  We advance the program of connections between final coalgebras as sources of
circularity in mathematics and fractal sets of real numbers. In particular, we
are interested in the Sierpinski carpet, taking it as a fractal subset of the
unit square. We construct a category of square sets and an endofunctor on it
which corresponds to the operation of gluing copies of a square set along
segments. We show that the initial algebra and final coalgebra exist for our
functor, and that the final coalgebra is bi-Lipschitz equivalent to the
Sierpinski carpet. Along the way, we make connections to topics such as the
iterative construction of initial algebras as colimits, corecursive algebras,
and the classic treatment of fractal sets due to Hutchinson.
