Gabriel's Horn and the Painter's Paradox 

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To a topologist, a coffee cup and a donut are the same. And so are the xy-plane and the xz-plane.

The music post horn-overlay is so soothing.

This is the only time i found the explanation of this paradox otherwise they would just prove and leave us thinking

Every math prof ever "I'm not an artist"

This is really good stuff. Hope you get a ton more subscriptions!

This channel should have wayyyy more subs/views given how good the quality of these videos are!

Brilliant. After watching many, albeit often good, videos on this matter, this is the one that finally made it clear to me how the paradox is resolved. It's key (for me at least) is consistency in the "paint's" description - mathematical or real - combined with bijection. The former seems more trivial but I think both views are equally valid. Additionally, the simple comparison of a sphere/finite volume of paint and an infinite surface brought the message home. Thanks for your time in making this. Very much appreciated.

Awesome video! If Gabriel's horn were physical you would also need all the atoms in the observable universe to build it and you still wouldn't be done. BTW that thing is pointy AF and it's kind of stabbing you in the thumbnail :)

I really love your videos dude!!!Keep it up

Brilliant!!!!!!! Wonderful thought process !!!!!

This is great content. You earned a new subscriber. Keep it up!!

Found you through Flammable Maths. Clicked for the maths, stayed for the Zelda soundtracks because that's literally what I use to study most of the time. You sould check out Teophany's "Time's End" albums if you haven't, they are based on Majora's Mask (as the name would suggest) and give me feels. Also great outro

the thumbnail is fricking epic, nice one

Mannn this editing is next level!!!

Here is my "loose description" of the solution to the final exercise of this video, since as pointed out, this is a more advanced problem than normal. Recall from my video on isomorphisms ( ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ZPbYriK_gCs.html ) that the isomorphisms of topological spaces are continuous bijective functions whose inverses are also continuous. So the final exercise is really asking: are the open unit 3-dimensional sphere, and the 2-d plane, isomorphic as topological spaces (or, are the two spaces /homeomorphic/)? Now, even if you have no background in that subject, knowing that isomorphic is the default idea of "sameness" of topological spaces, the idea of a 2-dimensional object (the plane) and the 3-dimensional sphere being topologically "the same" might bug you out, and you're correct to bug out! They are topologically completely different, and so such a function between the two, preserving all-there-is about topology, is impossible! Notice the function I gave is continuous, but not one-to-one. If I gave a one-to-one function instead, it would fail to be continuous and/or fail to have a continuous inverse. So how can we show this? Recall also from that video that there can be multiple notions of "sameness" of mathematical objects, and there is another notion of sameness in topology, which is homotopy equivalence. Homotopy equivalence is weaker than homeomorphic, so if we can show the two spaces aren't homotopy equivalent, that proves they aren't homeomorphic and so no such function exists (this is the same reasoning as: if two geometric shapes aren't similar, then they are not congruent). Now, all of this can be phrased very formally and precisely, but I'll leave that to someone else, and I'll be giving a more basic description. Here's why the 3d sphere and the 2d plane fail to be homotopy-equivalent: Imagine you are in your room. There is an immovable basketball floating in the center of the room, and around the basketball is a hoola-hoop hovering around it, like Saturn's rings. You could go up to the hoola hoop, lift it, lower it, or manuver it many other ways, so that it is no longer going around the basketball. You can move the hoola hoop however you want, in fact. The basketball hasn't locked it in place. Now, imagine that there is a hockey puck on your floor which is immovable, and there is a small hoola hoop around it too. You are no longer allowed to move the hoola hoop in 3 dimensions, you are only allowed to slide it on the floor. In this situation, there is no way for you to move the hoola hoop so that it is not going around the hockey puck. This illustrates that any loop going around a "missing point" in the 3d sphere can be continuously contracted into a single point, while a loop going around a "missing point" in the 2d plane can only be continuously morphed onto some other loop that is still going around that missing point - we cannot morph it through a point that is not there. The purely mathematical result is that the 3d sphere with a missing point has a trivial fundamental group (all loops can be morphed into any other loop), while the 2d plane with a missing point has the integers as its fundamental group (any loop going in some direction around the missing point n times can be morphed into any other loop going around the missing point n times in the same direction). Since the 2d plane with a point removed and the 3d sphere with a point removed have different fundamental groups, they are not homotopy equivalent, and hence they are not homeomorphic. It is then a very short argument to conclude that the 2d plane and the 3d sphere are themselves not homeomorphic. Since they are not homeomorphic, the function I asked for cannot exist.

nice video! subscribed!

6:38, the solution: paint is composed of finitely small atoms (never mind the surface tension, just speaking from a zero-tension fluid), which cannot fit beyond a particular limit, whatever that limit may be, which will cut off (likely) terms around the millionth digit of pi.

Ayee good way of showing the stereographic projection. Lit video

Rly enjoy your vids m8

That’s genius. Since any 3D shape is analogous to an infinite series of two shapes stacked, then any volume of mathematic paint can cover an infinite 2d area

It's so interesting seeing the difference between when a mathematician discusses physical reality and when a physicist discusses maths. In each case, their speciality is kept foremost. An engineer would probably say that at the narrow end, the paint seals the end so it's all protected. No problem

See this object, finite value at 2 dimensions contains an infinite at 1 dimension: circle have finite area to one given R (although the precision is given by the π decimals). One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. The Integral from 0 to R of 2πr, gives exactly πR^2, but trying to see the integration process as that increasing swirl see the 1D line Length going to infinite. At Gabriel Horn is the same. 3D finite Contains a 2D infinite .

I do not see why it is a paradox. The point is that you can indeed paint an infinite area with "finite paint" if you allow the layer of paint to be "infinitely small high". This is what happens at the end of the horn.

Thank you

Can you give more info on how that equation maps every point on the sphere to every point in an xy plane?

The Majora's Mask music in the background is perfect for this video. It is a finite amount of time (3 days) repeated infinitely (*Song of Time* Dawn of the First Day)

I think your resolution is right but the mathematical paint idea isn't telling the whole story - it should really be concerned with the measure of the set, not the cardinality. For example if we only care about cardinality, any uncountable set, even the cantor set, could cover all of R^n for any n. But using measure preserving transformations that can't occur

I tutored a student on Gabriel's Horn at work, application of Improper Integrals.

The DK 64 music made me so nostalgic. ☺

The thing is that when you apply the paint, it can't have a finite width, in the volume exemple, there are "many" point where the paint as infinitesimal width. Even if the surface area is infinite, if the width is small enough, the volume of paint can be finite.

Thanks to this vid, I think I have a much better intuition for the Banach-Tarski Paradox. Sure, with a physical sphere you couldn't take it apart and make two new ones, but with a mathematical sphere with infinitely many new points to take out of the first sphere, it makes sense that we could make a second one. Does that make sense?

The last excersise is pretty hard. I am pretty sure you need to know about fundamental groups in order to do it (invariance of domain).

(1,1) should be equidistant from both axis. Just an observation.

Do we need another pi units of volume of paint to cover the other side of the horn?

Hey! That's what I said - I'll just fill it with paint and paint it that way. Anyway, I may have a different solution. Once we know the volume, which is finite, why not just use the volume to extrapolate the surface?

You don't even need a function to paint a sphere on to a plane. If you are using a continuous substance, then its very nature means that there is no thickness that cannot be cut in half, therefore the surface area it covers can be doubled infinitely, and thus it can cover a surface of infinite area. Though, I always like alternative solutions, and find that is like seeing different pictures of the same object from different angles, and it fills out my conception of it.

Conversely, does a finite surface area have an infinite volume? I guess no. But I don't know how to prove it.

The localization of the volume is distributed into infinite half-plane, nevertheless it is finite.

Infinite surface area = finite volume. Paradox solved.

I wished Taco Bell served chocolate milk.

A fractal can have infinite perimeter but finite area(sorry for the english)

Do you write backwards on your board?

Is this equivalent to the ultraviolet catastrophe?

Take 3!

What if we use dark matter paint? maybe dm it’s not quantised after all....

I'm not seeing something here. At 9:43 the function seems undefined for say, (x,y,z) = (1,0,0). When you work it out, -ln(1-sqrt(1^2 + 0^2 + 0^2)) = -ln(1-1) = -ln(0) = undefined. I can't seem to get that function to work. What am I missing? Never mind you said open sphere.

Is that a Zelda song?

What’s the intuition behind the surface area? Why can’t you just add all the circumferences?

Noice

Isn't part of the explanation for being able to cover an infinite plane with a finite volume of mathematical paint simply that such hypothetical paint can be applied in an infinitely thin layer and can therefore be infinitely stretched, so-to-speak?

Physical laws aside... To cover an infinite surface requires infinitessimally divisible paint. Include physical laws: a-takes infinite time to reach the end. b- the end will be narrower than the paint atoms and simply block the horn at some point. But hypothetically/mathematically infinitessimals can produce infinities of finite numbers! There are infinite number of infinitessimals in the number 1, or any finite number. But physically it cannot exist infinitessimally thin, or 2 dimensionally. Just factor in that paint is 3d and surface area is 2d and the paradox is resolved. If the paint is just mathematically hypothetical, then its no more paradoxical than the horn in the first place, in fact its just a liquid interior of the solid-the same shape almost, and no different to continually adding a 9 to 9.99999 and never reaching 10, subsequently smaller fractions will converge to a finite number after infinite summation

from 3d to 2d to paint the inside. Surface includes the outside. Not so simple on the outside

I would say, an infinite amount of paint is not needed Firstofall, We know that the Volume is finite, therefore, we can fill it up with a finite amount of paint. Secondly, If we say that the layer of paint has a thickness of 0. This leads to a volume of paint of 0*infinity. That's not good but, Thirdly, If the layer of paint has a thickness greater than 0, at some point on we wouldreach a point where the Layer of Paint is wider than Gabriel's Horn. If we go from that point towards infinity, the Horn is filled with paint, and that is a finite amount of paint. If we we go from that point toward 1 we can calculate the the Volume of the Paint on that Part of the Horn, which is also finite, Therefore The amount of Paint is finite. Maybe with an limit it is possible to figure out the amount of paint if the thickness is 0

Problem, the physical solution assumes the horn is also continuous, which in reality it isn't, so the paradox is solvable in reality as the paradox doesn't exist.

But is pi finite?

but how can you say that pi is finite? I mean its a non terminating non-recurring decimal

8:48 thats the xz plane

Gabriel's horn also known as H*L where H is pi/x and L is x-1 and since for all finite x we can know the surface area up to x however the problem with Gabriel's horn is that it's L is infinite. However the volume of said object is finite and will be infinitely close to pi even if cut short by an infinite length. Hence the real problem is the infinitude of one of the properties of the object. Or in layman's terms, The object is too gosh darn long. Or in engineering terms, non-constructable. Physics terms, Impossible. In my terms. Illogical and purely mathematical. Yes it's illogical, however not mathematically but physically.

Gabriels horn cannot exist in the real world because you would need an infinte amount of atoms even if the material was 1 atom thick. The real world has a finite amount of atoms and therefore there is no paradox.

The channel seems interesting but your description of paint as a “continuous substance” in minute 8 is problematic for me: if you’re willing to accept that any bijection preserves the amount of paint, you could duplicate your paint easily, since any ball in the Euclidean space is in bijection with a disjoint union of two balls of the same radius. This even tells you that it is not a sensible definition of “amount of paint”. Any reasonable “continuous nature” of paint would be expressed in terms of (something like) measure theory, which in this case doesn’t seem to be quite appropriate, since you’re trying to compare (measurable) sets of different dimension (the horn and its interior).

This is no paradox and have never been a paradox. Yes, the volume is finite and the surface infinite. But, as the surface has 0 thickness, the surface has no volume.

Not true ln(inf.) =inf. is when integral of unit hyperbola 1/x is from (0, infinity) ln(infinite/0)=ifinite, when unit hyperbola 1/x is from (1, infinity), is finite answer example ln(google) :) ln(10^100)=230,25, x=10^100, is very small answer of natural logarithm.

It's not much of a paradox if one tries to compute the numerical value of pi (which is in itself an infinite process). What I mean is since pi is a non-terminating decimal, one will never achieve a perfect pi amount of anything, let alone paint. This means the volume only seems finite when one has a finite approximation of pi, otherwise the volume is computationally infinite, just as the surface area. The paradox, it seems, feeds on a misconception one may have about the size of pi and the labeling of 'finite.'

You say your drawing looks pretty bad? I say i'd kill for your artistic tallent. I'm seriously jalous of it.

As Pi is irrational it's also infinite. Thus isn't the volume also infinite as well?

But pi isn't a finite value.

Stop tryign so hard to flex all the time.

Gabriel’s Horn Paradox - I think that this is utter nonsense and a cheap trick to justify what can only be thought an error in the architecture of the mathematics which try to address the relevant aspects of the proposition. Neither the math nor the scheme it defines can be devoid of logic or it is meaningless, which it appears this is. Consider…IF the horn as defined by the rotation of the line on the graph is actually possessive of infinite surface area then by definition, it would be on the inner surface as well as the outer surface. I don’t care what mathematical machinations one might bring to bear, if the outside surface is infinite then the inside surface must also be. IF this is so, the volume cannot be finite, by virtue of the very nature of physical existence. If you claim the horn to be finite in volume, you do so by the expelling of logic from the scheme and by some manipulation of formulae which by definition function on some error. You cannot have it both ways. Claiming that the math proves it yet in doing so, violates the physics which are the product of that very brand of mathematics is a grotesque contradiction and far more significant in a final analysis than this sophomoric paradox.