The Sikusi Fractal is a self-similar geometric pattern created from a mechanical iterative process on a rectangular grid. Start with an empty m x n grid. Repeatedly fill every cell sequentially with the cycling integers 1 through x (row by row, left to right). Then, erase every cell whose value belongs to a chosen “removal set” A. The newly emptied cells are immediately refilled with the next cycle of 1 through x, and the erasure step repeats until no members of A remain anywhere on the grid.
The final arrangement of retained numbers creates nested structures. The generator renders this live on a canvas.
(Rendered by the Sikusi Fractal Generator. try different grid sizes and iteration depths!)
This construction is related to automatic sequences, base-x digit sums, and two-dimensional analogues of the Thue-Morse and paper-folding sequences. The retained cells after infinite iteration correspond to positions whose coordinates (in row-major order) avoid certain forbidden digits in base x (a feature of fractal self-similarity).
The digit-filtering rule uses constructions from combinatorics on words and cellular automata. But this specific iterative “fill, erase, refill” loop doesn't appear to be something that anyone has written about before. When I searched, and could not find any mathematical papers about this fractal, I felt inspired to build the generator!
The pattern first revealed itself to me while grinding my combat level in Runescape (I wanted to upgrade to upgrade to mithril armor and needed level 20 to do so).
The easiest way to do this, it seemed, was to fight the cows that spawn near Lumbridge. When defeated, each cow drops, in this order:
My inventory (a 7x4 grid) soon filled up with this repeating pattern of: hide, meat, bones, hide, meat, bones, hide, meat, bones.
Though my inventory was full, I wanted to keep grinding. But it is inconvenient to get rid of the meat and cowhide in the field (I would have needed to go elsewhere to cook the meat and tan the hides). However, I could dispose of the bones immediately by burying them (which has the added benefit that it levels the character's prayer level).
This opened up some space in my inventory so I could start grinding again. The cowhide and meat I'd collected stayed, but the empty spaces filled up again, with the same items (cowhide, raw meat, bones) in that same order. And when I again ran out of inventory, I simply buried the bones again.
The cycle continued until my inventory had only cowhide and meat.
I realized that this made a cool pattern, and that this pattern was generalizable, and that one may:
The screenshots below capture the exact stages of this discovery:
