Tiling the plane with congruent equilateral convex pentagons (Q598454)

Which convex \(n\)-gons can tile the plane? (An \(n\)-gon tiles the plane if congruent images can cover the plane without overlap.) All triangles and quadrilaterals tile the plane. All convex hexagonal tilings of the plane have been classified by \textit{K. A. Reinhardt} [Über die Zerlegung der Ebene in Polygone (German) (Diss. Frankfurt a.M., R. Roske, Borna-Leipzig) (1918; JFM 46.0832.04)]. The problem of classifying convex pentagons that tile the plane has been only partially solved. \textit{D. Schattschneider} [Math. Mag. 51, 29--44 (1978; Zbl 0408.51010)] made a contribution, and \textit{M. D. Hirschhorn} and \textit{D. C. Hunt} [J. Comb. Theory, Ser. A 39, 1--18 (1985; Zbl 0573.52019)] proved that an equilateral convex pentagon tiles the plane if and only if it has two angles adding to \(2\pi\), or it is a uniquely determined pentagon with special angles. In the present paper an alternative demonstration of this theorem is given. Even though ``The new approach does not eliminate sorting completely, it reduces it substantially to a small number of trigonometric equations that have been solved with `Maple'.''