A sphere has infinitely many infinitesimal flat surfaces.
For an infinitely large sphere, does it have infinitely many infinitely large flat surfaces? So an infinitely large sphere is a flat surfaces?
Thursday, March 18, 2004
I object to virtually every word of this question.
Friday, March 19, 2004
Ok, I'll elaborate.
> A sphere has infinitely many infinitesimal flat surfaces
Well, not really, unless you count a point as an "infinitesimal flat surface" (which nobody does). Or are we talking about something more like r^2 dTheta dPhi? Why is it flat? Are you assuming that smaller parts of a sphere are "flatter" than larger parts? Even if that were true, does it imply that the "infinitesimal surfaces" are flat?
> For an infinitely large sphere
A what? Assuming an embedding in R3, what is its equation? Given its center, can you name a single point on it? Supposing you can formulate a description, how do you know it's a sphere?
> does it have infinitely many infinitely large flat surfaces?
If it did, it would contradict what you said earlier ("infinitesimal "). Or is this your point?
You really need to think about this more rigorously. Start with your first statement - what exactly are you trying to express, and how do you know it's true?
Then, when you want to talk about infinity, make sure you still can. What you usually do is describe a sequence of objects O_n, and take the limit. But you have to make sure that the limit exists and makes sense, which is going to be a problem in this case.
Saturday, March 20, 2004
"Infinitesimal" reminds me of calculus.
One way to find approximate arc length is to use line segments along the curve, then sum up the lengths of all line segments. To get a better approximation, use shorter line segments, thus increasing the number of line segments. Let the number of line segments approach infinity, and the sum becomes an integral, which is the arc length. So one can assume "Infinitesimal" line segments made up the curved line.
Apply a similar argument to a curved surface by using double integrals. The curved surface is partitioned into "Infinitesimal" surfaces. Since the surfaces are "Infinitesimal," supposedly they are also "flat," otherwise they are not small enough. But then they become so small, they are virtually just points with "Infinitesimal" areas!
I agree with above post that you need to find a formula for a sphere with radius r. Maybe take a cross section of the sphere to simplify the problem. So the cross section is just a circle. Take the derivative of the formula to get the slope of the tangent line at any point. Take the derivative again to get the rate of change of the slope. Let r go to infinity and analyze the these derivatives, especially see if the slopes at two different points are the same, and if the rate of change of the slopes at different points are constant.
Monday, March 22, 2004
It took two hundred years from the first beginnings of calculus to the time when it was finally logically justified. Most of the difficulty was because it was very hard for people to NOT make the following fallacious logical leaps.
"A collection of straight lines can be made as close as possible to a curve" -> "a curve is made of infinitely small straight lines"
"as a becomes closer to zero, the sum becomes closer to the integral." -> "when a becomes zero, the sum becomes the integral"
THESE ARE WRONG! However where is a long and illustrious history of people in the 1700s who could not get around it. Mathematicians struggled with these kinds of fallacies for a long time, but in the end they could not be justified. Part of the problem was that they didn't have a good definition of what the real numbers were. They had been identified with geometrical quantities like the length of a line, but it turned out that geometry was not powerful enough to express the things that were being done with numbers.
Nowadays we define real numbers as the limits of series of rational numbers. So when we integrate a function we take the limit as the width of each slice decreases; if the series converges the limit is a real number.
This system excludes infinites. You can speak of the limit of some quantity as a sphere becomes arbitrarily large, but it is meaningless to talk of an infinite sphere itself. You can examine limits when quantities become arbitrarily small but there is not such a thing as an infinitesimal in the real numbers.
More recently, some people have constructed an alternate system (nonstandard analysis) which provides a rigorous way to talk about infinite and infinitesimal quantities, but you have to be more specific (I doubt that "infinitely large sphere" would be a maningful statement in nonstandard analysis either.)
The sphere question. There is actually a branch of geometry (projective geometry) that introduces points at infinity and identifies straight lines with circles whose centers are at that point. I don't think it speaks to infinitesimals though.
Tuesday, March 23, 2004
I should not have thrown around knowledge from my first year calculus course here.
Tuesday, March 23, 2004
sphere = a set of points whose distance from center is a given value (aka the "radius").
"infinitely large sphere" = the set of points whose distance from center is infinity.
You have bigger problems here than imagining its "surface."
Saturday, April 10, 2004
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