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Self-similarity displayed in fancy broccoli

Self-similarity and self-reference are key traits of the fractal and therefore the universe.

Self-reference means that recursive algorithms can be used to simplify the process of creating complex patterns. Self-similarity is what you get when you use self-reference and recursion to create a fractal shape.

Without self-similarity and recursion, the coding required to define the structure of a tree or that of our nervous and circulatory systems would be incredibly complex and it becomes impossible to conceive of how these could evolve via random processes. But with it, they boil down to simple branching instructions repeated a few times in a feedback loop.

The macroscopic is a manifestation of the microscopic and the same patterns appear at every level of scale. As the saying goes it's "turtles all the way down".

See Gödel to go down the rabbit hole.

Fractals are typically not self-similar
Fractals and Scaling: Self-Similarity Dimension
The Mandelbrot set is... self similar.

Divinyls - I Touch Myself