A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: The Infinite Vuvuzela Paradox

Mike_Bson wrote:This is the shape about which, OP, you are talking? If so (if not, then correct me), I agree the surface is infinite, but how is the volume? It just keeps going up, getting infinitely tall, and it gets infinitely wide at the bottom.

If you take that shape and revolve it around the x axis, and then only look at the part with x>=1, you have the correct shape. A better page to look at is this one Gabriel's Horn.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Mike_Bson
Posts: 252
Joined: Mon Jul 12, 2010 12:00 pm UTC

### Re: The Infinite Vuvuzela Paradox

jestingrabbit wrote:
Mike_Bson wrote:This is the shape about which, OP, you are talking? If so (if not, then correct me), I agree the surface is infinite, but how is the volume? It just keeps going up, getting infinitely tall, and it gets infinitely wide at the bottom.

If you take that shape and revolve it around the x axis, and then only look at the part with x>=1, you have the correct shape. A better page to look at is this one Gabriel's Horn.

That's pretty funny. Thinking intuitively, the volume would be infinite (the end never actually closes up), but the formula shows otherwise. Weird.

Now, when it says it would be pi units in volume (one unit being one little cube on the graph), wouldn't that be a little small?

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: The Infinite Vuvuzela Paradox

Mike_Bson wrote:wouldn't that be a little small?

Its not just the two dimensional area of the graph, its the three dimensional volume of the solid of revolution. The area is in fact infinite.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

SlyReaper
inflatable
Posts: 8015
Joined: Mon Dec 31, 2007 11:09 pm UTC
Location: Bristol, Old Blighty

### Re: The Infinite Vuvuzela Paradox

I just noticed the timestamp on the OP. I think this could possibly be the only mention of vuvuzelas on the Internet ever before this year.

What would Baron Harkonnen do?

ConMan
Shepherd's Pie?
Posts: 1690
Joined: Tue Jan 01, 2008 11:56 am UTC
Location: Beacon Alpha

### Re: The Infinite Vuvuzela Paradox

SlyReaper wrote:I just noticed the timestamp on the OP. I think this could possibly be the only mention of vuvuzelas on the Internet ever before this year.

I noticed that, too, and suspect that may even be why the thread was necro'd. I have no proof of this, unfortunately.
pollywog wrote:
Wikihow wrote:* Smile a lot! Give a gay girl a knowing "Hey, I'm a lesbian too!" smile.
I want to learn this smile, perfect it, and then go around smiling at lesbians and freaking them out.

makc
Posts: 181
Joined: Mon Nov 02, 2009 12:26 pm UTC

### Re: The Infinite Vuvuzela Paradox

Vesuvius wrote:However, with any depth at all, he's going to need infinite paint.
not at all, this is just difference of two volumes, each with slightly different radius, thus finite. of course, if he would keep painting at the same depth all along to +∞, _then_ it would be infinite.

Maquox
Posts: 3
Joined: Fri Jul 23, 2010 3:58 am UTC

### Re: The Infinite Vuvuzela Paradox

I hate how people break their heads with these questions, there is no paradox, you can't paint the "tip" of the vuvuzela as it not only doesn't exist, but is also smaller than an atom, thus smaller than the smallest molecule of paint, and even without that limitation, it would take infinite time for paint to reach the "tip"...

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: The Infinite Vuvuzela Paradox

Its true that its not a real world situation, and nor is it particularly paradoxical that surface area and volume aren't related in any obvious way, but I don't think its wrong for people to try and think out what the mathematics is saying.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

tastelikecoke
Posts: 1208
Joined: Mon Feb 01, 2010 7:58 am UTC
Location: Antipode of Brazil
Contact:

### Re: The Infinite Vuvuzela Paradox

ConMan wrote:
SlyReaper wrote:I just noticed the timestamp on the OP. I think this could possibly be the only mention of vuvuzelas on the Internet ever before this year.

I noticed that, too, and suspect that may even be why the thread was necro'd. I have no proof of this, unfortunately.

I proved your suspicion. I found it, and decided to revive it. Now I'm reviving it again.

thoughtfully
Posts: 2253
Joined: Thu Nov 01, 2007 12:25 am UTC
Location: Minneapolis, MN
Contact:

### Re: The Infinite Vuvuzela Paradox

SlyReaper wrote:I just noticed the timestamp on the OP. I think this could possibly be the only mention of vuvuzelas on the Internet ever before this year.

Vuvuzelas are big now. They're on Slashdot!

Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.
-- Antoine de Saint-Exupery

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:

### Re: The Infinite Vuvuzela Paradox

thoughtfully wrote:Vuvuzelas are big now.

But are they infinitely big?

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

thoughtfully
Posts: 2253
Joined: Thu Nov 01, 2007 12:25 am UTC
Location: Minneapolis, MN
Contact:

### Re: The Infinite Vuvuzela Paradox

phlip wrote:
thoughtfully wrote:Vuvuzelas are big now.

But are they infinitely big?

Not in surface area.

Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.
-- Antoine de Saint-Exupery

HungryHobo
Posts: 1708
Joined: Wed Oct 20, 2010 9:01 am UTC

### Re: The Infinite Vuvuzela Paradox

I'm guessing that we are assuming paint particles are infinitesimal and don't clump together

as with all things infinity leave common sense at the door.

talking about litterally infinitesimal particles leads to madness,. example: how many such paint particles would it take to fill a 10 litre bucket?

An infinite amount.

no matter if you kept pouring more and more into your bucket it would remain almost totally empty.

It doesn't even really make much sense to talk about volumes of these paint particles at all, how many are in an single cm^3, how about in a meter cubed, what's stopping however many are in a meter cubed from being squished down to a cm cubed?

Forget anything about painting an infinite surface area with these particles, filling a finite recular bucket would take an infinite amount of these paint particles.

So no real paradox, if you talk about infinitesimal particles of paint then talking about it's volume makes no sense.
If you're not talking about infinitesimal paint particles then the surface area they can reach and thus paint is finite.
Give a man a fish, he owes you one fish. Teach a man to fish, you give up your monopoly on fisheries.

sledDog
Posts: 3
Joined: Wed Dec 22, 2010 3:10 am UTC

### Re: The Infinite Vuvuzela Paradox

The volume is finite but it's not definite. You can't tack on a set volume because it will always get a little more, further down the line, but you can say it will never be greater than V (too lazy to do the integration) because the increments become consistently smaller in proportion to the width of the increments and approach 0. It has infinite surface area because as x approaches infinity, the increments decrease in such a way that they do not approach 0 but rather 1. Take a vase with a similar shape and set on it's small end and slice it parallel to the ground with thickness super-dooper-small. The volume of each slice from top to bottom approaches 0. If the vase extended for infinity, each slice would be smaller than the last and eventually be 0. Now take a tailor's measuring tape and measure the length of material in the vase to go down distance super-dooper-small. Each segment is smaller than the last but it doesn't approach length=0, it approaches length=super-dooper-small. If it extended to infinity, each segment would be a little less larger than super-dooper small but it will never be less than the width of the interval (unless you know a shorter distance between two points than a straight line). I can see the paradox but think of it this way: way down there at infinity, the vuvuzela is closed. But no matter how far you go, there will be more of it to paint.

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:

### Re: The Infinite Vuvuzela Paradox

That's... wrong, in almost every respect.

(a) The volume is definite. It isn't "always getting bigger" - it's a specific shape, it's not changing. The only way its volume could change is if it changed shape. But it's just a static shape of infinite length. Not (as you seem to be misunderstanding) a changing shape of finite but increasing length. No, it's already infinite, it's not growing. You may be getting confused by the fact that, to calculate the volume and surface area of the infinite shape, we look at the finite-but-growing shapes, and take the limit as it grows to infinity. However, even though the finite-but-growing shapes merely approach the final volume V in the limit, the infinite shape's volme equals V (and, since the shape is static, doesn't change, so it can't be said to "approach" anything).
(b) When we look at the finite-but-growing shapes (which, again, we take the limit of in order to calculate the volume/surface area of the original shape), the volume/SA added by each additional "slice" approaches 0 for both the volume and the surface area. It definitely does not approach 1 for the SA (it's approximately a cylinder, which is 2*pi*r*h, h is a constant and r goes to 0, so the whole thing goes to 0). The statements "goes to 0" and "goes to a veryvery small number" which you try to draw a distinction between are equivalent. If the limit of a sequence is an infinitesimal, then the limit is zero. The only difference is that if you claim the limit is infinitesimal then it's evidence you're confused about something. Both the surface area and volume of each slice goes to 0, it's just that the SA goes down by 1/x, and the volume goes down by 1/x2.
(c) The limit of the length of each slice is 1, but that's irrelevant, because we're not trying to find the length of the shape. We already know that's infinite.
(d) The "no matter how far down you go, there's always more to paint" line is also irrelevant - no matter how far down you go there's always more volume, too, yet the volume is still finite. So this line proves nothing.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

RonWessels
Posts: 43
Joined: Tue Aug 24, 2010 8:40 pm UTC
Location: Mississauga, ON, CA

### Re: The Infinite Vuvuzela Paradox

@phlip, I have to call "wrong" on you as well.

phlip wrote:... the infinite shape's volume equals V (and, since the shape is static, doesn't change, so it can't be said to "approach" anything).

Ignoring possible grammatical issues, you cannot talk about "the infinite shape's volume", since the infinite shape cannot exist. That is the problem with infinities - they are not a number that can be represented in the real world. They are a shorthand notation for limits as something gets larger and larger. Speaking about "the infinite shape" must therefore be a shorthand for a discussion of the limiting properties (ie. what the shape "approaches") as the length gets longer and longer.

phlip
Restorer of Worlds
Posts: 7572
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
Contact:

### Re: The Infinite Vuvuzela Paradox

RonWessels wrote:you cannot talk about "the infinite shape's volume", since the infinite shape cannot exist

Sure it does, and sure I can. Let S = {(x,y,z) in R3 : x2 + y2 <= 1/z2 and z >= 1}. This set is an infinite (by which I mean "unbounded; of infinite length") shape. The volume of S is (NB: not "approaches") pi.

No, you couldn't build this in real life, but we're not talking about real life, we're talking about mathematics. If your contention is that we can't talk about anything that's unbounded in size, then we can't talk about R3 at all. Or, hell, even R.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

xkcdfan
Posts: 140
Joined: Wed Jul 30, 2008 5:10 am UTC

### Re: The Infinite Vuvuzela Paradox

RonWessels wrote:@phlip, I have to call "wrong" on you as well.

phlip wrote:... the infinite shape's volume equals V (and, since the shape is static, doesn't change, so it can't be said to "approach" anything).

Ignoring possible grammatical issues, you cannot talk about "the infinite shape's volume", since the infinite shape cannot exist. That is the problem with infinities - they are not a number that can be represented in the real world. They are a shorthand notation for limits as something gets larger and larger. Speaking about "the infinite shape" must therefore be a shorthand for a discussion of the limiting properties (ie. what the shape "approaches") as the length gets longer and longer.

Do you not understand what math is? What the fuck are you doing here?

Do you not understand what civility is? -jr

redscourge
Posts: 3
Joined: Wed Jan 12, 2011 7:25 am UTC

### Re: The Infinite Vuvuzela Paradox

Correct me if I'm wrong, but can't you do something like this to approximate the surface area of this shape:

let f(x) be a the function of the surface area of this shape, and e(x) be a function known to have slightly less surface area, and g(x) be a function known to have slightly more surface area, where e(x) and g(x) are both convergent. So, e(x) < f(x) < g(x). Now, can't you just reduce the difference between e(x) and g(x) so that they approach zero, and then use that to find the surface area of f(x) even though f(x) is not convergent?

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: The Infinite Vuvuzela Paradox

redscourge wrote:Correct me if I'm wrong, but can't you do something like this to approximate the surface area of this shape:

let f(x) be a the function of the surface area of this shape, and e(x) be a function known to have slightly less surface area, and g(x) be a function known to have slightly more surface area, where e(x) and g(x) are both convergent. So, e(x) < f(x) < g(x). Now, can't you just reduce the difference between e(x) and g(x) so that they approach zero, and then use that to find the surface area of f(x) even though f(x) is not convergent?

The problem with that is that any figure that contains this one will also not have convergent surface area.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.