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Edge Tessellations and Stamp Folding Puzzles

Summary An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. In this article we prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon. A stamp folding puzzle is a paper folding problem constrained to the perforations on a sheet of postage stamps. Such sheets necessarily embed in an edge tessellation. On page 143 of his book Piano-Hinged Dissections: Time to Fold!, G. Frederickson poses the following conjecture: “Although triangular stamps have come in a variety of different triangular shapes, only three shapes seem suitable for [stamp] folding puzzles: equilateral, isosceles right triangles, and 60°-right triangles.” We prove that the four non-obtuse polygons mentioned above generate edge tessellations suitable for stamp folding puzzles. Our proof of suitability, which establishes Frederickson's Conjecture, exhibits explicit algorithms for folding each suitable edge tessellation into a packet of single stamps.