Edge Tessellations and The Stamp Folding Problem Ron Umble Millersville University of Pennsylvania Vassar College Math Chat February 23, 2010 Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 1 / 52
The Stamp Folding Problem How many ways can one con…gure the perforation lines on a sheet of postage stamps so that it folds into a stack of single stamps? Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 2 / 52
The Stamp Folding Problem How many ways can one con…gure the perforation lines on a sheet of postage stamps so that it folds into a stack of single stamps? The solution will follow from our main result... Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 2 / 52
Tessellations A tessellation (or tiling) of the plane is a collection of plane …gures that …lls the plane with no overlaps and no gaps Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 3 / 52
Tessellations A tessellation (or tiling) of the plane is a collection of plane …gures that …lls the plane with no overlaps and no gaps A regular tessellation has (non-trivial) translational symmetries in two independent directions Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 3 / 52
Tessellations A tessellation (or tiling) of the plane is a collection of plane …gures that …lls the plane with no overlaps and no gaps A regular tessellation has (non-trivial) translational symmetries in two independent directions Examples include edge tessellations, which are generated by re‡ecting a polygon in its edges Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 3 / 52
Main Result Theorem 1. Exactly eight polygons generate edge tessellations: Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 4 / 52
Main Result Theorem 1. Exactly eight polygons generate edge tessellations: Most symmetric examples of Laves tilings (Grunbaum & Shephard, Tiling and Patterns, p. 96) Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 4 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 5 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 6 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 7 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 8 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 9 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 10 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 11 / 52
Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 12 / 52
Solving the Stamp Folding Problem Perforation lines are lines of symmetry Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 13 / 52
Solving the Stamp Folding Problem Perforation lines are lines of symmetry Perforation lines miss stamp interiors Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 13 / 52
Solving the Stamp Folding Problem Perforation lines are lines of symmetry Perforation lines miss stamp interiors Polygons in Thm 1 with interior angles � 90 � generate solutions Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 13 / 52
Non-Solutions in Theorem 1 Polygons in Theorem 1 with an interior angle of 120 � fail because... Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 14 / 52
Non-Solutions in Theorem 1 Polygons in Theorem 1 with an interior angle of 120 � fail because... The bisector of a 120 � interior angle contains an edge of some stamp Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 14 / 52
The Edge Tesselation Problem Which polygons generate an edge tessellation of the plane? Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 15 / 52
Finding Edge Tesselations Let G be a polygon that generates an edge tessellation Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 16 / 52
Finding Edge Tesselations Let G be a polygon that generates an edge tessellation If V is a vertex of G , then θ = m \ V < 180 � Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 16 / 52
Interior Angles at a Vertex If G 0 is the re‡ection of G in an edge containing V , the interior angle of G 0 at V has measure θ ; inductively, every interior angle at V has measure θ V θ Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 17 / 52
Interior Angles at a Vertex If G 0 is the re‡ection of G in an edge containing V , the interior angle of G 0 at V has measure θ ; inductively, every interior angle at V has measure θ V θ θ Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 18 / 52
Interior Angles at a Vertex If G 0 is the re‡ection of G in an edge containing V , the interior angle of G 0 at V has measure θ ; inductively, every interior angle at V has measure θ V θ θ θ Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 19 / 52
Interior Angles at a Vertex If G 0 is the re‡ection of G in an edge containing V , the interior angle of G 0 at V has measure θ ; inductively, every interior angle at V has measure θ V θ θ θ Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 20 / 52
Crystallographic Restriction An n-center of a tessellation is the center of a group of n rotational symmetries Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 21 / 52
Crystallographic Restriction An n-center of a tessellation is the center of a group of n rotational symmetries Crystallographic Restriction: If P is an n - center of a regular tessellation, then n = 2 , 3 , 4 , 6 Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 21 / 52
Crystallographic Restriction An n-center of a tessellation is the center of a group of n rotational symmetries Crystallographic Restriction: If P is an n - center of a regular tessellation, then n = 2 , 3 , 4 , 6 Re‡ecting in adjacent edges of G is a rotational symmetry about the common vertex V Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 21 / 52
Crystallographic Restriction An n-center of a tessellation is the center of a group of n rotational symmetries Crystallographic Restriction: If P is an n - center of a regular tessellation, then n = 2 , 3 , 4 , 6 Re‡ecting in adjacent edges of G is a rotational symmetry about the common vertex V V is an n -center for some n = 2 , 3 , 4 , 6 Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 21 / 52
Admissible Interior Angles If G 0 is the image of G under a rotational symmetry at V , the number k of copies of G sharing vertex V is the order of the rotational subgroup at V . Thus k = 3 , 4 , 6 and θ = 60 � , 90 � , 120 � . G' G Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 22 / 52
Admissible Interior Angles Otherwise, G and G 0 rotate as a unit and the number k of copies of G sharing vertex V is twice the order of the rotational subgroup at V . Thus k = 4 , 6 , 8 , 12 and θ = 30 � , 45 � , 60 � , 90 � . G G' Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 23 / 52
Admissible Polygons Conclusion: Interior angles of G measure 30 � , 45 � , 60 � , 90 � , 120 � Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 24 / 52
Admissible Polygons Conclusion: Interior angles of G measure 30 � , 45 � , 60 � , 90 � , 120 � Number of edges e � 6 since the int angle sum 180 ( e � 2 ) � 120 e Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 24 / 52
Admissible Polygons Conclusion: Interior angles of G measure 30 � , 45 � , 60 � , 90 � , 120 � Number of edges e � 6 since the int angle sum 180 ( e � 2 ) � 120 e There are no regular tilings of the plane by pentagons Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 24 / 52
Admissible Polygons Conclusion: Interior angles of G measure 30 � , 45 � , 60 � , 90 � , 120 � Number of edges e � 6 since the int angle sum 180 ( e � 2 ) � 120 e There are no regular tilings of the plane by pentagons Therefore G is a triangle, a quadrilateral, or a hexagon Ron Umble (Vassar College Math Chat) Edge Tessellations and Stamp Folding February 23, 2010 24 / 52
Recommend
More recommend
