Here’s an interesting pattern that is not truly self-similar, but close. Start with a circle:
Stage 0
Pack in four circles with diameters that are half the diameter of the original circle:
Stage 1
Repeat (you’ll notice that here is where the figure is not 100% self-similar: the Stage 1 circles do not have four circles packed into them – just two):
Stage 2
Here’s my interpretation of the pattern in stained-glass:
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