I decided to fill each tile with a design that gives rise to a series of pursuit curves. By drawing many similar shapes inside, each slightly smaller than the last, the corners will lie on four graceful curves that model the path of an object pursuing another object. Here is a series of 150 quadrilaterals inside the fat tile:
Given that motif, it’s then simply (!) a matter of placing over 150 tiles into a rectangular space:
and filling each tile, taking care to aligning the curves to force aperiodicity. Voila!