Will it k-tile? Structural aspects of polytopes and lattices in multiple tiling Alexandru Mihai, Melissa Sherman-Bennett, Dat Nguyen, Alexander Dunlap Summer@ICERM August 7, 2014 Summer@ICERM K-Tiling August 7, 2014
Introduction to Multiple Tilings 1 k-tiling Polygons 2 A necessary condition for multiple tiling 3 A sufficient condition for multiple tiling with a lattice 4 A structure result for multiple tiling 5 Multiple tiling in three dimensions 6 Summer@ICERM K-Tiling August 7, 2014
Section 1 Introduction to Multiple Tilings Summer@ICERM K-Tiling August 7, 2014
Starting Definitions We say that a polytope P tiles R d with a discrete set of translation vectors Λ if ÿ ÿ 1 P ` λ p v q “ 1 P p λ ´ v q “ k @ v R B P ` Λ λ P Λ λ P Λ Summer@ICERM K-Tiling August 7, 2014
Starting Definitions We say that a polytope P tiles R d with a discrete set of translation vectors Λ if ÿ ÿ 1 P ` λ p v q “ 1 P p λ ´ v q “ k @ v R B P ` Λ λ P Λ λ P Λ Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A polytope P is centrally symmetric about the origin if for all x P P , ´ x P P . And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin. Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A polytope P is centrally symmetric about the origin if for all x P P , ´ x P P . And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin. Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A polytope P is centrally symmetric about the origin if for all x P P , ´ x P P . And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin. Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A polytope P is centrally symmetric about the origin if for all x P P , ´ x P P . And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.
Starting Definitions A polytope P is centrally symmetric about the origin if for all x P P , ´ x P P . And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin. Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r 0 , 1 s n The Minkowski sum is defined to be A ` B “ t a ` b | a P A and b P B u Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r 0 , 1 s n The Minkowski sum is defined to be A ` B “ t a ` b | a P A and b P B u Summer@ICERM K-Tiling August 7, 2014
Starting Definitions A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r 0 , 1 s n The Minkowski sum is defined to be A ` B “ t a ` b | a P A and b P B u 3 2 1 � 3 � 2 � 1 1 2 3 � 1 � 2 � 3 Summer@ICERM K-Tiling August 7, 2014
Polygons and the Fourier Transform The Poisson Summation Formula tells us that given any ”nice” function f on R d , we have ÿ ÿ ˆ f p n q “ f p ξ q n P Λ ξ P Λ ˚ ş where by definition ˆ R d f p x q e 2 π i x x ,ξ y dx, and where the dual f p ξ q : “ lattice is defined by Λ ˚ : “ t x P R d | x l , x y P Z , for all l P Λ u . Summer@ICERM K-Tiling August 7, 2014
Polygons and the Fourier Transform The Poisson Summation Formula tells us that given any ”nice” function f on R d , we have ÿ ÿ ˆ f p n q “ f p ξ q n P Λ ξ P Λ ˚ ş where by definition ˆ R d f p x q e 2 π i x x ,ξ y dx, and where the dual f p ξ q : “ lattice is defined by Λ ˚ : “ t x P R d | x l , x y P Z , for all l P Λ u . Thus, by the Poisson Summation Formula, we have ÿ ÿ 1 1 P p m q e ´ 2 π i x v , m y ˆ k “ 1 P p λ ´ v q “ | det Λ | λ P Λ m P Λ ˚ Now we use the fact that Fourier series expansions are unique, so all the nonzero Fourier coefficients on the right must vanish, because k is constant. And this leads to the proof of a very interesting theorem. Summer@ICERM K-Tiling August 7, 2014
Theorem A convex polytope P k-tiles R d by translations with the lattice Λ if and only if ˆ 1 P p m q “ 0 for all nonzero vectors m P Λ ˚ . Moreover, we have k “ Vol p P q | det p λ q| . Summer@ICERM K-Tiling August 7, 2014
Theorem A convex polytope P k-tiles R d by translations with the lattice Λ if and only if ˆ 1 P p m q “ 0 for all nonzero vectors m P Λ ˚ . Moreover, we have k “ Vol p P q | det p λ q| . Summer@ICERM K-Tiling August 7, 2014
Observations in the Fourier Plane Now what to do with these equations? 6 6 4 4 2 2 0 0 � 2 � 2 � 4 � 4 � 6 � 6 � 6 � 4 � 2 0 2 4 6 � 6 � 4 � 2 0 2 4 6 Summer@ICERM K-Tiling August 7, 2014
Observations in the Fourier Plane But how should we go about working with these curves? One of our initial conjectures was that the dual lattice Λ ˚ would be found at the intersections seen in the Fourier transform. Summer@ICERM K-Tiling August 7, 2014
Observations in the Fourier Plane But how should we go about working with these curves? One of our initial conjectures was that the dual lattice Λ ˚ would be found at the intersections seen in the Fourier transform. 5.0 4.5 4.0 3.5 3.0 2.5 2.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Summer@ICERM K-Tiling August 7, 2014
Observations in the Fourier Plane Now plotting the gradient field of the Fourier Transform we confirm that intersections are really saddle points. Summer@ICERM K-Tiling August 7, 2014
Observations in the Fourier Plane And we end up with a numerical approximation of the curves. 5 4 3 y 2 1 Summer@ICERM K-Tiling August 7, 2014 0 0 1 2 3 4 5 x
Section 2 k-tiling Polygons Summer@ICERM K-Tiling August 7, 2014
Motivation & Guiding Questions Combinatorial characterization Ñ constructive understanding Irrational multi-tilers An algorithm which, given a polygon P , determines if there exists a lattice with which P multi-tiles Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Some terminology: Given two parallel edges e and e 1 “ e ` � s , the vector � s separating them is called the side-pairing of e (or e 1 ). Can be thought of as the sum of the edges separating e and e 1 . e' s e Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Let Λ be a lattice and P a centrally symmetric convex polygon. Let e and e 1 be a pair of parallel edges of P . Definition e is called s “ e 1 for some Type 1 if e ` � s P Λ � e 1 P Λ and e “ ´ � Type 2 if � e e' s “ Aff( e 1 ) for some Aff( e ) ` � � s P Λ and e is not Type 1. s Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Let Λ be a lattice and P a centrally symmetric convex polygon. Let e and e 1 be a pair of parallel edges of P . Definition e is called e s “ e 1 for some Type 1 if e ` � s P Λ � e 1 P Λ and e “ ´ � Type 2 if � s “ Aff( e 1 ) for some Aff( e ) ` � � s P Λ and e is not Type 1. s e' Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Theorem (Bolle, 1991) Given a lattice Λ , a centrally symmetric convex polygon P k-tiles with Λ if and only if every edge of P is Type 1 or Type 2. Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Idea behind necessity: As one “leaves” one polygon translate by moving across an edge, one must enter another polygon translate. e e e' e e' e e' s s e' Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Idea behind necessity: As one “leaves” one polygon translate by moving across an edge, one must enter another polygon translate. e e e' e e' e e' s e' Summer@ICERM K-Tiling August 7, 2014
Bolle Characterization Easy facts: For a given lattice and a centrally symmetric convex polygon with 2 n edges, if: All edges of P are lattice vectors Ñ all side-pairings are lattice vectors Ñ all edges are Type 1 § Corollary: Can’t have all Type 2 edges § All lattice polygons have all Type 1 edges n ´ 1 pairs of edges are Type 2 Ñ n th pair of edges is Type 1 Summer@ICERM K-Tiling August 7, 2014
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