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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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    @blunders: Here's another link: http://www.oocities.org/cf/ilanpi/pi-exists.html2011-02-26

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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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    @Derek Jennings: +1 Thanks!2011-02-26