I have a problem, where I am asked to give a precise definition of the Sierpinski-triangle. The problem describes the usual algorithm to construct the triangle. However, I am clueless on how I should give a different definition.
Is there some definition using the union of the points of the Sierpinski-triangle?
How about: The unique compact subset $S$ of $\mathbb C$ such that $$S=\frac 12S \cup \left(\frac12S+1\right)\cup \left(\frac12S+\frac{1+i\sqrt 3}2\right)$$