# Thread: It can be paint or not

1. ## It can be paint or not

This one, I saw it a few time, so, probably you did also.

I am just curious to see what someone that know more than I do about math can tell me about this.

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If I take a square box with thin wall and with the length of it side =1, The area of one side is 1 and the volume is 1
If on top of the box, I put another box with the length of its side = 1/sqrt(2) , the area of one side is 1/2 and its volume is 1/(2sqrt(2))
If on top of the box, I put another box with the length of its side = 1/sqrt(3) , the area of one side is 1/2 and its volume is 1/(3sqrt(3))

And if I keep doing this to the infinite.

the total area will be 1+1/2+1/3+1/4+1/5+ .....

Since this diverge, the total area will be infinite. To paint it, I will then need an infinite amount of paint

BUT

Its volume is 1+ 1/(2sqrt(2)) + 1/(3sqrt(3)) + 1/(4sqrt(4)) + 1/5sqrt(5)) + ....

Since this converge to around 2.6123, therefore I can fill all the boxes with 2.6123 m3 (or CF) of paint.

How is that possible, since the paint inside the boxes is obviously painting the walls

2. ## Re: It can be paint or not

This is a classic conundrum, and it's just the way the math works out.

Standard example: Revolve y = 1/x, domain [1, infinity) about the x-axis. The surface area is infinite, but the volume is finite.

How is that possible? I dunno. It tells me to beware my biological/visceral-intuitions that combine "what paint can do" with notions of infinities and convergences. Instead, I trust more to my math-intuitions (limited though they be) when it comes to infinities and convergences. If you've some other perspective on how this seeming paradox should be understood, I'd be curious to hear it.

3. ## Re: It can be paint or not

I've been thinking at this a bit,

The amount of paint to paint the stack, is not a surface, but a volume ... I think that the answer is there

it T is the thickness of the coat of paint and if we adjust this thickness to be so that the volume of paint used to paint a box is less than the volume of the box,

therefore the volume of the paint needed will converge faster than the volume of the box ... as result that a finite quantity of paint will be required to paint the all thing

---- If this cannot be true, then it means that there is not enough paint in the volume of a box to paint the box ... therefore filling the boxes will not paint the walls ---

I think that make sens

4. ## Re: It can be paint or not

The volume of the box is decreasing much faster than the area.

What you said in your above post makes sense. Another way to see it is, when you paint, the thickness of the paint is not actually 0, but an arbitrarily small amount, T > 0. In order to be consistent in this scenario, the thickness needs to be the constant. However, as the volume of the box approaches 0, no matter how tiny you decided the thickness should be, the thickness of the box will surpass the volume, because T is constant but V --> 0.

5. ## Re: It can be paint or not

"If this cannot be true, then it means that there is not enough paint in the volume of a box to paint the box ... therefore filling the boxes will not paint the walls"
But this is what your example has shown is the case. So aren't you just restating the intutions that make this scenario so bizarre?
I mean, on any one of the boxes, as you said, we could make the thickness small enough so that it took less volume to paint than it did to fill. And since "in the limit" that thickness is zero, we're being more than generous in assigning any thickness to any box. So the volume required to paint can be made less than the volume required to fill for each box, but the sum of all of those painting-volumes ends up being more than the sum of all of those filling-volumes - or at least, "in the limit as thickness goes to zero" it does. Or so it seems.

Somehow jumping to the limiting case of zero thickness made "that which is always less" (painting-volume) diverge to infinity, but which "that which is always more" (filling-volume) converged to a finite value. Worse, the difference between them should grow as thickness shrinks, meaning, faster you shrink the thickness, the smaller the painting-volume is compared to the filling volume.

It's just damn bizarre.

6. ## Re: It can be paint or not

The thickness cannot change at all... otherwise the scenario is inconsistent. It'd be like switching from meters to cm.

7. ## Re: It can be paint or not

Why can't thickness change? Since my notion of "painting" ultimately has no sense of thickness, I don't see why I can't assign whatever positive thickness I like, different for each box, and still remain consistent with my intuition about what it means to paint a surface area.

8. ## Re: It can be paint or not

Because we are making the assumption, in our calculations, that the amount of paint required is directly proportional to the square of the side length... in that case the thickness must be a multiplying constant. Otherwise we can't equate paint needed with area, and the paradox falls entirely, because area would then be simply a mathematical abstraction. In order for area to represent anything physical, the thickness needs to stay the same.

If the painter can vary his thickness, then he indeed does only need a finite or convergent amount of paint.

9. ## Re: It can be paint or not

Well, a simpler way to resolve this paradox, is that, in order to let the side length approach 0, the painter HAS to vary his thickness. And if the painter can change his thickness, then the amount of paint required converges, because the terms would decrease faster than just the area of the box, owing to the lessening of the thickness.

You can think about it either way, but whichever assumption you make, it resolves the paradox, I think.

10. ## Re: It can be paint or not

If you use T constant, no matter as thin you make it, then the amount of paint to paint the boxes is

6T + 6T/2 + 6T/3 + 6T/4 .... (There is 6 sides on a box)

Which diverge ...

The problem with this is that the volume of a box gets smaller than the amount of paint required to paint a box (No matter how thin is the coat). Therefore filling the volume of the box with paint don't paint it !!!!!!

Except if:

Using a constant thin coat where the thickness would be at the limit so 0. and since the surface is infinite then the volume of paint require would be :
( 0 x infinite)

which don't exclude the possibility of a finite quantity of paint ..

But is it the right way to look at this ???

11. ## Re: It can be paint or not

If the thickness CAN change, then the series which represents the required paint converges. So no contradiction.

If the thickness CANNOT change, then the box-maker will have to stop making boxes once the side length is supposed to be lower than the thickness (which it will be eventually.) So because the process can't continue with physical sense, there's no contradiction either.

12. ## Re: It can be paint or not

If the thickness don't change, (except if the thickness is 0) then at the limit when the size of the box get to zero, then the thickness of the coat is infinitely larger than the box.

This is not to me the definition of painting ... Painting should remain the process of putting a thin coat of paint on an object,...Not to drown a tiny object in an ocean of paint

The logical process is to change the thickness.

And of course, if the coat is constant, and the thickness becomes infinity larger than the object, ... it takes an infinite amount of paint

13. ## Re: It can be paint or not

I think I see the explanation for part of this problem.

This is not what's going on:
"If the painter can vary his thickness, then he indeed does only need a finite or convergent amount of paint."

This is what's going on:
"If the painter uses ANY thickness, no matter how it varies, or how thin (yet positive), but THAT'S CONSTRAINED TO STAY INSIDE THE BOX, then he indeed does only need a finite or convergent amount of paint."

If we're just painting one side, then have these values (with hopefully their obvious meanings)

$s_n = \frac{1}{\sqrt{n}}, \ A_n = \frac{1}{n}, \ VBox_n = \frac{1}{n\sqrt{n}}, \ t_n > 0, \ VPaint_n = t_nA_n$

Here's the observation: to stay inside the box, the thickness must be less than the width of a side.

Thus $0 < t_n \le s_n$, so $0 < t_nA_n \le s_nA_n = VBox_n$, so $VPaint_n \le VBox_n$.

The consequence of that observation is that, on the "thickness that goes to zero" model, we'll get convergence no matter HOW small we make that thickness, or HOW much we vary it, so long as it's constrained to stay inside the box.

So the thickness paradigm, no matter how conceived, other than being positive yet the paint staying inside the box, always solves the paradox. You won't have a paradox if you treat surface area as a limiting case of a volume having a dimension go to zero. You must treat surface area as fundamentally different form volume, from the beginning, in order to even produce the paradox. However, the SOURCE of the paradox is our "but surface area is just volume with a really really thin thickness" - and that's exactly not the case if there's going to even be paradox. *IF* we're allowed to pretend that's the case, then there is no paradox mathematically (amount of paint used converges). *IF* we try to deny that, then we no longer have a credible intuition to appeal to regarding volumes, areas, and paint, so there is no intuitional paradox. Thus the problem dies either way!

14. ## Re: It can be paint or not

Just to throw in my 2 cents. After a certain point it would not be physically possible to "paint" the surface of the box. How could one put pain on a box that is smaller than an electron. This is not possible on the atomic level. Note this also keeps the paint outside of the box, it can't be filled, because the paint is too big.

15. ## Re: It can be paint or not

To me, there is no paradox, since it is always possible to paint a box with a volume of paint smaller than the volume of the box.

What makes it appears to be paradoxical, is that someone see it with a constant paint coat ... and this is not the question, the question is is it possible to paint a wall with an amount smaller or equal to the volume of the box ... and the answer is yes ... and therefore a finite amount of paint can be used to paint the stack

By the way, ... How do you get those math symbols to show ... That would be usefull to know

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