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[Realmfighter]

This contest will be so much fun.

[Christes]

Okay, let me make this clear:

we still don't know what pi is.  We don't have the formula.  I'm trying to prove this sucker with ancient methodology, and they didn't have pi yet--as I said in my post, those things are literally the only things I can use.

Attempted methods in the next post.

.

I would normally try to help on this, but I have a *differential* geometry exam tomorrow and I should go back to reading for it.

Also, I have little experience in straight-up Euclidean geometry, so I doubt I would be much help anyway.  How do you even define volume with those methods?

Cheers

[Vector]

This contest will be so much fun.

... I think I'm now kind of embarrassed by the speed at which I scrolled through this.


@Christes: with great difficulty and not much rigor =/  This is 2,200 years before our current mathematics.

If worst comes to worst, I'll just ask a classmate before class.  Thanks, though.


Methodology:

1. getting a contradiction through basic substitution (worked in the other direction)
2. getting a contradiction through messing with the Archimedean property and well-ordering (most of my effort was here)
3. getting a contradiction through flailing everywhere
4. trying to stick epsilons in uncomfortable places
5. trying to somehow get epsilon larger than the difference between the volume and the prisms, for all epsilon
6. trying to get the volume of the prism larger than the cylinder in which it is inscribed, to get a contradiction

Next step: break the prism into smaller prisms and beat the problem over the head with the method of exhaustion (the precursor to calculus).

I suspect I may also need ratios.

[MaximumZero]

Vector, please don't take this the wrong way, but

why would anyone torture themselves like you are currently doing?

[Heron TSG]

Okay, let me make this clear:

we still don't know what pi is.  We don't have the formula.  I'm trying to prove this sucker with ancient methodology, and they didn't have pi yet--as I said in my post, those things are literally the only things I can use.

.
That would have been important to know. Zero usage of Pi? Again, I don't know much about this at all. Could you possibly prove that π is equal to the circumference divided by the diameter? It sounds like R is given, and therefore the diameter could be found, and I guess you could use the noodle method

• to find the circumference, though I doubt that it's scientific enough. Even if you got this, though, it'd be hard to explain why you were doing that, other than to find π, the concept of which does not exist.

*It involves cutting a bunch of cooked spaghetti noodles to the length of the diameter and seeing how many it takes to go around the circle. A ruler is then used to find out how long the series of noodles is divided by the length of one noodle. It worked in 9th grade, but I suspect it'd be difficult to write down.

I wish I could help you out, but this math is beyond me for now. Good luck. 

[Shinziril]

Looking at the Wikipedia articles for the various methods, method of exhaustion looks the most suspicious to me, probably due to its similarity to "taking the limit" (without actually taking the limit, since you aren't allowed to do that).  Are you allowed to simply circumscribe another prism around the cylinder?

Hmm . . . for that to work, you'd need there to be e > a(P) - a(C) > 0 for all e > 0.  Can you prove that? 

[Nivim]

^{(Warning - while you were typing 4 new replies have been posted...)}
 Why does it seem to me like the height-volume doesn't matter all that much? It could be infinite or infinitely small and it wouldn't change whether or not the prism has less area, and thus less volume, than the cylinder.

[Vector]

Vector, please don't take this the wrong way, but

why would anyone torture themselves like you are currently doing?

Actually, it's kind of fun and you can see all these amazing work-arounds ancient mathematicians used to overcome problems they had with philosophy and abstraction.  My professor tells a lot of funny stories, and there's nothing nicer than doing boardwork with a group of other guys so quickly that anyone who comes by to look can't understand even the first step by the time you're done.

But mostly, it's about all of the completely crazy techniques used to cut solids up.  For me, this is way easier to understand than modern methods.  More fun, too, and somehow... more elegant and satisfying.  A lot of the machinery for calculus seems very messy (which is necessary because it's a far more flexible tool), but the machinery for this is just... very simple, very clever, very elegant.  The ancient mathematics is more aesthetically appealing.

That would have been important to know. Zero usage of Pi? Again, I don't know much about this at all. Could you possibly prove that π is equal to the circumference divided by the diameter? It sounds like R is given, and therefore the diameter could be found, and I guess you could use the noodle method

• to find the circumference, though I doubt that it's scientific enough. Even if you got this, though, it'd be hard to explain why you were doing that, other than to find π, the concept of which does not exist.

*It involves cutting a bunch of cooked spaghetti noodles to the length of the diameter and seeing how many it takes to go around the circle. A ruler is then used to find out how long the series of noodles is divided by the length of one noodle. It worked in 9th grade, but I suspect it'd be difficult to write down.

I wish I could help you out, but this math is beyond me for now. Good luck. 

Haha, that's still usage of pi >_<  And we don't have that sort of geometry of curves yet, either, so the spaghetti method definitely wouldn't work... especially because you can't measure pi with a ruler >_> <_<

Thanks for trying, though =)

Looking at the Wikipedia articles for the various methods, method of exhaustion looks the most suspicious to me, probably due to its similarity to "taking the limit" (without actually taking the limit, since you aren't allowed to do that).  Are you allowed to simply circumscribe another prism around the cylinder?

Hmm . . . for that to work, you'd need there to be e > a(P) - a(C) > 0 for all e > 0.  Can you prove that? 

No, because it's false by definition--P is inscribed in C.

I've been thinking about circumscribing another prism around the cylinder, but my main problem with this is that the volume is just far too badly defined for me to figure out what I'm doing.  As such, I no longer feel bad about being unable to solve this >_>  It's the ancient Greeks' fault!  They set up the problem poorly!

And yeah, method of exhaustion still definitely looks best.  Pfaugh.

Why does it seem to me like the height-volume doesn't matter all that much? It could be infinite or infinitely small and it wouldn't change whether or not the prism has less area, and thus less volume, than the cylinder.

Yeah, this is true.  The problem is getting from something we understand in 2 dimensions to something we don't yet understand in 3 dimensions.

I'm going to go back and reread the proof in 2-space.

[eerr]

Vector, please don't take this the wrong way, but

why would anyone torture themselves like you are currently doing?

That's what the professor told her to do : (

[eerr]

I think I got it, but it's kinda hard to check, so just a guess really.

hint: you need to reuse something.

Spoiler (click to show/hide)

Have you tried proving the polygon with one more side, has more area than the polygon with less sides?

the smaller polygon is proved smaller than the big polygon, and the big polygon is proved equal to or less than the circle.

[Vector]

Yeah, I've done that... that's what the part where you can get the difference between the circle's area and the polygon's area arbitrarily small has to do with.

Thanks, though.

[eerr]

So you can't prove one polygon bigger than the other?

[Vector]

So you can't prove one polygon bigger than the other?

I'm saying I already did it.

[eerr]

So you can't prove one polygon bigger than the other?

I'm saying I already did it.

not circle<larger polygon
transforms into
circle>=larger polygon

largerpolygon>smaller polygon
circle>=largerpolygon>smaller polygon

circle>smaller polygon

but ... many times or some bullcrap?

[Vector]

... Yes, I already did that.

But how does that apply to proving that V is not greater than h[a(C)]?