One can easily relate all regular polygons to the triangle:

triangle = 1 triangle square = 2 triangles pentagon = 3 triangles hexagon = 4 triangles

and so on and so forth…

A circle is basically just a regular polygon with an infinite number of sides.

Having said that, we could say that the circle is composed of infinitely many triangles of infinitely small size.

so infinity/infinity = pi*r^2

divide by r^2 and get that pi = infinity/(infinity*r^2)

We know pi is the fixed value 3.14… so wouldn't my logical progression prove that infinity can take on different sizes?



closed as unclear what you're asking by Zubin Mukerjee, Daniel Robert-Nicoud, user147263, user223391, Chappers Jun 29 '15 at 22:34

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    $\begingroup$ You can't manipulate infinity like that. $\endgroup$ – Zain Patel Jun 29 '15 at 21:48

The problem is that any argument involving infinity needs to be incredibly precise. (Well, all math needs to be precise, that's sort of the point; but in particular, when talking about infinity, naive intuition is dangerous.) Unfortunately, everything you've written is incredibly vague.

Basically, limits are complicated - it took a long time for calculus, for example, to be made rigorous. Besides studying calculus rigorously you should read about indeterminate forms, perhaps starting at the wikipedia page: https://en.wikipedia.org/wiki/Indeterminate_form.


You have to be really careful when making an argument that has a process continuing to infinity. You have to decide exactly what that means, and how to interpret the result. For example, this question shows a "proof" that $\pi=4$ based performing this type of process. The mistake is that the argument does not rigorously prove that what they are showing is true for every finite step in the process is also true in the limit at infinity.

While your ideas here are not wrong, per-se, when you say that something has infinitely small size, and you have infinitely many of them, you have to figure out what that means.


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