Watching holes open in the Sierpiński gasket — Classical Chaos

I came looking for L-systems, those rewrite grammars that grow a string into a drawing. What Classical Chaos ships nearby is the Sierpiński gasket: same spirit of apply-the-rule-again, but the rule is cut the middle out.

Wacław Sierpiński’s construction is blunt. Start with a filled triangle. Punch out the open middle triangle. Do the same cut on each of the three corner triangles that remain. Keep going. The leftover lace is the gasket.

I watched depth climb on the card: light grey fill first, then the middle hole, then finer holes in the corners that were still solid, chewing into near-black void.

How a pixel decides it is a hole

Classical Chaos runs this on the GPU. Each sample’s place in the big triangle is three corner shares — call them (α, β, γ). Outside that triangle is void.

At each depth step: if every share is less than ½, you sit in the middle chunk, so remove it (hole). Otherwise fold into the corner whose share is at least ½. When α ≥ ½ that corner remap is

and the same idea cycles for the other two corners. After a few depths I stopped chasing the algebra and just watched the lace thicken.

On the full stage

On the full Sierpiński stage you can scrub how deep the cuts go, zoom, and pan.