The Painter's Paradox - These Weird Objects Will Blow Your Mind

[@upandatom]

6:02 What is the length of this line?

[@Luca_5425]

About 3 and a half things

[@shashankkumar7698]

10 and a half

[@user-iu1xg6jv6e]

10 2/3 apples square

[@zaccarlotta554]

1 hour is not longer than 2 yards

[@Hailfire08]

The length is such that light takes x amount of time to cross from one end to the other :)

[@mayaral8637]

"Imagine a cow that isn't perfectly spherical" Physicists: What is this? biology?

[@astroceleste292]

* AHHAHAHAGA

[@W1ngSMC]

Well, definitely not topology.

This was never said in this video. You probably commented under the wrong one?

[@Censeo]

@ but your comment isn't said in the video either

[@mynameisatypo]

@@Censeo nice one

[@hoptanglishalive4156]

So if it takes forever for a single note to leave Gabriel’s Horn, should we conclude that Judgment Day will never come?

[@stephenrichards5860]

How would anyone (other than Gabriel) know that the Horn had been blown?

[@MarcelinoDeseo]

Does the infinite surface area of Gabriel horn implies the horn's infinite length?

[@mbrusyda9437]

@@MarcelinoDeseo I mean, the integration to infinity kinda does..

[@vigilantcosmicpenguin8721]

@@stephenrichards5860 More importantly, how does Gabriel even hold it?

[@stephenrichards5860]

@@vigilantcosmicpenguin8721 since the horn and Gabriel are improbable, why do you care

[@leroidlaglisse]

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.

[@BlueSapphyre]

Even if the paint had no thickness, an infinitely long object could not be physically created to paint in the first place.

[@leroidlaglisse]

@@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.

[@simowilliams6990]

@@leroidlaglisse No, how would that be paradoxical? Just unusual.

[@jorgepeterbarton]

@@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

[@leroidlaglisse]

@@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. :)

[@vari1535]

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.

[@Patrick_Bard]

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.

[@znotch87]

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?

[@dmuntz]

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.

[@suomeaboo]

Taking special relativity, wouldn't 1 second be equal to 299,792,458 meters (a light-second)?

[@jacobburr3570]

DMT is one hell ova drug

[@skhotzim_bacon]

Holy crap now go run through a gravitational field. Try to metric tensor your way through that one. See you in a lightyear

[@wren_.]

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

[@davidcroft95]

"the paradox lies entirely in our interpretation" no sentence has been so true 👏🏻 (it's also the favourite quote of my astrophysics professor)

[@QuantumFluxable]

it's a lot like zeno's paradoxi in that way

[@davidcroft95]

@@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

[@cosminstanescu1469]

Does this apply to the dual nature of light?

[@davidcroft95]

@@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

[@LordOfTamarac]

*black hole complimentary has entered the chat

[@MedlifeCrisis]

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!

[@redunleasher2147]

Which channel?

[@recklessroges]

@@redunleasher2147 Probably Numberphile?

[@almasrafi4102]

You mean Numberphile? But that was also intuitive too😥😥

[@derickd6150]

Oh wow. Nice to see Medlife crisis here!

[@TheTransitmtl]

@@derickd6150 He comments on almost all her video's

[@marksainsbury2422]

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 ...

[@dr.hoover345]

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.

[@Think_Inc]

The asterisk at 1:42 and the quote at 4:32 were priceless! *XD*

[@00BillieTurf00]

thanks for pointing it out, hilarious indeed, hadnt seen it impressed as I was by the mindblowing beauty of this principle.

[@karenjeandiez6331]

wow! "That made sense"

[@akira_rtt]

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

[@TheBoxysolution]

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.

[@saggitt]

What if the thickness of paint goes down the deeper you go into the horn, but it is never zero? :)

[@TheBoxysolution]

@@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.

[@ValkyRiver]

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.

[@TheBoxysolution]

@@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.

[@ValkyRiver]

@@TheBoxysolution Vsauce explains it here: m.kzbin.info/www/bejne/nJe4n4GXhrmZkKc

[@Mad-Lad-Chad]

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.

[@christopherhernandez3937]

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.

[@shlusiak]

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".

[@richardmellish2371]

Yes, and it needs to get thinner and thinner to fit inside the smaller and smaller cubes or sections of the horn.

[@joet3935]

I propose that infinitely thin paint lacks volume, and would then be unable to fill a can or cube.

[@shlusiak]

@@joet3935 infinitely thin paint over an infinite area may in fact have a concrete value of volume though.

[@bermchasin]

@@joet3935 interesting.

[@joet3935]

@@shlusiak Thats like folding a 2D plane to fill a cube. How many shadows do you have to stack to make a volume?

[@dru4670]

"What is this!? Physics 😏 " Physics explains our universe, mathematics describes all possible universes is how i usually put it. 😂

[@alexv3357]

Physics explains what's possible, maths constrain what's imaginable

[@QuantumFluxable]

@@alexv3357 maths is just philosophy on a higher difficulty setting

[@jovian304]

@@alexv3357 I'm stealing this

[@howardlam6181]

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.

[@cabbage5114]

@@alexv3357 requesting permission to use your statement incase I ever get into a maths vs physics discussion.

[@ryanfriedrich6634]

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.

[@nosuchthing8]

No

[@MeriaDuck]

"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?)

[@upandatom]

not yet!

[@Christian_Prepper]

@@upandatom *Who else has no clue what she's talking about, but still enjoy watching her & listening to her accent?*

[@kioarthurdane]

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!

[@rallok2483]

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.

[@atomatopia1]

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

[@truevelvett]

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

[@atomatopia1]

@@truevelvett Thanks! That’s kind of how it clicked for me too

[@Zuzezno]

This

[@adityaanantharaman7963]

Mathematics overtakes/overwhelms Physics at the Planck Length. As always, excellent! 😊

[@IceMetalPunk]

Yep! Infinity is nice and all, but physics says everything is finite if you get small enough :P

[@snakezdewiggle6084]

@@IceMetalPunk incorrect ! Physics says, everything is quantifiable, except for those that are not. ;)

[@monad_tcp]

@@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).

[@monad_tcp]

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.

[@Founderschannel123]

@@IceMetalPunk well black holes are a example to it since its volume is infinite however is surface isnt i guess?

[@GrowlingM1ke]

Literally yesterday was doing the Calculus in a nutshell course on brilliant and I was wondering about the exact same thing XD

[@codyofathens3397]

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?

[@thargy]

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. 👍🏻

[@robertfletcher]

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.

[@JaykPuten]

We can experience the infinite... When you're sleeping without a dream, completely unconscious and without thought, you just need to realize that's how conscious you were since the beginning of time, and will be after your finite time Or I'm just being really nihilistic And somewhat sarcastic

[@cyb3r._.]

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

[@lifeinthevoid1595]

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 🤔

[@louiscallens4183]

”What is this! Physics?” Great quote ;) I thought I knew all about this paradox but you just proved me wrong!

[@bobgroves5777]

Physics? ... Now, a practical introduction to Dimensional Analysis.

[@kalyngriffin1518]

This channel is so underrated. I absolutely love this content.

[@chaosorr]

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.

[@ahmetdogrusoz6951]

I am sure hundreds of comments in the many months since the posting of the video said the same thing but... You CAN paint an infinite surface with paint of finite volume if you can spread the paint infinitely thin (0 thickness). Because the amount (volume) of paint to use a finite area (within the infinite surface) will be zero if paint is spread infinitely thin. And if you assume the paint is NOT spread infinitely thin, then the paint on an infinite surface creates (is of) an infinite volume. You can not paint the inside of Gabriel's Horn not (just) because there is not enough paint but because there is not enough volume inside the infinite surface for paint (with non-zero thickness). Or if paint is of no volume (zero thickness), then you CAN paint because you will never run out paint even if you have a finite amount (VOLUME) of it because you are spreading it infinitely thin. This is, I think, the same thing that was implied (or explained) by the video's mention of the issue of the thickness of the cubes and cube painting. It also amounts to the same thing as the incomparability of volume and surface, because the comparison would actually require (presume) a "paint thickness" which, if zero, creates an 'infinity-times-zero equals finite value' quagmire (but no paradox).

[@InternetDarkLord]

Almost....You can paint Gabriel's Trumpet by painting the paint layer decreasing thickness, but it never reaches zero. Imagine the thickness of the paint layer is 1% the diameter of the horn. It gets thinner and thinner (with a limit of zero) but never reaches zero thickness. We can even compute the thickness of the paint each distance along the horn and the total finite volume of the paint. The boxes show the flip side of thickness. As the boxes get smaller and smaller, they will eventually be less than one millionth of the millimeter on each side. If the paint is just one millimeter thick, you therefore have cubes almost entirely paint, one millimeter cubed. The rest of the cubes will just be an infinity of paint cubes 1mm by 1mm by 1 mm, with a negligible volume cube at the center. No paradox here.

[@diggy5179]

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!

[@Squossifrage]

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!

[@engelsteinberg593]

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.

[@PyroMancer2k]

The problem with the example though is the same problem with trying to paint the object. Just like paint has thickness and thus if you tried to paint the surface it would taking up more volume that the inside of the object contains. The object in question just like the paint has to be made of something and that item can't be shrunk indefinably. In the case of the cubes eventually you get to he point where the walls of the cube would need to overlap with the walls on the other side due to their thickness thus having two objects occupying the same space and time which is impossible. This reminds me of the Blackbody radiation problem. Originally it was assumed that, much like this paradox does, things could be infinitely divided into smaller and smaller amount. The problem with this is when the numbers were run for black body radiation it showed they could produce infinite amounts of energy, which is impossible. This became known as the Ultraviolet Catastrophe. This was solved by Max Planck who showed energy comes in discrete amounts that can't be reduced to smaller units. When you go to quantum mechanics there is a minimum amount of space, a Planck Length. This then becomes the smallest size a cube could become and thus puts a hard limit on the number of cubes. Even in the case of the horn the thickness of the neck of the horn gets smaller and smaller but eventually it can't get smaller than a plank length and thus the horn ends. This yet another interesting thought experiment on the relation of things but ultimately holds little bases in reality. Like so many other "Paradoxes" in history. They are the result of looking at things in the wrong way. When they popup they can be useful to highlight issues in our current understanding.

[@mjohnson2807]

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

[@Jopie65]

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.

[@nosuchthing8]

No

[@nosuchthing8]

There is no such paint. It would take an infinite amount of paint.

[@Jopie65]

@@nosuchthing8 The cubes would become infinitesimally small, so there are no such cubes either. It's just a mathematical thought experiment.

[@nosuchthing8]

@@Jopie65 I'm only concerned with Gabriel's horn

[@Jopie65]

@@nosuchthing8 Gabriëls horn becomes infinitely long and thin

[@LyonsTheMad]

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.*

[@happmacdonald]

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.

[@TheAdwatson]

You could even do the Kessel Run in less than twelve parsecs.

[@shadowcween7890]

@@TheAdwatson Star Trek?

[@NEMountainG]

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!

[@amonia1766]

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 :)

[@skhotzim_bacon]

I thought about the coastline of Britain

[@Fudgaroni]

8:54 girl straight up went vsauce mode xD

[@Lucky10279]

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.

[@paulgoogol2652]

I just immediatly decided the length was x.

[@jaelin9107]

@@paulgoogol2652 my immediate conclusion, too. The line is one line in length.

[@radward7173]

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.

[@craigvdodge]

“Don’t worry I haven’t gone insane.” *sad American noises*

[@eastonw.534]

10:34, we’ll, if the paint didn’t have thickness, it has no volume and shouldn’t be able to fill an area. Correct me if I’m wrong.

[@mbstp]

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.

[@Delibro]

Does this make any sense to someone? Of cause the metric system is useful to convey information.

[@MarkWaner]

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

[@sanmar6292]

When you start applying practicality, all of those paradoxes get solved by planck uncertainty anyway.

[@cirelancaster]

However eventually the thickness of the outside paint dwarfs the object itself, rendering the object impractical.

[@timanderson5717]

Build a second horn, fill it with paint and then dip the first one in it.

[@MarkWaner]

@@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...

[@tabchanzero8229]

@@sanmar6292 When you start applying practicality, you'll find that you can't make an infinitely long object.

[@rbkstudios2923]

Jade: A piece of time is longer than a piece of length Einstein: I got that reference

[@danielmunoz-cj7hj]

hahahahahaha The best reference

[@sly1024]

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

[@rhysun]

That was a beautifully crafted video. I could actually feel my mind being expanded whilst watching it! Thank you!

[@headjump803]

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...

[@dubsed]

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!

[@Zoomeep]

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?

[@jali7913]

"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.

[@elminster8149]

Love this stuff, well done Jade.

[@algorithminc.8850]

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.

[@dannythemedic]

if 4:26 could be a framed poster, I would buy one

[@adrihooijer536]

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

[@dattatreyadas01]

3:54 Me, explaining my friends, a physics theory.

[@Dudleymiddleton]

The swings and roundabouts of maths, basically! :) Great to see you back, Jade, awesome video as always!

[@wiseSYW]

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!

[@HerrFinsternis]

which is why you can't divide by zero :)

[@SrssSteve]

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.

[@AlexandarHullRichter]

@@SrssSteve that's actually the best reason why 10/0=infinity you can take 0 from 10 infinite times.

[@SrssSteve]

@@AlexandarHullRichter You *can* take 0 away from 10 infinite times, but you will still have 10. That’s why infinity is not the answer.

[@j3ffn4v4rr0]

@@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.

[@MrDoboz]

If your paint is infinitely thin, you are good to go. A liquid will take the shape of the container, and if your container has infinite surface area, your paint is going to have infinite surface area, as it's now the same shape. Also, if you want to be really crazy, you can pour out the unused paint, and maybe paint another one of these objects with it. And another one. And another one? As many as you want, because you don't leave behind even a finite amount of paint. One little caveat is though, you would probably never finish filling up the first one. Remember, it's infinitely long, so it would take literally forever to fill even with the speed of light. Also if you press it just a bit too hard, you might create a black hole somewhere down the line

[@itsROMPERS...]

A simpler related problem is this: If you have an infinite series of squares, starting out with sides of: ½ + ¼ + ⅛ ..., continuing infinitely with each box being half the size of the one before it, what's the size of the square they make up? Since it's an infinite series, you might think the answer is "infinite", but since they also halve infinitely the answer turns out to be "1".

[@Vide0Browser]

1 is just the smallest unreachable value of that series. It can never ever reach 1. Never ever!

[@pomilkatoch]

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.

[@pomilkatoch]

Love your videos though.

[@stargazer7644]

You just have to paint it infinitely fast. Then it will be done in an infinitely small period of time.

[@roypatton1707]

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?

[@joshuaewalker]

That would mean the infinite line has surface area which, by definition, it does not.

[@JdeBP]

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? (-:

[@roypatton1707]

@@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.

[@joshuaewalker]

@@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.

[@joshuaewalker]

@@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.

[@dark_knight2357]

Math's coolness goes to infinity, while our ability to understand is finite!

[@squeaksallan8195]

I rase you: maths coolness = imagination, understand = apply

[@isobar5857]

Great comment.

[@shadowrider2432]

The line is almost the width of my television.

[@lukeerikblue958]

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!)

[@rollomaughfling380]

Great job, Jade! Wish you could mathematically work out a way to make a new video every day! ;)

[@prathampanchal9260]

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

[@daphenomenalz4100]

What is this? Mathematics. Loved this line ngl

[@juzoli]

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.

[@slofty]

_"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.

[@tabchanzero8229]

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.

[@juzoli]

@@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.

[@slofty]

@@juzoli Incompleteness Theorem.

[@juzoli]

@@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.

[@VlianVlian]

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.

[@groumi]

@8:55 "But we if we did?" Classic Michael Stevens line!

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.

[@michaelelkin9542]

Amazing video. I think I finally understand why. From calculus any line segment contains infinetly many points of infinetly small size, just as any area contains infinetly many lines used to measure it. Just like any finite volume can be split into infinetly many flat or 2D sheets. Just the basics of how calculus works, and also means that any higher dimension has infinetly many slices of any lower dimension. So the tiniest 4D object would contain infinite volume in 3D, Thank you.

[@gobblinal]

That's the truthful answer. If you use calculus then any amount of volume "contains" infinite surface area. And then there's no paradox.

[@cliffordhodge1449]

Something similar to this basic concept is actually useful (and presumably used) in everyday life by many people. For example, you may be aware that smaller animals require more calories per unit of weight than large animals because a mouse, e.g., has more surface for its total volume than an elephant. Similarly, when shopping, I try to buy larger bananas because I know that if I buy small ones, a larger percentage of my purchase price is paying for the peel as compared with a large banana. So if you think about the series of cubes or Gabriel's Horn as being comparable to a series of mammal stages or banana stages, with the object steadily shrinking in size, the "ratio" of surface area to volume will approach the point of having denominator zero at which point the animal's body would have no heat to be retained and the banana would have no value as a fruit snack. For such everyday practical purposes, the fact that area and volume are incommensurate measures is really not a problem, since we are still free to assign some sort of general "unit of bigness" such that at the start of the series the surface is assigned so many and the volume is also assigned some generic measure of magnitude that we are calling 'bigness'. In any event, as the object shrinks it will be found that the surface shrinkage is not a simple linear function of volume shrinkage.

[@ConceptJunkie]

Nothing's a waste of time when watching you, Jade. You do a great job of describing things. Even when you're describing something I'm thoroughly familiar with, you still make it interesting.

[@carlsagantribute8688]

Very good, Jade. Keep it up! (and Atom)

[@raykeefe9253]

Quite right. The key is that as the cubes become too small for practical application of real paint. And this works because it is conceptual rather than practical. But it is important because it helps us to understand concepts and these are the foundations for solving tomorrow's problems. Including the ones we didn't know we would be facing.

[@PukeSkinwalker]

Volume and surface area are two different things. Volume can and has a limit and that limit tends to be at the surface area. The paradox is a calculus problem. If you can gain the surface area with a derivative or integral of the surface area problem and it changes from a divergent integral to a convergent then it is going to create a paradox. There are a lot of formulas here. There is likely a system of nature that this occurs like if you took the derivative of the force of electromagnetism. Maybe that derivative is 0.

[@queens.dee.223]

I love this video! Others have listed the seemingly infinite number of things that are wonderful here, and I agree! All I have to contribute is this comparison. we do have an intuitive model for something similar, and that's a tube of toothpaste! In a world where infinites are possible, a full finite-length tube of toothpaste will contain a finite volume of toothpaste coating the finite interior area. When the tube is squeezed empty from the end, we have an infinitely small volume -- zero-ish if you will -- of toothpaste coating the same finite area. Great video, and a wonderful way to start the day!

[@michaellv426]

Wonderful! Also, all those cubes can be painted in just one step, if you first open them and place them inside each other, - now you only need to fill this outer cube with paint, - it would require even less paint than filling an empty cube. As a result, a finite quantity of paint can be used to "paint" an infinite amount of surface

[@martinze11]

I'm a layman with a life-long interest in science. Now that I am retired, I have the time to watch these videos. The variety is wonderful. Has youtube always been like this? I find that the more that I think that I know, the more that I have to learn. The big difference between reality and fiction is that whatever you think reality is, it's something else.

[@Kapomafioso]

10:00 I was about to type something along those lines. As a physicist, I'd immediately associate "painting" with "coating with a paint of finite thickness d" - that way you basically convert "area" to "volume" (roughly speaking, multiply covered area by this finite thickness d). In that sense, there's a problem immediately: pouring paint into Gabriel's horn would sooner or later come to a point when the walls of the horn are closer to each other than this thickness d, so by our rules we wouldn't consider that "painted" anymore (and that's true no matter how small, albeit finite, d is). In the case of the cubes, the inside walls would get too close together as well. The outside walls could be painted, but we would need infinite amount of paint (the cubes would get smaller, but we would need to coat them with paint with finite thickness, so at that point we would use some roughly constant amount of paint per cube, which sums up to infinity). Of course, in the real world, paint would stop flowing when the diameter gets small enough. To really "paint" the whole thing, we would need inviscid paint and we'd have to constantly redefine the thickness as it flows down the horn (the thickness of the paint we consider "painted" would need to tend to zero faster than the diameter of the horn), and in that case, the horn would be considered "painted". If we are really allowed to paint in infinitely thin layer, then sure - you can paint the whole thing by pouring a bucket of (ideal, inviscid) paint inside - just like the original horn, the liquid paint now has infinite surface area.

[@davidblazek2418]

7:09 surface area is always bigger than volume of any objects (numerically speaking)!! Because units are incomparetable!!

[@j3ffn4v4rr0]

3-dimensionality (volume) varies with the cube, whereas 2-dimensionality (area) varies with the square...therefore, things that are shrinking (like the skinny end of Gabe's Horn) lose volume faster than they lose area. So, you can use an infinite series and "tune" the rate of shrinking, like Torricelli did with the Horn, to arrive at apparent paradoxes like this. Try it with a 4D object (a hyper-Gabriel's Horn, perhaps?) and the variance is even greater between hypervolume/volume/area.

[@RF-fi2pt]

Gabriel Horn have the same paradox as a more simple case of Circle Area in the plane. Area of circle is a FINITE constant πr^2, but the 1D swirl line starting at center have INFINITE length to achieve the total radius. Finite although Irrational like π. And that operation is done by integration the polar coordinates of the circle drdθ, from the center to r. At the horn, finite volume contains infinite area. At the circle, finite area contains infinite length.

[@XY-vf7qy]

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.

[@joesands3350]

The problem with mathematicians using calculus is that they treat or make you believe "approaching to infinity" is an actual number. Unfortunately, we live in a "physical" world - unless someone can prove we live in a simulation. Let's assume that the smallest distance between any 2 objects is a plank length ~ 1.6x10-35m. Given this limit then you can calculate a very accurate surface area, volume etc without invoking the ridiculous concept of infinity! AND as a bonus you get a realistic answer! p.s. the inner surface of the cube will only have the same surface area of the outer cube in the case of zero thickness = impossible. Defining exist = "consistent with the rules & laws of the mathematical world" is simply absurd (even in computer simulations!) How about an object existing = an object with physical location (I'm not referring to concepts existing in one's mind) Whilst I really enjoy the abstractness of mathematics (as you quote eloquently timestap 11:25) when applied to the physical world it should be tempered with realism e.g. One of my pet peeves is stating that the singularity in a black hole has infinite mass & zero volume - a perfect example of where the mathematics fails us

[@TheJCHarkins2]

Question for a future consideration: 2nd law of thermodynamics states all energy is converted in some form to another form of matter. Big bang was the primordial creation of the universe. Where did the initial energy to create the universe come from? I have seen Futurama's cyclic universe time line episode which to an extent makes sense but the question is where did the original system begin? Dimensional flotsam of a higher/lower dimension? God? I'm intrigued by the concept and makes me a bit uncomfortable thinking of the ramifications that there is a question that could in possibility be lost to the aether of time and space.

[@MikeRosoftJH]

The answer is: we don't know! And we don't even know if the question even makes sense. What caused the Big Bang? What was before the Big Bang? One - admittedly unsatisfying - answer is that there's no "before". You could as well ask what is to the north from the North Pole. And saying "God" doesn't answer the question, it just pushes it one step back: where did God come from? And if one were to say that God has always existed without being created, why not remove the extra step and say that the universe has always existed? (Though it is possible that the "always" only extends to some 13.8 billion years in the past.) There are some speculative theories of cyclic universe, where the Universe is indeed eternal, and the universe as we know it is just one phase in the big scheme of things. And the law of conservation of energy is a consequence of the laws of nature being constant in time (see the Noether theorem). In general relativity the law needs to be modified.

[@YayComity]

Jade has a special ability to humanize the coolness of math without dumbing it down. This is a great one!

[@kenny-kvibe]

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.

[@ohnonomorenames]

Jade is the adult version of Play School - Non-Australians think Sesame Street for general reference - The enthusiasm and friendliness makes me want to sit a 4 year old down to watch it. But the complexity of the ideas is amazing. Jade trust that you can come along and learn something and all she is doing is acting as a guide. It is truly amazing! To make topics that the greatest minds 300 years ago struggled to grasp approachable to high school math students is brilliant. Jade I think your brilliant, your energy is infectious and you are creating amazing content. Best wishes.

[@craigdavies2598]

Me from the uk:not understand

[@halflight8811]

I would like to say one thing,there are sooo many cool channels on youtube teaching paradoxes, basic highschool topics, soo seamlessly that it goes into your mind and fits like a puzzle piece and there are the hidden underrated ones....I just wish I knew each channel, each day I spent here, If find more and more cool topics......Thank you.

[@chrisg3030]

I love your painty paradoxes, having revisited your Aristotle's Wheel. Maybe combine the two.

[@DrDaveW]

A good example is the area of an island compared to the length of its coastline. Area is straightforward. Coastline, though - where do you "draw the line"? Round the headlands and inlets? Round the rocks? The grains of sand? the electrons orbiting the atoms and molecules in the rocks and sand? The sub-atomic particles?

[@triadmad]

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.

[@Cubinator73]

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)

[@msamour]

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.

[@Xanman64-p6q]

It comes down to being able to fit infinite 2D objects into a 3D space because you can set one of the dimensions to 0. Think of it as stacking flat paper in a box - if the paper has a thickness of 0 you never fill the box. If you cannot ignore the thickness of the paper, it is 3D and therefor will fill the box at some point. Same thing happens at 1D vs 2D. A finite plane contains infinite lines. And a finite line contains infinite points.

[@haph2087]

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.

[@SidharthCAnil]

I like how her eyebrows moves up when she says something

[@intotheunknown8100]

Every time you come back with something wonderful. Finally, I can understand this horn.

[@jonthecomposer]

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!