 Lambert quadrilateral construction of a cone torus

A Lambert quadrilateral with acute angle $\pi /14$ and sides $x,y$ incident to the opposite right-angle vertex satisfies the displayed relation. Reflect four copies around that vertex to make a symmetric hyperbolic quadrilateral whose four corner angles are $\pi /14$. Identifying opposite sides yields a torus with one cone angle $2\pi /7$. The two central axis loops have lengths $2x,2y$ and meet once at right angles. Cutting along one loop gives a pair of pants with two equal geodesic boundaries and the cone point; reglue the boundaries with a Fenchel–Nielsen twist to obtain the two-parameter cone-torus family. Its seven-sheeted torsion-free covers have genus four.