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Why is Euclidean geometry scale-invariant?

Proof that Pi is constant (the same for all circles), without using limits

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marked as duplicate by Ross Millikan, Chandru1, Qiaochu Yuan Feb 26 '11 at 19:38

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Did you mean why is the ratio of circumference of a circle to diamter independent of the circle? – Aryabhata Feb 26 '11 at 19:08
What definition of $\pi$ are you using that might make it not a constant? What would cause it to vary? Vote to close as not a real question. – Ross Millikan Feb 26 '11 at 19:09
I think the new version of the question is a good one. – Mike Spivey Feb 26 '11 at 19:28
Duplicate:… – Qiaochu Yuan Feb 26 '11 at 19:38
@blunders: Take a look at this also:… – Derek Jennings Feb 26 '11 at 19:47
up vote 1 down vote accepted

Because when you change the scale, both the diameter and the circumference change by multiplying by the same scale factor, since they are both of dimension 1. Thus, their ratio is independent of the scale factor. Since all circles are similar, the ratio is the same for all circles.

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+1 Funny thing is, your answer makes sense to me -- the answer and question linked to as a duplicate... does not - oh, well. – blunders Feb 26 '11 at 19:50
@blunders: You may appreciate these answers (added again here as the identical link is a bit lost in the comments to the question):… – Derek Jennings Feb 26 '11 at 19:59
@Derek Jennings: +1 Thanks! – blunders Feb 26 '11 at 20:09

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