The probably most well-known occurrence of the Sierpinski Triangle is as the odd entries of the Pascal triangle
Some month ago however, there was an article about mathematical models of sandpiles along with some images of computer simulations; it struck me to see the same nested triangles as in the Sierpinski triangle (cf e.g. here).
Then I recently wanted to list all pairs of disjoint subsets of some finite set; in order to be able to use bit operations, I iterated over all pairs of subsets encoded as binary numbers and checked whether BIT_AND-ing yielded zero.
Much to my surprise, the Sierpinski triangle showed up again, when I visualized the outcome of the bit operation for each pair.
where else does the Sierpinski triangle, i.e. its fractally nested, regular triangle pattern, appear?