Algorithmic DNA Tile Self-Assembly (2D): Growing the Sierpinski Pattern

DNA origami folds one long scaffold strand into a shape, but a different branch of structural DNA nanotechnology builds shapes from many small, identical-looking tile molecules that only "know" the local rule written into their sticky-end sequences. This 2D lattice view grows such a tile assembly one attachment at a time: each new tile reads the bit carried by its west and south neighbors and locks in their XOR, the same copy-and-combine logic used in Winfree and Rothemund's landmark algorithmic self-assembly experiments. Starting from a seeded edge, this simple local rule reliably produces a global Sierpinski triangle — dial in an attachment-error rate to watch a single mismatched bond corrupt every tile assembled downstream of it, or switch on proofreading tiles to see redundant bonding suppress that same error.