Introduction Self avoiding and non-crossing paths About the shape - PowerPoint PPT Presentation

Introduction Self avoiding and non-crossing paths About the shape Tilings Tile recognition : Given a word w ∈ Σ ∗ 1. check if it is a boundary word Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths About the shape Tilings Tile recognition : Given a word w ∈ Σ ∗ 1. check if it is a boundary word : w is closed Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths About the shape Tilings Tile recognition : Given a word w ∈ Σ ∗ 1. check if it is a boundary word : w is closed and does not intersect itself Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths About the shape Tilings Tile recognition : Given a word w ∈ Σ ∗ 1. check if it is a boundary word : w is closed and does not intersect itself 2. check if it has the “right” shape : BN-factorization (later) Tile generation : 1. Build a closed word having the BN-factorization 2. check that it does not intersect itself 1 self avoiding and non crossing paths 2 about the shape 3 tilings Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings Problem Given a path w ∈ Z × Z of length n, determine if w is self avoiding or intersects itself. Many solutions : ⇒ O ( n 2 ) sparse n × n matrix = list of n sorted points in = ⇒ O ( n log n ) hash table = ⇒ O ( n log n ) worst case Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings Problem Given a path w ∈ Z × Z of length n, determine if w is self avoiding or intersects itself. Many solutions : ⇒ O ( n 2 ) sparse n × n matrix = list of n sorted points in = ⇒ O ( n log n ) hash table = ⇒ O ( n log n ) worst case Problem Given a non-intersecting closed path encoded by a word a , ¯ w ∈ { a , b , ¯ b } of length n, determine whether w is written clockwise or counterclockwise. Solutions : left to the reader as an exercise Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The binary representation of numbers in N × N can be stored conveniently in a dictionary (trie, radix-tree) G = ( N , R ) : Root := (0 , 0) the root has three child nodes : (00 , 01), (01 , 00), (01 , 01) each other node n = ( x , y ), with x , y ∈ { 0 , 1 } ∗ has 4 child nodes : ( x 0 , y 0) , ( x 0 , y 1) , ( x 1 , y 0) , ( x 1 , y 1) 0,0 (0,1) (1,0) (1,1) 0,1 1,0 1,1 (0,0) (0,1) (1,0) (1,1) (0,0) (0,1) (1,0) (1,1) (0,0) (0,1) (1,0) (1,1) 0,2 0,3 1,2 1,3 2,0 2,1 3,0 3,1 2,2 2,3 3,2 3,3 (0,0) (0,1) (1,0) (1,1) (0,0) (0,1) (1,0) (1,1) 4,2 4,3 5,2 5,3 6,6 6,7 7,6 7,7 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings Partioning the plane Definition ⇒ p − q ∈ − → Two points p , q ∈ Z × Z are neighbors ⇐ Σ The neighborhood relation T can be added on the tree G for every level 3 point so that G becomes G = ( N , R , T ). level 2 level 1 level 0 Figure : Partition of the plane Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The radix tree with the neighborhood relation 0,0 (0,1) (1,0) (1,1) 1,0 1,1 (0,0) (0,1) (1,0) (1,1) (0,0) (0,1) (1,0) (1,1) 2,0 2,1 3,1 2,2 b a a b (0,0) (0,1) (1,0) (1,1) 4,2 4,3 5,2 5,3 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings Observation For each coordinate, the father of each node x i is ⌊ x i 2 ⌋ , and consequently, when x i is odd, we have ( x i + 1) = ( ⌊ x i 2 ⌋ + 1) · 0 . It leads to the following computation Lemma Let G ( k ) be the complete graph representing B ≤ k × B ≤ k for some � = ( x , y ) be a node of N k . If one of the four k ≥ 1 , ǫ ∈ Σ , and z conditions holds : (i) ǫ = a and x [ k ] = 0 , (ii) ǫ = a and x [ k ] = 1 , (iii) ǫ = b and y [ k ] = 0 , (iv) ǫ = b and y [ k ] = 1 , then f ( z � + ǫ ) = f ( z � ) . Otherwise, f ( z � + ǫ ) = f ( z � ) + ǫ . Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings Propagating the carry ... (1,1) (0,1) a 1011,0101 1100,0101 (0,1) (1,1) (0,1) a a 10110,01011 10111,01011 11000,01011 Figure : Adding 1 to the first coordinate in the radix tree representation. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm It remains to construct dynamically the graph G w = ( N , R , T ) a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,0) (0,1) (0,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,0) (0,1) (0,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,0) (0,1) (1,0) (0,0) (0,0) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,0) (0,1) (1,0) (0,1) (0,0) (0,1) (0,0) (2,1) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,1) (0,1) (1,0) (0,1) (1,1) (1,0) (0,1) (0,0) (0,0) (0,1) (1,0) (0,0) (2,2) (2,1) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (1,0) (0,1) (1,1) ( ! 1,0) (0,1) (1,0) (0,1) (1,1) (1,0) (0,1) (0,0) (0,0) (1,0) ( ! 1,0) (0,1) (1,0) (0,0) (1,2) (2,2) (2,1) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (0, ! 1) (1,0) (0,1) (1,1) ( ! 1,0) (0,1) (1,0) (0,1) (1,1) (1,0) (0, ! 1) (0,1) (0,0) (0,0) (1,0) ( ! 1,0) (0,1) (1,0) (0,0) (1,2) (2,2) (2,1) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Radix-trees with neighborhood links About the shape The linear algorithm Tilings The linear algorithm a ¯ w = aabb ¯ ba (0,0) (0,1) (1,0) (0, ! 1) (1,0) (0,1) (1,1) ( ! 1,0) (0,1) (1,0) (0,1) (1,1) (1,0) (1,0) (0, ! 1) (0,1) (0,0) (0,0) (1,0) ( ! 1,0) (0,1) (1,0) (0,0) (1,2) (2,2) (2,1) (2,0) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Let w be a word on { a , b , a , b } , then - a“left turn” of w is a factor of w in { ab , ba , ab , ba } , - a“right turn” of w is a factor of w in { ab , ba , ab , ba } . Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Let w be a word on { a , b , a , b } , then - a“left turn” of w is a factor of w in { ab , ba , ab , ba } , - a“right turn” of w is a factor of w in { ab , ba , ab , ba } . Definition (Alternative) Let w be the counterclockwise boundary word of a polyomino. A salient corner is a left turn, and a reentrant corner is a right turn. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Lemma (Daurat and Nivat 2003) Let w be the counterclockwise boundary word of a polyomino. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) = 4 . Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Lemma (Daurat and Nivat 2003) Let w be the counterclockwise boundary word of a polyomino. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) = 4 . Manhattan Taxi driver’s proof : Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Lemma (Daurat and Nivat 2003) Let w be the counterclockwise boundary word of a polyomino. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) = 4 . Manhattan Taxi driver’s proof : N W E S Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Lemma Let w be a closed non-crossing path written counterclockwise. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) = 4 . Lemma Let w be a closed path. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) ≡ 0 mod 4 . Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Lemma Let w be a closed non-crossing path written counterclockwise. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) = 4 . Lemma Let w be a closed path. Let R ( ww 1 ) be the set of its right turns and L ( ww 1 ) be the set of its left turns. Then L ( ww 1 ) − R ( ww 1 ) ≡ 0 mod 4 . Exercice : hexagonal grids. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Euclidean convexity) A subset S ⊂ r 2 is said convex if for any pair of points x , y ∈ S, the line segment joining x to y is entirely included in S. y x Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Minsky et Papert 1969, Slansky 1970) A subset S ⊂ Z 2 is digitally convex if it is the discretization of a convex Euclidean figure. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Minsky et Papert 1969, Slansky 1970) A subset S ⊂ Z 2 is digitally convex if it is the discretization of a convex Euclidean figure. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Minsky et Papert 1969, Slansky 1970) A subset S ⊂ Z 2 is digitally convex if it is the discretization of a convex Euclidean figure. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Minsky et Papert 1969, Slansky 1970) A subset S ⊂ Z 2 is digitally convex if it is the discretization of a convex Euclidean figure. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Minsky et Papert 1969, Slansky 1970) A subset S ⊂ Z 2 is digitally convex if it is the discretization of a convex Euclidean figure. Remark : connexity may fail ! Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition A discrete figure is a finite 8 -connected subset of Z 2 , without holes. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition A discrete figure is a finite 8 -connected subset of Z 2 , without holes. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition A discrete figure is a finite 8 -connected subset of Z 2 , without holes. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition A discrete figure is a finite 8 -connected subset of Z 2 , without holes. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Kim) A discrete figure F ⊂ Z 2 is digitally convex if it is the digitization of its Euclidean convex hull. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Kim) A discrete figure F ⊂ Z 2 is digitally convex if it is the digitization of its Euclidean convex hull. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Basics Definition (Kim) A discrete figure F ⊂ Z 2 is digitally convex if it is the digitization of its Euclidean convex hull. Remark : 8-connexity is included in the hypothesis ! Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings HV-convexity Definition A subset S ⊂ Z 2 is H-convex if all its rows are connected. (a) (b) (c) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings HV-convexity Definition A subset S ⊂ Z 2 is H-convex if all its rows are connected. Definition A subset S ⊂ Z 2 is V-convex if all its column are connected. (a) (b) (c) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings HV-convexity Definition A subset S ⊂ Z 2 is H-convex if all its rows are connected. Definition A subset S ⊂ Z 2 is V-convex if all its column are connected. Definition A subset S ⊂ Z 2 is HV-convex if it is H-convex and V-convex. (a) (b) (c) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Previous result Theorem (Debled-Renneson, R´ emy and Rouyer-Degli) Given an HV-convex discrete figure F. Deciding if F is digitally convex is decidable in linear time. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Previous result Theorem (Debled-Renneson, R´ emy and Rouyer-Degli) Given an HV-convex discrete figure F. Deciding if F is digitally convex is decidable in linear time. Algorithm based on the segmentation of curves in discrete lines, which requires only arithmetic computations. Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . ���� · (0 1 0 1 1) · (0 0 1) · (0 0 0 0 1) 2 . = (1) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . � �� � ) · (0 0 1) · (0 0 0 0 1) 2 . = (1) ���� · ( 0 1 0 1 1 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . ���� ) · (0 0 0 0 1) 2 . = (1) ���� · ( 0 1 0 1 1 � �� � ) · ( 0 0 1 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . � �� � ) 2 . = (1) ���� · ( 0 1 0 1 1 � �� � ) · ( 0 0 1 ���� ) · ( 0 0 0 0 1 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Here, the boundary word w is written clockwise. Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 w = 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 . � �� � ) 2 . = (1) ���� · ( 0 1 0 1 1 � �� � ) · ( 0 0 1 ���� ) · ( 0 0 0 0 1 Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Self avoiding and non-crossing paths Daurat-Nivat relation : R − S = 4 About the shape Lyndon+Christoffel = Digitally convex Tilings Proposition A word w is NW-convex iff its Lyndon factorization w = l n 1 1 l n 2 2 · · · l n k is only composed of Christoffel words. k In practice there are linear algorithms for building the Lyndon factorization of any word (J.P. Duval) checking if a word is a Christoffel word By combining the two, where each Lyndon factor must be a Christoffel word, this leads to a linear algorithm which is 10 times faster than previous ones. Then it suffices to check NE, ES, SW-convexity by permuting the alphabet NW : (0 , 1) ; NE : (0 , 3) ; ES (3 , 2) ; SW (2 , 1) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Figure : Maurits Cornelis Escher (1898-1972) Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Figure : Hexagonal tiling Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Figure : Hexagonal and Square tilings Sreˇ cko Brlek Words2011: words and digital geometry

Introduction Escher tilings Self avoiding and non-crossing paths Tiling by translation About the shape Algorithms Tilings Double Squares Sreˇ cko Brlek Words2011: words and digital geometry