Illustrating the Leau-Fatou flowers

[bo198214]

but I'd like to begin to chew this material... everyone here seems to understand terms and definitions but I'm a total beginner.
What the hell is, formally, a flower?

You can find a proper definition of a petal in Milnor in the beginning of §7.
A flower is just all the petals of a fixed point.

Looking at the pictures it seems the image of a path. In particular it seems it has to do with images of the family of paths \[\gamma_n:\mathbb S^1\to \mathbb C\]

No, nothing like that - it is just an *illustration*! But with this illustration one can well see the attracting and repelling petals.