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This question explains that scale-invariance (or more accurately, similarity) is an important property of Euclidean geometry. Are there any other ways to define scale-invariant geometries in any sense? And how do they differ from Euclidean geometry?

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If you accept Euclid's first four postulates, then as the question you mention discusses, being able to scale an object (a triangle) implies the parallel postulate and Euclidean geometry.

So any other geometry will be significantly different from what we learned in school.

Here are a few examples of some other scale invariant geometries:

These are just a few; an exhaustive list is impossible.

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