Tiling the plane with congruent equilateral convex pentagons

It was shown by Hunt and Hirschhorn (J. Combin. Theory. Ser. A 39 (1985) 1) in 1983 that an equilateral convex pentagon tiles the plane if and only if it has two angles adding to 2π or it is a uniquely determined pentagon with special angles. Their proof is based on studying all the points of intersection of 100 curves. In this paper we provide an alternative demonstration of this result. Our approach is based on an observation deduced from Eulerʹs theorem for plane graphs. 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’.