Is self-similarity a form of symmetry, is it the other way around, or is it something else?

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I'm aware that self-similarity is a form of symmetry, however I'm interested in getting a more in depth explanation of the relationship.

Could you consider symmetry to be a form of self-similarity?

(Yes, the two definitions are nearly equivalent)

What's the difference between self-similarity and symmetry?

(Symmetry can refer to any preserved property while self-similarity refers to the whole being encoded in the fundamentals)

(1) things in parentheses are my guesses at the answer

I'm interested in a compare/contrast similar to the above so that I can better distinguish self-similarity from symmetry.