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

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*Creator*
Jade Tan-Holmes

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Sheila Miller and Jade Tan-Holmes

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Daniel Kouts

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Junior Arruda

[mayaral]

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

[hoptanglishalive]

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

[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.

[vari]

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.

[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.

[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!

[davidcroft]

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

[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.

[dr.hoover]

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.

[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.

[dru]

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

[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

[marksainsbury]

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 ...

[dotHTM]

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!

[Think_Inc]

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

[ryanfriedrich]

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.

[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".

[christopherhernandez]

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.

[cybr._.]

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

[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?)