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I am not a mathematician but I am an artist and fractal enthusiast who spends lots of time creating art using this geometry. I recently discovered online you can make fractals in MS Paint (there's plenty of info online about this) and one of the first fractals I created was the Viscek fractal. First I did the cross form, then the saltire form, but then I had an idea: why not combine the two? So first I start with a square, then add a square to each side like you do with the cross form. Next, I duplicate this new cross shape and add it to the corners of the cross, creating somewhat of a tile pattern.

somewhat of a tile pattern

I repeated this for an unknown amount of iterations until I eventually ended up with a stunning final result.

stunning final result

Surely a shape this simple must have been discovered before, right? I doubt an amateur like me was the first one to think of this. Either way, I still think it's neat I came up with it on my own even if I wasn't the first.

Peter Kagey
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katzenklavier
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Did you just invent a new fractal?

Probably. The process you are using can generate infinitely many different ones, so if you start with anything other than the simplest initial pattern you are likely to build a fractal no one has ever seen before.

This one is compelling. Elegant. Stunning, as you say. I think you should make one in cross stitch. That does not make it particularly interesting as mathematics, whatever it is as art. But don't let that discourage you. Keep playing.

Ethan Bolker
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    Thank you Ethan! Yeah, I wasn't expecting a Fields Medal anytime soon lol, but it is a neat novelty. I am excited to work with this process more, and more like it that are created with two alternating rules instead of just one that is repeated. Have a great day! – katzenklavier Apr 20 '21 at 20:36