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Edge Tessellations and the Stamp Folding Problem



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EDGE TESSELLATIONS AND THE STAMP FOLDING PROBLEM MATTHEW KIRBY AND RON UMBLE Abstract We prove that a polygon generating a tessellation of the plane when re ected in its edges is either a regular hexagon a 60 90 120 kite a 120 rhombus a rectangle an equilateral a 30 right an isosceles right or a 120 isosceles triangle Of these the four whose interior angles measure at most 90 solve the Stamp Folding Problem How many ways can one con gure the perforation lines on a sheet of postage stamps so that it folds conveniently into a stack of single stamps The solution of this Stamp Folding Problem follows from our main result which was discovered by Millersville University students Andrew Hall Joshua York and the rst author during an undergraduate research seminar directed by the second author in the spring of 2009 Theorem 1 A polygon that generates a tessellation of the plane when re ected in its edges is either a regular hexagon a 60 90 120 kite a 120rhombus a rectangle an equilateral a 30 right an isosceles right or a 120 isosceles triangle A tessellation or tiling of the plane is a collection of plane gures that lls the plane with no overlaps and no gaps An edge tessellation is generated by re ecting a polygon in its edges Edge tessellations are the most symmetric examples of Laves tilings as one can see from the complete list in 3 The name Laves tiling honors the German crystallographer Fritz Laves 1906 1978 Corollary 1 A polygon generating an edge tessellation that solves the Stamp Folding Problem is either a rectangle an equilateral a 30 right or an isosceles right triangle Figure 1 Edge tessellations with interior angles at most 90 Date April 15 2010 Key words and phrases Symmetry tessellation wallpaper pattern 1 2 M ATTHEW KIRBY AND RON UM BLE On page 143 of his book Piano Hinged Dissections Time to Fold 2 G Frederickson conjectures that the various types of triangles in Figure 1 are the only ones that seem suitable for stamp folding puzzles To prove this fact



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