- #1
JDude13
- 95
- 0
Whilst messing around with some geometry pertaining to the n-sided regular polygon, I stumbled upon this equation which I could not find anywhere on the internet.
[itex]\pi = \lim_{n \to \infty} n \sin \frac{\pi}{n}[/itex]
But if we take this to be true then, by substitution, this is also true:
[itex]\pi = \lim_{n \to \infty} n \sin (\sin \frac{\pi}{n})[/itex]
And ad infinitum:
[itex]\pi = \lim_{n \to \infty} n \sin(\sin(\sin(\sin(...))))[/itex]
However this makes no sense to me... at the heart of this seemingly infinite sea of sine functions is there a pi/n core? Or is there no centre and is each sine function ultimately a function of nothing?
Or have I been misusing the maths?
EDIT:
Just blew my own mind with
[itex]p = \lim_{n \to \infty} n \sin \frac{p}{n}[/itex]
[itex]\pi = \lim_{n \to \infty} n \sin \frac{\pi}{n}[/itex]
But if we take this to be true then, by substitution, this is also true:
[itex]\pi = \lim_{n \to \infty} n \sin (\sin \frac{\pi}{n})[/itex]
And ad infinitum:
[itex]\pi = \lim_{n \to \infty} n \sin(\sin(\sin(\sin(...))))[/itex]
However this makes no sense to me... at the heart of this seemingly infinite sea of sine functions is there a pi/n core? Or is there no centre and is each sine function ultimately a function of nothing?
Or have I been misusing the maths?
EDIT:
Just blew my own mind with
[itex]p = \lim_{n \to \infty} n \sin \frac{p}{n}[/itex]
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