  Building on work of Brenner and Monsky from 2010 and on a Hilbert-Kunz
calculation of Monsky from 1998, we exhibit a novel example of a hypersurface
over $\overline{\mathbb{F}_2}$ in which tight closure does not commute with
localization. Our methods involve a surprising tiling argument using
Sierpi\'nski triangles, as well as an inspection of a certain dynamical system
in characteristic two.
