Someone showed me a photocopy of a neat…uh… thing,
in geometry.
Start with a circle, and draw an equilateral triangle around the circle, so that the triangle’s side’s midpoints are tangent to the circle. Now draw a circle around the triangle, so that the corners touch the circle.
Draw a square…then a circle.
A pentagon…then a circle.
Keep the progression of polygons and circumscription up until you reach a limit to the number of sides you can add to the polygon…infinity.
The outermost circle is a fraction over eight times the size of the innermost circle.
Is this a new constant, or just a combination of pi and e?
Wouldn’t the figure keep getting bigger? The polygons would inch ever closer to the circle as you added sides, but they’d always add a tiny bit to the size of the figure, wouldn’t they?
Yeah, I get lost at step 12. It looks like he’s saying y[sub]2[/sub]=y[sub]1[/sub]/pi. But what exactly is he trying to do? What is y[sub]2[/sub] supposed to be?
Rickjay: yes, if I understand this, each time you draw a new n-gon or new circle, you are making the figure slightly larger. However, that doesn’t mean that the size becomes unbounded.
Take the formula 1 + (1/2) + (1/4) + (1/8) + …
Each subsequent addition adds “a little bit” to the sum, but the total sum is not infinite, but bounded, and in fact is always less than 2 (although getting very very very close to 2 as the number of terms increases.)
So it is not out of reasonableness that the geometric progression described would not expand infinitely.
I’m way too far out of touch with this stuff to comment much further, however.