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I'm just using L-systems. I do not know how to prove anything about them. I want to know whether it is possible to formulate an L-system for this Pierced Diamond Fractal. All my trials have failed.

Thank you in advance.

enter image description here

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I think it is possible, at least it seems to be possible to generated the shape with a graph directed iterated function system of similarities, which I believe (a reference would be appreciated, I couldn't find one in a quick internet search...) has equivalent expressive power to L-systems:

red rule: red rule

blue rule: blue rule

seed rule: seed rule

result: result

(sorry for the crude appearance, it's just a quick sketch in my https://mathr.co.uk/blog/2007-10-03_graphgrow.svg )

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