Infinite gift and the painter’s
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- เผยแพร่เมื่อ 30 ก.ย. 2024
- In this animation, we show an approximation of the mathematical object colloquially referred to as the "infinite gift." This gift is somewhat paradoxical in nature because it is infinitely long and requires an infinite amount of wrapping paper to cover, yet it only encloses a finite area. This object is a discrete analog of the more famous "Gabriel's horn," which is an object studied in most integral calculus classes as it is an object enclosing finite volume but has an infinite surface area.
One major purpose of this video is to investigate the different, but related, infinite sums of 1/n and 1/sqrt(n cubed). These sums both have interpretations in terms of the infinite gift, and one of them diverges while the other converges.
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Here is a slower, wide format, high definition alternate version of this video:
• Visualizing the Infini...
For related information, I would begin with the Wikipedia article about Gabriel's horn and the painter's paradox: en.wikipedia.o...
Here is one of the first places I remember seeing this object (there are earlier ones, but I can't track them down): fe...
#manim #infinitegift #paintersparadox #gabrielshorn #harmonicseries #infiniteseries #convergence #divergence
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And thinking about how they would all fit inside of each other
They all fit inside eachother? A bigger cube fits inside a smaller cube? Sentence unclear.
@@larkohiya Picture every cube having an open top and turning gravity on.
Wait, is this a proof that an infinite area can be "folded in" to fit into a finite volume?
That is even not that difficult to construct or imagine such case. A bit more difficult is to construct a surface that fits inside a qube and "touches" every point inside. That is, a dense embedding of 2d space into 3d.
@@serhiidyshko So it is trivial to fit infinitely many surfaces in a finite 3d space. However, there isn't a fundamental difference in the "strength" of "2d paint" (for painting the surfaces) vs a "3d paint" (for filling the volume), as you could fill a volume by painting a properly folded infinitite surface?
The stack's height is infinite, visible surface area is infinite, but the volume is finite. Math is fascinating
Right?
that's really weird. if the height was infinite you would think the volume would be the same.
Oh, please tell me how the volume is finite. I'd absolutely love to know.
@Oleg_K. basically, convergence.(duh)
There is a convergence test which is called a p-series test that is applicable to series whose terms are in the form 1/n^p, and the series converges if and only if p > 1.
Now in our case, nth cube's height is 1^sqrt(n) or 1/n^0.5, so the height of the stack equals the sum from 1 to infinity of 1/n^0.5, which diverges to infinity by the p-series test.
The surface area of the nth cube is no smaller than 4 * (1/sqrt(n))^2 or 4/n, the surface area of the stack is greater than the sum from 1 to infinity of 4/n, which is also 4 times the harmonic series, which also diverges to infinity, and thus the stack's surface area also diverges to infinity.
The volume of the nth cube is (1/sqrt(n))^3 or 1/n^1.5, the stack's volume is equal to the sum from 1 to infinity of of 1/n^1.5, which converges by the p-series test.
@@JordHaj Wow. This felt incredibly counterintuitive at first. Thanks for the elaboration.
"It's a cake you can eat, but not frost."
-Vsauce
Omg i remember that video. He basically described in detail the concept of an infinite series from AP Calculus BC, but never mentioning the term “infinite series” by name
Grandma: "What do you want for Christmas?"
Me: "Something small. Nothing too special"
Grandma:
someone needs. to make a satisfying render of these cubes with open tops falling into each other
It would never end
@@AverageBishop- good
@@SirNobleIZHbut that means it would never render
@@xaigamer3129 loop the video and have a seamless transition
they could render a looping version of it where it zooms in at some point@@xaigamer3129
Volume is zeta(1.5) in case anyone was wondering.
what's zeta?
@@proton.. Riemann zeta function.
The painter's paradiox is one of my favorite and fascinantes me the most ! I can't handle how the surface isn't "contained" in the volume... Cool visualization !
_"... how the surface is (not) contained in the volume ..."_
Once you understand that the hypothetical color you use to paint the surfaces must be "infinitesimally thin" (so to speak - as area has only two dimensions) you'll find that you can put an infinite amount of such a color in a 3D space as each "layer" has zero thickness and therefore an infinite amount of "2D color" will fit in a "3D color can".
@@mittelwelle_531_khz I'm not sure to have understood your answer ^^' To be clearer, what I struggle to understand is why one can not take the limit. I explain : imagine you create a smaller cube inside each cube. Lets note "d" the distance between each smaller cube and each original cube (this distance obviously decreases as the side length of the big cube decreases). Now you put paint in between the two cubes. It's quite clear that the resulting volume is smaller than the original volume, so it is finite. But as "d" goes to zero, this volume goes also to zero (well) but I have the feeling that this volume should "converge" to the area...
In fact, by writing the comment I realized that what I say is equivalent to pretending that the area of a rectangle "converges" to its side length as its height converges to zero... And it's not true obviously but goes against some intuition I think...
@@mehdimabed4125 I see your approach, so maybe let's start with something more general:
There are sums of elements getting smaller and smaller for which the upper limit still is infinite. Or more mathematically: if you claim it will never grow above a value you tell me the I can tell you how many summands you have to add to exceed that number.
One sum of this kind is
1 + 1/2 + 1/3 + 1/4 ... etc.
If you add enough terms, though each one is smaller than the one before, you can reach each number you like. It may not be "intuitive" at first case but if you want I can give you the full mathematical prove.
Summing up all the surface areas is a sum of this kind. (The multiplier 4 doesn't change that - if you can reach any number N you can also reach 4 times that number N - you may need to add many many more terms than just four times the terms you had to add to reach N, but you definitely can reach 4×N too.
Is it already here where your problem starts, that an infinite number of smaller and smaller summands can grow beyond any limit? Do you want me to show you how this can be proven?
The growth of volume falls into another category, the one where the summands can never grow above a fixed value. It's not exactly what we have in the example adding the ever smaller volumes, but such a sum eg. Is this:
1 + 1/2 + 1/4 + 1/8 ...
Ie. each summand is half of the one before. It's relatively easy to see that the above can never exceed 2 because what you add in EACH step is just HALF of that what is still missing to reach 2.
The volume sum falls into this category too though it adds up to a little bit more than 2.
There is a Numberphile video about Ptolemy's Theorem. They explain the concept of inversion there and they don't explicitly talk about it, but they show that via an operation called inversion every point on a line with infinite length can be mapped to a circle with a finite circumference, which still is absolutely mindblowing to me. It is quite long and not directly about this concept of infinities in finite spaces, but if you have the time, I can recommend it.
I heard a great solution to the painters paradox. It was that it's trying to combine 2 units that are inherently different. Surface area is not volume. It's like asking how many meters are in 1 liter of water. Because we all know that paint has a thickness, but when we look at a wall, it doesn't feel like the paint has thickness, the painters' paradox persists.
Said more succinctly, any infinitesimally small volume of paint is enough paint to cover an infinite area.
@@ObjectsInMotionyour comment just made it click for me, thanks.
Was about to write a comment about the area equation missing an additional area that is left uncovered by each successive cube (would just be (1/(root1))^2 which is just 1, as it would just be the view from above at infinity) but then remembered the series diverges so it's a moot point anyway. Great video!
I want to be a mathematician and physicist. I love you bro ❤️ u help me to understand from the core
I hope you achieve your potential. God blessed me by directing me into mathematics. When I was younger my focus was science. I would not have thought I could have success in mathematics. Cheerful Calculations. 🧮
the volume is actually Zeta(3/2) where Zeta is the Riemann Zeta function. also reminds me of Gabriel's Horn
That’s a weird concept. You can’t paint the outside because it has infinite area, but you can paint the inside because it has finite volume.
The surface area is the same on the inside and outside. You can paint an infinite area with any amount of paint, as the 2d surface has 0 thickness.
@@TomGalonskayeah just thought about pouring paint into the boxed cubes - that's possible. Just because the area is "infinite" doesn't mean it's impossible to paint. Think about the coast line paradox, the outline seem infinite, if you zoom in, which look like fractals, but obviously the outline of the island is finite.
Isnt it beacause Volume is infinite layers of 2d?
By "1/√1" do you mean '1'?
Well defined volume with unlimited surface area lmaoo
I think you could paint it if the thickness of paint you use decreases. That's basically what happens when you fill the volume with paint, the thickness of paint is reducing.
You cannot paint an infinitely long string
@@egg5802not with that attitude
Is this like a 3D version of how the perimeter of an island can technically go to infinity as you decrease the minimum measurement length, but the area remains finite?
Yes :)
In this case,,the outside is bigger than the inside……except volumes and areas are different animals
Sir, we have enough paint to fill the entirety of your home, but definitely not enough to paint your infinite walls.
Mathematician: "The surface area is greater than or equal to this sum."
Me: "I think this thing is strictly greater than."
Mathematician: "Yes. We are correct."
Surface area can't be greater than volume, because they have different units.
@@carultchYou can compare their magnitudes, as long as we agree on the measuring stick. A cube with side length 2 meters will have an area of 24m^2 and a volume of 8m^3. If we agree that the measuring stick is a meter, then 24 > 8 and the surface area has greater magnitude. However, if we change the measuring stick to a centimeter, then the area will be 240 000 cm^2 and the volume will be 8000 000 cm^3 so the volume will have greater magnitude.
Easy just fill the inside and you will only use a small amount of paint to cover up all of the surface area (from the inside)
Its a joke please noone try this
I know its impossible to have an infinite tower but still dont try
:)
I hate the painter's paradox so much, like 'i will have one infinitely long horn please'
Oh yes ... mhm ... totally. Yeah I don't get it, but it's probably really cool
Basically, u can fill up the inside of the infinite gift with paint, but can never paint its outer surface even with as many finitely many refills as u like - mind boggling
Would I be able to ask what the painters paradox is? I haven't been to a math class in a hot minute lol
The short blurb at the end tried to hint at it - in theory you can’t paint this object on the outside because the surface area is infinite. But the volume is finite so you could fill the object with paint.
You can learn more about it by looking up the wiki article on Gabriel’s Horn.
Think like this:
You have a big book of 500 pages and its volume is 1 liter and its surface is 500 cm^2. If you want to paint only the surface of this book you will need a little paint. Now if you want to paint all the 500 pages of this book you will need to buy much more paint.
This is to see that a book with a specific volume (for example 1 liter, 1000 cubic cm) can have very small surface whet it is closed or can create a very big area when the pages are opened and spread on a desk. The volume of the book doesn't change but its area can change.
On this video of infinite boxes the volume of all infinite boxes converges, which means that it is not an infinite volume and so we could fill all this volume with paint. But the whole surfaces of all these infinite boxes is an infinite area and so no matter how much paint you have you can't paint an infinite area. Mathematically it is proven that the surface of these boxes "diverges", which means that it goes to infinity. And you can't have infinite quantity of paint to paint an infinite area.
So the paradox is that theoretically a specific volume (let's say 1 liter) can create an infinite surface.
Practically you can increase the surface of a book cutting its pages and spreading them on a desk. For example radiators are made with big surface to transmit heat from surface of radiator to the air faster. Of course practically you can't have a book with unlimited pages because the thickness of pages can't be unlimited small. So in nature you can increase the surface of objects very much, but not indefinitely.
Oh, I understand now, thank you for that
@@charoslian2461very good explanation, i can use this
Удивительно
Площадь больше чем объём
Парадокс
Imagining you're inside this structure, you'd see the volume shrinking til it becomes a Zero Dimension point. Outside of it you can see the surface approaching a 1D line
Does it work the same way as fractals in how they have finite volume but infinite perimeter?
Same!
i love infinity :3
i hate it
reminds me of koch snowflake. finite area, infinite perimeter
Wile the volume is finite and the surface is infinite, the statement that the finite volume of paint cannot cover infinite surface is incorrect, as comparing apples with peaches. Mathematical surface has zero thickness, hence finite volume can "cover" infinite surface.
Exactly there is no paradox at work here, problem lies in the fundamentals, you just need an infinitesimally small volume of paint, to cover an infinite surface
You can put every gift n inside n-1 gift
OK class, quiz tomorrow.
300th comment
Very trippy
So. You mean that it has an infinite area but finite volume? So you could fill it with paint but you can't paint the surface. That's just crazy man
You could paint the surface, you just have to spread the paint thinner and thinner, which it already is doing anyway for it to be inside the shape.
What about the hight? Does it converge?
That diverges too. Didn’t have time to fit it here but it’s in the video linked in description.
The height is the sum of 1/sqrt(n) which divereges by the p-series test.
I find it similar to the Gabriel's Horn.
UGH I HATE IT... but I love it at the same time
Wouldnt you have "painted" all the cubes, if you poored the volume ammount of paint over the cubes?
That’s why it’s a paradox :)
Shouldn't the Area have a +1 at the end as you can still look at it from a top-down view.
Maybe a +2, that is why I used >= instead of equal for the area though... didn't need exact computation, just that it is larger than an infinite sum :)
Infinite wrapping paper, finite gift
It's a square Gabriel's horn
Thank you,sir
That's counterintuitive, a large cube obviously holds enough paint to paint its outside... until you realise that when the cube become small, the volume of paint contained is not enough to paint its outside, as cube area/volume -> infinity.
I love when these are something I actually know. I was able to pause and figure out the convergence/divergence before you said it! Makes me feel like a total genius 😅
👍😀
TH-cam shorts continues with making its loops ever more subtle.
As the stack started to form, I was reminded of a Vsauce video I watched. Then you mentioned Gabriel's horn, and I was happy that I remembered 😊
So if you filled it up with a paint the whole inner surface would painted because it’s full of paint but if you just painted the inner surface you’d need infinite paint
Wat.
I love this.
Why is there 4 insead of 6? 6×a² = area of cube. And there is visible top of cube.
Well not all sides of cube are visible. So I just used the four sides and said greater than.
@@MathVisualProofs oh. I knew that not all sides are visible, but that top made me confused. Thanks
What about the faces that are covered by the cube above it?
Yeah. Just ignores it because area already diverges. But bottom area is 1 and top area essentially equals 1 overall.
like gabriels horn
So it has a fixed area but an infinite surface area??
Yea
If we take the exponent e^(-x) and look at its graph on the interval (0; ∞), the area will be finite (exactly 1) but the length of the curve will obviously be infinite. This fact, in my opinion, is as surprising as the fact presented in the video, that is, very ordinary
Has some fractal vibes to it. A Koch curve has an infinite length, but a finite area, if you create an object by rotating the curve, it would also have infinite area and finite volume
Integraling the Volume gives a fraction, but when variable is a polynomial of n >= -1, it soars (When it's 1/x, you just get ln x, which diverges with higher x)
This breaks down at the molecular level of the paint. Also of the blocks, which have to be made of something. Once we reach the Planck scale all intuition is off, as far as practicalities are concerned. Sure, mathematically it holds up, but don't try to "paint" it, even conceptually.
This follows a similar conception as Gabriel horn.
It is made by stacking infinitesimal discs on a f(X)=R=1/X.
The area is infinite but the volume is finite.
Nothing special - like an infinite line which surrounds only a finite area.
S.th. special would be a finite surface which contains an infinite volume ...
Waiting for it: Herbert
Entra la niña llorando diciendo:
MAMA ME CONFUNDIERON CON LA HERMANA DE PHINEAS DE PHINEAS Y PHERB!!
A lo que la madre le dice:
¿Porqué?
La Niña glitcheandose en el suelo:
¡Talvez es porque so-
Gabriel's trumpet is a mathematical expression of a function that when rotated around the x axis creates a shape with infinite surface area, but definite volume. Hence the "Painter's paradox" when you could fill it easily, but never paint its surface.
Infinite gift of mathematical equations that work on paper but not in reality 😩
You lost me when you put the giant E up there
It’s actually a Greek ‘S’, and all it means is “add all these numbers together”. It stands for ‘summation’.
So... the surface area is always building, but volume will rarely go above 2.whatever because it'll eventually turn into a srting with a width less than the size of a molecule....
Cool, another Blender project that will break my computer.
Maybe not. Let’s see it!
@@MathVisualProofs Well, it's been 4 days and finally my computer turned back on.
Well if you fill it with paint did you not paint the entire surface area? Assuming the walls of the container are infinitely thin I suppose. I feel like this isn’t necessarily a paradox, it’s like a fractal. If you have a fractal with a finite area and infinite perimeter, obviously if you just dump a blob of paint on the entire thing, it covers the infinite perimeter. Real life items have “infinite” perimeters as well, cause you can always go into deeper detail to define the perimeter. Take a beach for example, how do you get the perimeter of a beach? Assuming the water is frozen in time, you could pick big landmarks to define a general perimeter, or you could follow every single grain of sand to get a much larger perimeter, or you could follow every atom in every grain of sand, or subatomic particle, or some other smaller than quantum measurement, to get a basically infinite perimeter. That doesn’t mean you can’t traverse or cover that infinite perimeter.
To quote Mr Spock, this is not logical. Let’s assume the outside of the structure has negligible thickness. This explanation means that you can paint the inside but you can’t paint the outside even though it has the same surface area.
theres a continuous version of this problem called Gabriel's horn
I love how mathematical calculations can be used to point out paradoxes but WHAT IS THE PURPOSE OF THAT I’M DUMB
I've always felt that Gabriel's horn demonstrates the operation is exceeding its domain of applicability.
That you can have a surface area that is greater than the volume it encloses is proof that the solution is nonsense.
surface area calc left out the horizontal surfaces. The bottom is 1 and the upper surfaces summed also equals 1. so the Area formula requires a + 2 constant term.
Just decided to do > since it’s a short :) and greater than infinity is still infinite.
The paradox here is that they are probably saying stuff that is sensible, but since I don't know math it sounds like gibberish and made up combinations of complcated words.
It seems like the theoretical box sides are also Infinitesimally thin
amazing that you every one of those boxes would fit inside the first one
But what happens if you make another structure (1/sqr(n)) * 1.5... then fill it with paint and dunk the original un paintable horn in it?
how to build tower of thinning layers
yay jtoh
Huh?! Why you tell us stories about gifts🎁? 😮 Every mathematician knows about "Gabriel's horn"🙄🤔...
I think that the infinite sum of the areas is infinitly big in area,but finite in volume. I think it,because a cube is a stack of areas,like a stack of cheese slices filling a volume,but with no hight.
Yeah it's not a paradox as paint still has three dimensions. You can't compare apples and oranges and go ooooh look a paradox. If paint only had two dimensions you couldn't fill the cubes.
- Have you started your homework yet?
- Almost! I just gotta stack these boxes first...
Something wrong because imagine the same structure and add side length 2 cube to the bottom, than all this structure will fit inside and there will be 8 units of space more which can be filled with paint in order to paint it
2.614 is approximately phi squared.
Is that relevant to this?
I’m not a mathematician, but I’m into phi.
Yes.
I don't see the paradox. Unless you consider Zeno's paradox a paradox. Ow right. There are no paradoxes in Math. You have to close one eye I suppose. 😅
Math. Not even once.
Wait, the area on the inside would also diverge right? But the volume converges AND connects to the entire area. So which one is it?
Make the boxes clear, fill it with paint, then pour it out boom
This is kind of like the coastline "paradox" where you can have an island with a finite area but if you try to measure the perimeter with more and more precision, you will keep getting a longer coastline, essentially an infinite perimeter.
So if you want to determine the world record for the island with the longest coastline, you first have to agree on the level of precision for measuring it.
Similar for sure!
The gift that keeps on giving
Turns out this part of math doesnt matter much. It an good to know, usless beyond specific applications.
Two different measures (I mean two different set functions) + thinking of them as one measure = magic.
I guess it's not really surprising. You can derive an infinite amount of are out of a volume, so it actually makes sense.
You didnt nclude the top area of the top most cube but it tends to 0 so no big difference
City planning by-laws though, amirite
Is it similar for 2 dimensions with squares and its area and perimeter
I see area would still diverge but it's perimeter might be finite
It's not really hard to imagine. Since a surface is infinitely thin, you can fit infinite surface area inside any finite volume.
Painter's Paradox more like Painter's Nightmare
Discretizes Gabriel’s horn
Nvm watched till the end and realized you mentioned it
Same as Torricelli's Trumpet right?