I know that the theorems of Euclidean geometry are preserved under rotations, reflections, translations, and compositions of the former three. But what about dilations? Intuitively, the theorems should not change if we measure distances using inches versus centimeters versus light-years, etc. So, my real question is, can someone tell me in what sense Euclidean space is the same no matter if we measure distances using different scales? I don't know how to put it formally, but I hope my informal explanation suffices.
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2In fact the theorems don't specify units of distance at all. There are a lot of ratios. The Pythagorean Theorem doesn't even involve squaring in the modern sense; it actually puts squares on each side of the triangle. – David K Nov 08 '22 at 04:48