Digit sequences of rational billiards on tables which tile ${\mathbb{R}}^2$

We classify the periodic digit strings which arise from periodic billiard orbits on the four convex $n$-gons $\mathrm{\Delta }$ which tile ${\mathbb{R}}^2$ under reflection, answering problem a posed by Baxter and Umble. $\mathrm{\Delta }$ is either an equilateral triangle, a $45-45-90$ triangle a $30-60-90$, triangle or a rectangle.