what is so strange about triangle with length of sqrt(2) and circumference of circle?

Thank you, everyone; particularly carbonzit, MrAl.

MrAl said:

Hi there PG,

#1
That would be true if things worked that way, but they dont quite work that way. The problem is a mix of dimensions. We can have something described one way in X dimensions that reaches a limit, while something else related to that in Y dimensions doesnt reach a limit.
We're talking about an 'object' one way in three dimensions, then talking about that same 'object' another way in two dimensions. One is volume (3d) and one is surface area (2d) so the limits dont have to coincide. We cant forget that we're dealing with a fictitious object here, not a real life object. We can only construct an approximation anyway. You have to remember that theory is not real life. The universe was here before theory, or at least Man's theory as it is known today.

#2
You can make pi look rational if you envision a circle as a multi sided object rather than a perfect circle. This is good for practical applications, but i dont see what good it would do to try to make pi perfectly rational. pi converts linear systems to circular systems, which have their very own rationale. Radial lines map perfectly to rectilinear lines in theory only. As soon as we try to build something like this to prove it we find it just doesnt work.
Mathematically it works, and again we might call in the purposed theory that we (as mankind on earth) have too much information, more information than reality, at least for the time being. This means we'll find lots of things that work mathematically but dont work in real life. To put it another way, if math was a solid object it would be bigger than the universe.

Click to expand...

Forget about calculus for a moment. Suppose that Gabiel's horm is made of stretchable rubber. In math when a limit is taken of something we don't mean that that 'something' becomes the thing limit tells us. e.g. When the side of a polygon reaches infinity it becomes a circle. It's the limit. But in reality that polygon will never become a circle but it can always go on seek perfection. Now coming back to the horn. So, when you say that when the horn has been stretched to infinity its volume becomes finite and surface area infinite. [PAUSE] While writing this posting I think I have made some progress and can understand it and should some follow-on questions rather than refuting the well established truth! Well, I wasn't refuting anything. Sometimes, you have to challenge the truth to understand it.

I have used the volume formula from the image below and in both cases I get infinite volume. Why is so? I was expecting to get a finite result at least in case of limit [0 --> 6]. Please help me with it. Thanks.

**broken link removed**

[LATEX]$\int\limits_{0}^{6}\pi \left( \frac{1}{x}\right) ^{2}dx=\allowbreak \infty $[/LATEX]

[LATEX]$\int\limits_{0}^{\infty }\pi \left( \frac{1}{x}\right) ^{2}dx=\allowbreak \infty [/LATEX]

Hi again PG,

Who said we had to start at x=0? Start at some positive value greater than zero. That illustrates the concept just fine. The image you posted shows the integration starting at 1.

BTW, how did you embed that graph diagram into the page?

Thanks, MrAl.

**broken link removed**

Hi

I have always believed that every mathematics formula, equation, etc. can be conceptually understood. In my view math is just a compact form of expressing quantification of a phenomena and our concepts of nature.

Please have a look on this Google Doc (the file was uploaded by me).

This is how I would try to explain it. In case of volume we are able to find a limit which the horn can get as close as it wants without ever really reaching it. Therefore, when its side is stretched to infinity its volume will never cross that certain value. In the linked PDF you see that when upper limit of integral is made larger from, say, "6000000000" to "999999999999999", there is very little change in volume. So, we know the boundary it can neither touch nor cross but can get as close as it wants.

But in case of surface area when the upper limit is made larger in the same proportion as in case of volume there is significant change in the surface area measure but still surface area do not just gets out of bounds (I mean the different is not in 1000's; check the highlighted part in the PDF the different between values is only "29" (130.918 - 101.321) though in once case upper limit has been significantly large. If we had used the same limits for two cases in the case of volume, then the different would be of very, very small degree ). ButBut in case of surface area we cannot find a limit. So, we say that it has an infinite surface because we do not know any boundary in this case.

What I say above might be little confusing but I hope you would extract and connect the relevant parts to make sense out of it. Thank you.

Regards
PG

PS: I have checked it; there doesn't exist any relation between volume and surface area of the horn.

Hi again,


Not sure what you mean by "doesnt exact any relation between volume and surface area".

Google doc doesnt quite illustrate this property as well as it could have. Try the following experiment:

Compute the integral from 1 to 2, call that y12.
Compute the integral from 2 to 3, call that y23.
Compute the integral from 3 to 4, call that y34.

Now using those results do the following:
Compute the ratio y23/y12, call that R1.
Compute the ratio y34/y23, call that R2.

Note that R2 is greater than R1.

In other words, for equal width pancake slices of the horn each successive slices area is proportionally larger than the previous.