Self-Similarity

Self-similarity displayed in fancy broccoli

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

“Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.” -James Gleick

Here is an example of self-reference.

That was actually an example of self-referential humor, or a "meta" joke. If you don't find them funny, you're gonna have a bad time reading this wiki. Or you'll just laugh a bit less than if you did. The cosmic joke will be a bit of a groaner for you, since it's definitely meta humor.

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, most of nature, 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.

Self-replication is self-referential process that causes life to evolve.

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 and Scaling: Self-Similarity Dimension

The Mandelbrot set is... self similar.

If you want to be technical, many fractals have self-similar traits rather than true mathematical self-similarity. That's why this this wiki emphasizes the point that fractals are a good metaphor to help you visualize many concepts even when they aren't literally fractals.

Fractals are typically not self-similar

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