What is the significance of the Sierpinski Triangle and Barnsley Fern?


I understand the basic concept around fractals and how they function as mathematical equations. Bear in mind I don’t know the actual nitty gritty behind the equations of these two fractals but I understand why fractals work on a shallow level. A lot of explanations I see behind fractals use the triangle and fern as two well known examples. Why these two? Is there something significant behind these two shapes when it comes to fractals? Does the existence of these fractals imply something that pertains to the real world or is it just a cool little numbers trick to show off on paper or a computer?

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It’s just that they are good examples of how very simple rules can give rise to very complex behavior. In the case of the Barnsley Fern, it demonstrates how very little information needs to be encoded in DNA to explain seemingly complicated structures: you only need six parameters, and even that is “more than necessary” because some parameters can be reduced to fewer ones with changes of variable.

Presumably, even a very small genome could produce a phenotype with fern-like structures because of this. This is wild to me because ferns seem so darned intricate… But their blueprints are super simple, apparently!