disagree because time vs. length are clearly different measurements that have no correlation whatsoever. Surface vs volume have some correlation, for example, they are both used to measure space, different aspects of space, but it's not the same difference to compare time and space. Another correlation is that one could say that both space measurements (surface and volume) use the same principle calculating a 2D area, one multiplies it for how many faces, the other imagines a stacked version of the shape and multiplies for its height.
And time and space are the same kind of dimension in spacetime that you can rotate into each other. So you could ask how much yards is an hour. How much meters is a second?
I was having a chat with a friendly hypercube the other day, and she assured me that time and length are compatible--time can be measured in centimeters. Frankly, I was skeptical, until the hypercube pointed out that a square, living in a 2D world experiences time in exactly the same way that we create cartoons or motion pictures. The square was able to run 10 meters in about 5 seconds, which to me appeared to be about 1 cm worth of "frames" so the square could run 2 meters per second, or 10 meters per cm (measured along the 3rd dimension, i.e., time). The hypercube told me I couldn't see it, but when she watches me for 10 seconds, she measures 2 meters along the 4D axis, and tells me that time is 5 seconds per meter. I couldn't argue with this, even after spending 1200 km thinking about it.
Taking special relativity, wouldn't 1 second be equal to 299,792,458 meters (a light-second)?
DMT is one hell ova drug
Holy crap now go run through a gravitational field. Try to metric tensor your way through that one. See you in a lightyear
just told the new hypercube hire to paint Gabriels horn lmao. next I’m going to tell him to make a 3-D model of the Klein bottle. He’ll never suspect a thing
I like how you showed the paradox doesn't exist in the physical world, because of the minimal thickness a layer of paint must have. Even numberphile failed to explain that.
Even if the paint had no thickness, an infinitely long object could not be physically created to paint in the first place.
@@BlueSapphyre of course. But I mean the paradox also holds without involving infinity. One can build a horn that is so big that it's surface is arbitrarily huge (say 1000 km square) while its content is 1 ml. That still is paradoxal. No ? But even in that case, there's a solution to the paradox : the molecules of paint won't be able to reach the bottom of the horn.
@@leroidlaglisse No, how would that be paradoxical? Just unusual.
@@leroidlaglisse i feel like this aspect not mentioned enough: yes you cant paint it, due to infinity. But as you say also cant fill it, due to it approaching an infinitessimal width.
Infinity and infinitessimal kind of balance out.
Assuming paint has discrete elements.
If paint is continuous, that atoms, or planck length dont exist. Maybe its made of black holes....just divide it by zero or whatever you need to make it cover an infinite surface and fit inside the infinitessimal neck of the horn
But that said- its no more paradoxical than...numbers themselves.
Taking something like an inverse exponential curve it does the same...numbers go on infinitely,
So basically "what happens if one dimensions approaches infinity and the other approaches zero" if you drew it on a graph we are all used to that from grade 10 math class
@@simowilliams6990 you are perfectly right. The word "paradox" has several meanings. I was using the weak definition : "seemingly contradictory". Which is the same definition we use for the classical Gabriel's Horn paradox. It is formally not a true paradox, as Jade brilliantly explains in the video. It's just an apparent paradox, for us mere mortals. :)
"the paradox lies entirely in our interpretation" no sentence has been so true 👏🏻 (it's also the favourite quote of my astrophysics professor)
it's a lot like zeno's paradoxi in that way
@@QuantumFluxable yeah, exactly! Every paradox relies on a interpretation (or a model, if you prefer).
Or in other words, paradoxes are not false or nonsense, they are just limits of our interpretation/model
Does this apply to the dual nature of light?
@@cosminstanescu1469 yeess, but not really. Light (and every quantum particle) is always a wave, but in some experiment the "waveness" is not evident and it seems it acts like a non-quantum particle
*black hole complimentary has entered the chat
I challenged this problem in my Cal 2 class: The interior volume is finite, therefore the interior can be painted, since paint is a 3 dimensional substance. Such a painted horn shape will reach a point where the paint thickness is greater than the half the radius, and therefore that section on is equivalent to the filled volume. Furthermore, the horn will reach a small size where it cannot contain paint molecules (regardless the scale).
I appreciate the purpose of the problem, but it's literally putting the horse before the cart. Someone discovers something interesting, but has to put the interesting-ness into terms that ordinary people (even other mathematicians) can appreciate, often obscuring the original point or creating pseudo-context for the observation.
Richard Feynman had a story about feuding with mathematicians, where shortly after the discovery of the Banach-Tarski paradox, a group of math students claimed they could duplicate a sphere and someone suggested "an orange" as the model. The math students began explaining the theory, and Feynman stopped them, protesting that an orange was not a continuous object like a pure sphere, that it's made of atoms and the analogy falls apart.
Love your content, great video, just this specific thought experiment bothers me for being a poster-child of "see, math can be interesting!"
Keep up the good work!
If the paint is a 3 dimensional substance then you cannot paint the outside of the entire horn either. Eventually the paint particles will repel each other enough and the horn will go between the particles. One side of the horn might be touching paint, but not the rest of the surface in the same spot. Assuming the paint is infinitely thin on the outside is the same as reducing the size of the paint particles on the inside for smaller cubes, you either do both or neither for consistency.
This is brilliant! I recently rewatched a Physics Girl video on Mirrors and reflection which made a similar point to yours: "the paradox lies entirely in our interpretation". In the "Reflection" video, the intuitive interpretation that most of us apply doesn't account for (we don't realise) the fact that there's a perspective shift that happens. We 'miss'/erase/skip over this key event and then interpret the reflection in 'everyday', 'obvious', intuitive terms based on the fact that we're used to seeing other people facing us.
Our natural intuition or biases blind us and it takes something special to step outside of these or to realise that these might be what's causing the problems. You've broken this example down wonderfully ...
I watched a video about Gabriel's horn from a well known channel and I didn't understand it, but you've explained it so well that even this maths fool got it!
Which channel?
@@redunleasher2147 Probably Numberphile?
You mean Numberphile?
But that was also intuitive too😥😥
Oh wow. Nice to see Medlife crisis here!
@@derickd6150 He comments on almost all her video's
I love teaching this in my calculus classes, and although I can show the mathematics with no problem I am always looking for good ways to explain the paradoxical part in nonmathematical terms. I have pointed out before that surface area and volume are not comparable because they are different dimensions, but I think your analogy of comparing time and length is very illustrative. I'm going to use that in the future.
The way I look at this is: Say you take one one-foot cube with negligible wall thickness and place a second cube within that cube that is half of the outermost cube’s size. You can continue to add cubes that are larger than all inner cubes and yet still smaller than the outermost cube. Essentially, any 3D volume has an infinite amount of 2D space inside of it
Hm if the cubes are permeable and you fill the outermost one with paint, then all surfaces would've been painted too. I guess that makes sense since the paint itself will have infinite 2D space inside of it too. Your analogy really drove it home for me
@@truevelvett Thanks! That’s kind of how it clicked for me too
This
"What is this!? Physics 😏 " Physics explains our universe, mathematics describes all possible universes is how i usually put it. 😂
Physics explains what's possible, maths constrain what's imaginable
@@alexv3357 maths is just philosophy on a higher difficulty setting
@@alexv3357 I'm stealing this
maths are simply analysis tools in the world of physics. Math models are constructed to model physical models so that stuff can be predicted(interpolated/extrapolated based on observation) given a set of variables/initial conditions. Those models can even be machine learned with lots and lots of variables fitted to construct the mathematical model.
@@alexv3357 requesting permission to use your statement incase I ever get into a maths vs physics discussion.
Either we use paint that has a particular volume (p1), or we use paint that does not have a volume - only a surface area (p2). If we try to paint Gabriel's horn with p2, it will take forever. But it will also take an infinite amount to fill the volume of the horn with p2, since it does not have volume. Likewise, if we use p1 to paint the surface area of the horn, there will be a point where we will "clog" up the horn with paint, meaning that p1 can only reach a finite amount of the horn. Hence, both filling and painting the horn with p1 takes a finite amount of time.
What if the thickness of paint goes down the deeper you go into the horn, but it is never zero? :)
@@saggitt Imagine first drawing the function f(x)=1/x to get the initial formula for the horn of Gabriel, then another function h(x)=.99/x to represent the remaining volume after the surface has been coated with paint. Rotate the two functions around the x-axis, subtract the volume of h(x) from the volume of f(x), and you should still be left with a finite volume, since the volume of f(x) is pi and the volume of h(x) is slightly less than pi. Hence, if the paint has any volume whatsoever, it will still require only a finite amount of paint to coat the surface of an infinitely large surface.
You can paint the infinite amount of cubes in a finite time, represented by t:
Paint the first cube in time t/2, paint the second cube in time t/4, paint the third cube in time t/8; In general, paint the nth cube in time t/(2^n)
The time required would be t/2 + t/4 + t/8 + t/16 + t/32 + t/64... which equals t, not infinity.
@@ValkyRiver Why are you assuming that the time it takes to paint the surface of one cube is equal to half the time it took to paint the previous one? The size decreases by 1/n, not 1/2. If we assume the time to be proportional to the size, then the time it takes to paint a given cube n should hence also be a divergent series, like T = t/2 + t/3 + t/4 + t/5+... Thus, the total time T would also be infinite.
@@TheBoxysolution Vsauce explains it here:
m.czcams.com/video/ffUnNaQTfZE/video.html
I don’t know how you don’t have more views. You keep me interested in these concepts that would put me to sleep if it was someone else teaching it.
What you said at 6:29 made me happier than it should have xD I do a ton of DIY projects, and a lot of my measurements are difficult to describe. I rarely have a rule or tape measure on hand for example, but I also rarely need a specific length. Rather I just need all the pieces to be the same length, whatever that happens to be. So I'll use what ever is near me that I can grab. So many of the people I know have always been so surprised that I do this and that it works so well. Exciting to see this explained.
The asterisk at 1:42 and the quote at 4:32 were priceless! *XD*
thanks for pointing it out, hilarious indeed, hadnt seen it impressed as I was by the mindblowing beauty of this principle.
wow! "That made sense"
I was with this question in mind after seeing a video talking about how it's impossible to really tell the perimeter of countries. In a nutshell, it depends how close you measure, just like the fractal you showed.
Thank you so much for this video, it's so clarifying
about painting Gabriel's Horn, I think I have come up with some good ways to think about it (or "solutions" to the "paradox")
here are the different scenarios/interpretations:
1. paint can be spread infinitely thin - if this is true, then you would indeed be able to coat the entirety of the horn, since surface area and volume are both uncountably infinite (although since the paint could be spread infinitely thin, no volume of paint would be consumed anyways)
2. paint on an object has a thickness - if this is true, there will eventually be a point in Gabriel's horn, no matter how large the horn is, where the paint on "opposite sides" (directly across the center axis at that depth) of the horn will intersect, thus making the rest of the horn (which has infinite surface area) just being filled with paint (finite volume) instead of being "painted" in the traditional sense
3. surfaces "soak up" paint (there is a requirement for the volume of paint used to coat the surface; the surface soaks up the paint without increasing in thickness) when they are coated - if this is true, then you will never be able to fully coat the horn, since all of your paint will be soaked up by the infinite surface area of the "bottom" (the tip) of the horn
Thank you so much! This is a really cool way of talking about things that normally need calculus, but without it! I'm really excited to show this to my middle school and high school students! (And by show this I mean actually do some math with it - perfect for our chapter on sequences and series!)
"What is this, physics!?" - Up and Atom 2021
Also, "to oppugn", didn't know that word existed :)
(watched it on Nebula first, but you can't comment there can you?)
not yet!
@@upandatom *Who else has no clue what she's talking about, but still enjoy watching her & listening to her accent?*
Literally yesterday was doing the Calculus in a nutshell course on brilliant and I was wondering about the exact same thing XD
I've considered getting brilliant, but I have a sort of innate aversion to getting anything from a commercial. Lol. Is it actually good, or just hype?
I love following youtubers who have a clear passion for the things they are talking about! It opened so many new areas for me that I was previously not really interested in but can clearly see why someone is so passionate about. That put me to some strange places already, like classic black and white horror movies (by following the avgn) and some strange sports and such...
Wikipedia explained it shorter: "The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area - it simply needs to get thinner at a fast enough rate".
Yes, and it needs to get thinner and thinner to fit inside the smaller and smaller cubes or sections of the horn.
I propose that infinitely thin paint lacks volume, and would then be unable to fill a can or cube.
@@joet3935 infinitely thin paint over an infinite area may in fact have a concrete value of volume though.
@@joet3935 interesting.
@@shlusiak Thats like folding a 2D plane to fill a cube. How many shadows do you have to stack to make a volume?
Thank you! Out of the several videos I've seen on this topic you are the only one to have explained it correctly. That it isn't a paradox and that you can't compare area and volume like that. Bravo!
My favorite thing about this "problem", as you pointed out, is that you get different answers based on your assumptions. If you are assuming real paint on some sort of real object and you ignore the glaring problem of an infinitely long object actually existing, you could never paint it. Of course you couldn't fill it either since it would take an infinite amount time to fill. If you use mathematical (0 volume) paint then you can both fill and paint it, assuming you magically poof the paint in since you still have the issue of the time it takes to fill.
Again Thank You!
Mathematics overtakes/overwhelms Physics at the Planck Length.
As always, excellent! 😊
Yep! Infinity is nice and all, but physics says everything is finite if you get small enough :P
@@IceMetalPunk incorrect !
Physics says, everything is quantifiable, except for those that are not. ;)
@@IceMetalPunk Then why most physicists refuse to acknowledge fields in the general theory of relativity are actually discrete, only the result is that particles are continuum of probabilities. (they insist its the other way around).
The Math obviously works both ways, but as a computing scientist, thinking that the Universe is discrete makes more sense, and that continuous analysis is just an useful tool, not the reality itself (at least its more intuitive to me), its not like things are actually infinite and we can have infinite energy in this Universe.
@@IceMetalPunk well black holes are a example to it since its volume is infinite however is surface isnt i guess?
This goes perfectly well with the videos explaining how all infinites are not equal, and convergences! Shoot your shot and do a collab with Veritasium.
No
there are 2 interpretations I have about 2 different scenarios:
If we consider the paint to have a thickness then as you just said, filling a transparent shape with paint doesn't make it look painted from outside.
If we consider the paint to be infinitely thin then any positive volume of paint would be able to paint an infinite surface area.
I'm pretty happy with myself, after about 20 seconds I thought out this entire episode. The only concept I missed was filling that objects volume to coat the surface area at the same time. Interesting episode
”What is this! Physics?”
Great quote ;)
I thought I knew all about this paradox but you just proved me wrong!
Physics? ... Now, a practical introduction to Dimensional Analysis.
I absolutely love this video, Jade! Whenever I thought about “hmmm what about this?”, you showed an animation depicting it and gave a nice explanation. Keep up the fantastic work!
To solve the paradox: You can't fill the horn with paint and therefore not paint the inside of the horn. Why? Because filling the horn without spilling paint would take an infinite amount of time. You fill the horn from one side and the paint is running down the horn until it reaches the bottom. Only when the paint has run down, you can pour in some more paint. But since the horn is infinitely long, it never can be filled with paint and you end up with a situation that you can add less and less paint, since the level to lower takes longer and longer, and you' would never be able to let the final drop of paint into the horn without it to overflow.
I love the admission that while the metric system is great for doing something we almost never do, converting amongst units, it can be unhelpful doing things we do every day such as conveying information. A short person is 1 m and change, but a very tall person is 1 m and change. Once we have the specifics, we will have a really good idea how many kilometers tall they are. But that also does not matter.
Does this make any sense to someone?
Of cause the metric system is useful to convey information.
Seen so many versions of this explanation I almost didn’t watch - so glad I did!!! Your clear focus on area and volume not being comparable finally made it click in a way no other explanation has. 👍🏻
I would have to agree with the incompatible measuring improving my understanding. With the hypothetical example that Jade gave of the boxes being clear, I would have to say that we a still seeing the boxes in terms of volume, because theoretically, light particles are measured in volume and eventually the squares will get smaller than a light particle, which makes the "color" of the surface irrelevant.
You are so impressive... and the way you explain stuff in an easily understood manner...can't praise you enough cos just can't find good enough words 🤔
We can also understand this paradox (or non paradox) by taking example of a dough ball, this would have a fix/finite volume, now we can keep rolling it and the surface area will keep on increasing with surface area reaching infinite as thickness approaches zero
8:29 This objects also exist in our world, as the coastline paradox shows. Beaches have an finite volume, but when you try to be absolute precise, it has an infinite perimeter. Great video :)
I thought about the coastline of Britain
Really would love a video on planks length! I think you bring up a good discussion about relativity of measurement in the video and would love to hear more about it from a more technical perspective!
1) If paint takes up volume, there is going to be a point in which the diameter of the horn becomes smaller than the diameter of a single atom, which means that the paint can't go any deeper than that.
2) If the paint is infinitely thin, then the amount of paint needed to coat the inside wall would be an infinitesimally small thickness times an infinitely large area, which is undefined. But we can take limits, nonetheless.
My interpretation of the infinity thin paint example is that the paint already has infinite surface area.
To give an explanation, imagine you cut up a cube into 2 rectangular prisms and put them next to each other. The volume would not change, but the surface area would increase. As we cut the cube more and more, the surface area will continue to increase. Now, imagine what would happen if the cube was cut into infinitely thin slices. The surface area would be infinity.
The same principle would apply to the paint.
6:55 in fairness we do have the speed of light as a pretty solid, fundamental and universal conversion ratio for those in the cosmic speed limit. Using this, an hour is indeed much, much longer than 2 yards- about 580.8 *Billion times greater.*
Invokes Jade's complaint: "what is this, physics?" xD
Our cosmos is full of facts that as of yet have no mathematical foundation, such as the speed of light-in-a-vacuum/causality. We call these "empirical" facts because they must be measured to learn what they are. They cannot be deduced from any simpler sets of axioms we are aware of: they basically establish their own axioms for the time being.
Questions in pure mathematics cannot include these axioms unless they are explicitly introduced. That's the only way we can discuss "infinitely long objects" or painting them to begin with: we have to choose which axioms to accept (eg, maybe "paint" must have thickness or maybe not, depending on what we wish to mathematically explore) and which to discard as undecided.
You could even do the Kessel Run in less than twelve parsecs.
@@TheAdwatson Star Trek?
Jade: What's the length of this line?
Me (who just finished explaining to a chemistry student why units are so necessary to measurement): it depends on the unit.
I just immediatly decided the length was x.
@@paulgoogol2652 my immediate conclusion, too. The line is one line in length.
Well done! 👏👏👏 That was by far one of my most favorite videos. A bit of deja Vu as well. Keep up the phenomenal work!
Great video as usual, Jade!!! There's a lot to be said about both delivery and factual research.
I really feel like it SHOULD be more "normal" for math to expose inconsistencies in what our perception of logical application is. Not necessarily because there are some crazy "secrets," but because math, unlike reality, is not based on what we experience, but what we can apply it to. It is also purely logical. I pretty much feel like if we didn't expect at least a few (even small) surprises, math wouldn't be doing its job!
That was a beautifully crafted video. I could actually feel my mind being expanded whilst watching it! Thank you!
Jade: A piece of time is longer than a piece of length
Einstein:
I got that reference
hahahahahaha The best reference
I didn't think about this, but so true. Einstein would disagree: you CAN compare space and time, they're both in the space-time continuum. :D
When we consider the hypothetical, mathematical paint that can be applied with no thickness (or more precisely thickness approaching 0), we find that when applied, it has no volume. So you can take the full bucket of paint, and then apply some paint to a surface. That applied paint has no volume so the bucket is still full. An infinite surface area is paintable from any finite volume of no-thickness paint. I think that this eventually comes right back to the original point made in the video - comparing the volume with area just doesn't have any meaning.
I think bringing up paint gives breadth to the fiction that there's a paradox because paint is measured both in volume (the amount in the can) and area (the amount you can cover with the paint in the can.
I suspect that ink, which can be used to draw lines and fill areas might also create the same confusion. But to draw the perimeter of the Koch Snowflake, the limits will be not be the amount of ink in the bottle,, but the fineness of the nib and the size of the paper (and the snowflake drawn upon it).
Yay! You're back! 🎉
edit: 10:34 “what is this, physics?” genuinely had to pause the video until I'd stopped laughing 🤣
... but if the paint is infinitely thin, it has no volume, right? so we're not actually using any paint at all, so there is still no paradox! CHECK MATE, ABSTRACT MATHEMATICIANS!
What about being infinitely divisible, you can divide a finite volume and a infinite area and you cab get from a cube, so if the paint is infinetly divisible there is possible to paint the horn and fill it with the same amount of paint.
Great video as always!!
As for the interpretation: When you paint a surface infinitely thin, then with one drop of paint you can paint an infinite surface.
No
There is no such paint. It would take an infinite amount of paint.
@@nosuchthing8
The cubes would become infinitesimally small, so there are no such cubes either.
It's just a mathematical thought experiment.
@@Jopie65 I'm only concerned with Gabriel's horn
@@nosuchthing8
Gabriëls horn becomes infinitely long and thin
I think the issue with paint is that it doesn't just coat a surface, but it also takes up volume. The volume of the paint on a surface is the size of that surface multiplied by the thickness of the paint coating, which may be thin but never zero. You can't have infinitely thin paint. At the very least, if all surface tension was removed and we didn't care how transparent the paint was, it would still need to be 1 molecule thick to cover a surface, and then it would still be taking up volume.
The fact that an object with finite volume can have an infinite surface lets us paint a 1m^3 Gabriel's Horn with 1 mL or 1 mm^3 of paint as both of their surfaces are the same (infinite). So...
If you where to fill it with paint and then empty it, it could be completely painted with and infinitely small volume of paint or in other words 0mL of paint.
This channel is so underrated. I absolutely love this content.
Wouldn't the infinitely thin paint lead to a rather funky "dividing by zero"-scenario? That could allow a finite volume of paint (no matter how small) to cover an infinite area... I think?
"Infinitely thin" paint would actually not be infinite. It would have a thickness that converges to zero, because if it were zero, there wouldn't be any paint. The thickness of the paint can be any number close to zero, but never zero itself. The infinity in this is the number of steps you take by making the layer of paint ever thinner. Thus, division by zero avoided.
Hello Jade! Love your content and energy you put in your explanations.
😀
Actually, you can paint the surface of Gabriel's horn with a finite amount of paint if you accept a coat of paint that's not uniform in thickness: Assuming you had infinite time to paint, and assuming the coat of paint can be arbitrarily thin (in particular, thinner than molecules and atoms), then you just need to make the coat of paint thinner and thinner (fast enough) towards the "end" at infinity.
Say, Gabriel's horn is centered around the interval [1,inf) on the x-axis. Then the radius of the horn around a point x is given by r(x)=1/x. If we paint the horn such that the thickness of the coat of paint around the point x is T(x)=ε/x, then the volume of the coat of paint is given by V(paint)=π(ε²+2ε)
if your paint have zero thickness, you can cover an infinite amount of surface with a finite volume of paint.
in other words, dividing by zero gives you infinity!
which is why you can't divide by zero :)
If your paint has zero thickness, you can’t cover anything with it. Just like n/0 is not infinity, mathematicians say it is undefined; it is more like never. 10/2 is 5, which is: 2 can be taken away from 10 5 times. 10/0 will never happen since taking 0 away from 10 will *never* give you a result.
@@SrssSteve that's actually the best reason why 10/0=infinity you can take 0 from 10 infinite times.
@@AlexandarHullRichter You *can* take 0 away from 10 infinite times, but you will still have 10. That’s why infinity is not the answer.
@@SrssSteve You can't uphold infinity as a concept, and still employ the term "never" as a limiting factor. That's a confusion of contexts.
I guess part of the answer lies in the question "is painting something outside the same as painting it inside". If you try to paint Habriel's horn inside - you get to the point where the diameter of the horn is smaller than paint's thickness. But outside you don't meet such a situation. As the paint thickness is constant - outside paint's volume is gonna be infinite
When you start applying practicality, all of those paradoxes get solved by planck uncertainty anyway.
However eventually the thickness of the outside paint dwarfs the object itself, rendering the object impractical.
Build a second horn, fill it with paint and then dip the first one in it.
@@timanderson5717 It's not easy to do, because to dip it into the first horn, you have to find the end of the second one, which is not there...
@@sanmar6292 When you start applying practicality, you'll find that you can't make an infinitely long object.
For me it's easiest to think about a probability density function like a normal distribution (or similar function). It can extend in both directions infinitely, but the area under the curve had a finite sum.
This reminds me of the guy that made a video illustrating how it was impossible to measure the length of the coastline for the UK. Because the more you zoom in, the longer it gets. You would need thousands of volunteers to go with measuring tapes and measure the entire coast.
Even though the volume is finite, it will take infinite time to fill the object as the paint or the painter will never reach the end of an object spread infinitely. This also resolves the point about filling from inside and not being able to paint it, you just won't have the time to fill it.
Love your videos though.
You just have to paint it infinitely fast. Then it will be done in an infinitely small period of time.
“Don’t worry I haven’t gone insane.”
*sad American noises*
6:53 Since a yard is defined as exactly 0.9144 meters. And since a meter is defined as the length of the path traveled by light in a vacuum in 1/299,792,458 of a second. One may argue that two yards are approximately 6.1 * 10^-9 seconds. That makes the statement that 1 hour is longer than 2 yards completely correct
Superlative channle and video!!!. I have been teaching physics and engineering for more than 45 years and I love to learn from you. I shall share my dear. Cheers from Patagonia, Argentina.
Great video! Although the thickness of the paint layer was mentioned, what about the thickness of the physical shapes? In your drawn examples, the internal and external surface area are equal, but this can never be so in real life - the internal surface area will always be smaller.
3:54
Me, explaining my friends, a physics theory.
So this paradox has a pretty simple solution that I feel like you hinted at but didn't put it together in a concise way. The paint, too, can have infinite area. Just divide the cube of paint in half an infinite number of times and it becomes easier to see that you can create an infinite area with a finite volume for the paint as well as the boxes.
Your analogy is somewhat confusing because we are not only dividing layers of paints but also adding it in the previous paint block so it should not only create infinite surface area but also infinite volume
If you think my point is wrong please correct it
@@ayanahmedkhan2580 I wasn't using an analogy, I was giving another way to think about the answer to the problem. The problem with the math is that area doesn't take depth into account, so if you removed a layer of paint with zero depth from your cube of paint, you would have exactly the same volume of paint that you started with after removing one layer. You seem to be thinking of paint as a series of molecules with a finite number of layers it could possibly have, which is what paint actually is in reality. So for this problem, you have to remember that area does not have any depth, only volume has depth, and so by that property alone, in a purely mathematical scenario, an infinitely small volume, so far as it's over zero, can be used to cover an infinite large area.
And so you are correct in thinking that a finite volume of paint in reality would not be able to cover an infinite area of space. However, you also have to remember that reality also cannot produce objects with an infinite area with which to paint on. The answer to this paradox is very abstract, has no basis in reality and is purely mathematical.
Great video, as always - really love this channel for explanations. I would argue no need to apologize for whatever units you've chosen, though. Use whatever system you like ... so long as you let people know what that is (Imperial, Metric, Non-standard) ... many have arbitrary aspects. Perhaps some units/systems are more useful to some applications, and others to others ... but I personally never cared for the snobbery of any particular system. Clarity and consistency for communication purposes likely matter the most. Love this channel.
Love this stuff, well done Jade.
Gabriel's Horn can be thought of as a 3D asymptote, where, at a certain point, the inner surface would be too small for the paint molecules to fit, but that doesn't mean there is no surface area in there, right?
Or would the walls eventually meet and then continue as a line, giving you both an infinite outer surface and a finite volume?
That would mean the infinite line has surface area which, by definition, it does not.
The important point to remember, actually pointed out in the video but lost on some of the commenters, is that this all comes down to how many and what (unrealistic or semi-realistic) things one is willing to postulate. They can include (1) infinitesimal paint (2) paint that travels at infinite speed (3) zero-width walls. Indeed, one can get interesting and postulate things like (3a) walls whose thickness is in a fixed ratio to the horn diameter at that point, (1a) paint whose individual molecule volumes come in an ever decreasing infinite series of some kind, and even (2a) paint whose speed is governed by "dark energy" repulsive forces rather than poured under gravity. How fast is the paint moving 14Gpc down the horn? (-:
@@joshuaewalker
But it wouldn't be a line. It would be an infinite number of points almost occupying the same space. That would make it "thicker" than a line.
@@roypatton1707
Unless the "horn" collapses to an infinitely long 2-dimensional plane defined by exactly two parallel lines then there will always be volume if it is "thicker" than a line.
@@JdeBP
They point out in the video how nonsensical it is to compare different units, e.g. an hour is longer than a meter. I think it is equally nonsensical to ask a physical question regarding an imaginary, mathematical concept. You can't paint or fill the cubes (or the horn) because they don't exist and can never exist. If you posit imaginary paint that can always fill the volume of the imaginary cubes no matter how small they get then the answer becomes "an infinite amount of imaginary cubes will require an infinite amount of imaginary paint to fill them". There will always be another cube in the series, so you will always need to get more paint. It doesn't matter if the "size" or "amount" of the volume is going "up" towards infinity or "down" towards infinity it is still trending towards infinity.
BUT if the paint coat thickness decreased in proportion to the size of the cube, THEN the amount of paint used would be finite. It's only because the paint coat is implicitly assumed to be of a constant thickness (and at some point this will be much thicker than the cube, it will be a blob of paint with a dot of cube in the middle) that the paint used becomes infinite.
This discrepancy between surface area and volume is why fine powders and dusts are explosive - large exposed surface area to ignite with very little mass needing to be heated to ignition point.
5:14 - if that inner surface is infinite it means it doesn't have an end, so the paint (@ 5:22) can never touch that end, therefor making it infinite in volume aswell, but because paint has its dimensions (volume & surface) it makes both dimensions finite, because they'll (paint's dimension) both reach a point where they'll be greater than the "coverable/fillable" dimensions of that object.
Whenever I was learning maths, I was always looking for some intiition and then I would often stumble upon such boundaries between the maths and the interpretation. I think these videos may close the gap to understanding for many people.
The swings and roundabouts of maths, basically! :) Great to see you back, Jade, awesome video as always!
That line is (on my phone screen) about as far as light travels in a vacuum during 2.3 times the period of on transition between the two hyperfine levels of the ground state of a cesium-133 atom.
At least, that’s how metric defines the units, not using length, but using other constants that can be measured more accurately without coming up with more precise prototype objects.
Sadly this doesn’t really break dimensional analysis, it just uses it to replace length with other units, speed and time.
I remember a teacher doing the math for the Gabriel's Horn thing when I took calculus. For a long time I remembered how to do it, but since it's now approaching 50 years since I sat in that class, along with the fact that in real life I never had to manually do calculus equations, I've forgotten how to solve the equations.
I love your videos Jade. I won’t pretend to be able to wrap my mind around a lot of the higher IQ stuff, but you never fail to open my eyes and make me think and observe in ways I did not previously… and you are able to do this in a way that is both pleasantly engaging and compelling in nature! This is a true gift as few people can pull this off successfully… sooo… thanks!
thank you for the kind words John!
Math's coolness goes to infinity, while our ability to understand is finite!
I rase you: maths coolness = imagination, understand = apply
Great comment.
When she exposed the paradox at 10:50 i was pretty much confused. But it is true that the Horn inner Area*paint thickness=volume of paint -> A*t=V
Then we have 2 cases:
1) t equals any positive number not approaching 0.
If so, think about the part of the horn close to the mouth (approaching infinity).
This part cannot be painted because the available inner volume is less than the volume of the paint you need to use.
So you are ideally cutting the thin part of the horn and the area becomes finite.
So you will be able to paint everything.
2) t approaches 0
A*t= V becomes inf*0=L
This statement can apply mathematically (for instance 1/x * x for x goes to infinity is equal to 1 that is finite). In this case you are not using paint to paint the surface because the thickness is 0!!!
As well as the other case you can paint everything.
I think the horn makes it easy to get your head around. You fill the horn with (finite) paint and the full (infinity) area vill be covered.
If you look done the inside of the horn you will see that it will convert to zero diameter, so the paint thickness will convert to zero thickness to be able to fit inside. So it will cover an infinity surface but with an thickness converting to zero so no problem.
"But you said 'paint' so I assumed that you meant real 'paint'. You can't paint 'paint' any thinner than a single molecule of the paint. Therefore a film of paint on a 2D surface does have a rigorously definite volume and can be directly compared against the volume of paint that can be held in a shape."
-- Paraphrasing Richard Feynman about cutting oranges.
"You can't paint paint any thinner than a single molecule..."
Yeah, what is this, mathematics?
But if we’re going there, it is also impossible to have a surface that is infinitely big because it would require an infinite number of molecules.
@@stargazer7644 I wouldn't say that's a problem per-se. While our observable universe does appear have a finite number of atoms, we can't yet disprove the idea that the universe is infinite. It's possible that we have an infinite number of atoms at our theoretical disposal to use.
@@KantiDono Any atoms that might be beyond the limit of the edge of the observable universe are moving away from us faster than the speed of light due to the expansion of the Universe and are forever beyond our reach, theoretical or not.
10:39 if we consider that layer of paint which is painted on that object was infinitely thin then that paint would not have any volume. If that paint didn't had any volume then how you can fill an object with finite volume with stuff which does not have volume? So conclusion is volume is nothing but infinite number of infinity thin layer of surface areas stacked over each other
What is this? Mathematics.
Loved this line ngl
You most certainly can equate time with measurements of physical distance (length of a yard or meter, inch, etc). If you take the distance of oscillations in an atom per second and prescribe to a length, you can reference time literally by overall distance covered during the duration of oscillations.
If the paint is infinitely (arbitrarily) thin, that you can coat an arbitrarily large (but finite) area with an arbitrarily small volume of paint.
As the area goes to infinity, and the thickness goes to zero, we can se up a relation between the area and the thickness to get anything from zero to infinity. Not sure about negative amounts of paint though.
e.g. For a square of side length l, we use a thickness of 1/l^3. The volume of the paint layer is now 1/l, which approaches zero as l goes to infinity.
Great job, Jade! Wish you could mathematically work out a way to make a new video every day! ;)
Math is bigger than reality. If something exists in math, it doesn’t mean it is possible in reality.
But anything what DO exist in reality MUST exist in math as well.
_"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."_
Chaotic dynamics are fully void of deductive structure.
You can also make a case for the opposite if reality is more than just physical reality and includes the sphere of ideas.
If something is real, it doesn't mean it can be mathematically expressed (and as such does not exist in math).
But everything that does exist in math is real.
@@tabchanzero8229 Well, we certainly know that not everything exist in reality what exists in math.
But is there anything in reality what can NOT be expressed by math? Give me an example.
@@juzoli Incompleteness Theorem.
@@slofty That basically says that not all math problems can be solved using math.
But what I’m saying is that also not all math problem represents a real life thing in the universe.
So it is entirely possible that to unsolvable problems doesn’t have a real life counterpart in the universe to begin with, and everything in the universe can be described by math.
For example we know about unsolvable problems in math. But we don’t know about anything in the universe which couldn’t be described using math.
At some point the so called boxes opening will become smaller than a paint molecule or it's associated emultion. This of course is how filtration works.. This can be a fun thought experiment sure, but even down to the smallest subatomic particle, we live in a discrete world.
Most of the "paradoxes" that I've heard of seem to simply amount to, "I don't understand/can't figure this out, therefore it's a paradox".
I feel like there needs to be a new word to describe that and then we reserve the word paradox for something that actually isn't resolvable.
Very good, Jade.
Keep it up! (and Atom)
Beautiful! Excellent narrative too, many thanks for these videos!
I've always seen it best described with an islands coastline versus it's landmass - all depending on how accurately you're looking at it makes the length of coast technically infinite.
6:19 Yes, if you line up a yardstick along the edge of a table top nothing forces you to say you are using the yardstick to measure the table any more than you are using the table to measure the yardstick.
It seems the "infinite" surface area just means a number that is beyond our understanding or beyond our current mathematic language to describe.
Paint can cover infinite area as well, so i think the proper way is to essentially obtain the volume of an object and subtracting the volume of the internal chunk as to just have the "skin" of the object. That way, we can approximate the volume of paint that will be able to cover an equivalent "infinite" surface area.
Viscosity of the paint would matter as well as it would probably help determine how thin the coat can be
You need paint which can be spread infinitely thin to fir in the bottom of Gabriel's Horn or to fill this infinitely small box. If you can spread it infinitely think you only need one drop of paint.
By first defining what "to paint" means we can solve this problem without entering the realm of physics:
Let's (arbitrarily) define to paint as putting a 0.1mm layer on top of our horn, this leads to a new body that is not like the body before. Even at the smallest point of this new body never gets smaller than 0.2mm in diameter, so this new body has an infinite volume.
Or define to paint as putting paint from an infinite supply on any surface without taking time. There it's painted.
the way to determine amount of paint used would be the thickness of the paint used in the volume formula so you would have a end point when the size of the cube was smaller than the thickness of the paint.
If a color does not consist of atoms and can be painted infinitely thin, the paint in the smallest of the infinitely small cubes is sufficient to paint all of them, because you can find a transformation between individual points of volume and surface.
Reminds me of the Banach-Tarski paradox ...
When the paint is made up of atoms, there is a point where the cubes become smaller than the pigments and then cannot be filled or painted.
Time can have a physical length if it is taken as an interchangeable variable in 4d space-time. Likewise, a meter, yard, cubic yard, fathom... can have a duration.
Infinite 2-D surface area is 0 in volume , so it makes sense that infinite surface area is needed for some finite quality of volume.
This is also known in the second dimension as coastline paradox. You don't even need the Koch snowflake. You can measure the coastline of an island with arbitrary precision. The limit of the sum of the segments between two measurement points where the numer of measurement points going to infinity is infinite.