Singular and Calabi–Yau varieties linked with billiard trajectories and diffusion operators

The images under a certain map of periodic billiard trajectories inside the fundamental region of the affine Weyl group of the root system A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{2}$$\end{document} are closed curves and configurations of lines described by a one-parameter family of polynomials. The polynomials are related with eigenvectors of symmetric diffusion operators connected with the deltoid. Several associated singular varieties and Calabi–Yau threefolds defined over the rationals are constructed.