Christoffel and Fibonacci Tiles S ebastien Labb e Laboratoire de - PowerPoint PPT Presentation

Christoffel and Fibonacci Tiles S´ ebastien Labb´ e Laboratoire de Combinatoire et d’Informatique Math´ ematique Universit´ e du Qu´ ebec ` a Montr´ eal DGCI 2009 September 30 th , 2009 With : Alexandre Blondin Mass´ e, Sreˇ cko Brlek and Ariane Garon S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 1 / 24

Outline The Tiling by Translation Problem 1 Discrete Figures Tilings Beauquier and Nivat Hexagonal and Square Tilings Double Square Tiles 2 Christoffel Tiles Fibonacci Tiles Conclusion 3 S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 2 / 24

Discrete Figures and Polyominoes • Discrete plane : Z 2 • Definition : A polyomino is a finite, 4-connected subset of the plane, without holes. ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 3 / 24

The Tiling by Translation Problem Let P be a polyomino. We say that P tiles the plane if there exists a set T of non-overlapping translated copies of P that covers all the plane. P is called a tile if it tiles the plane. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 4 / 24

The Tiling by Translation Problem Let P be a polyomino. We say that P tiles the plane if there exists a set T of non-overlapping translated copies of P that covers all the plane. P is called a tile if it tiles the plane. Problem Does a given polyomino P tile the plane? S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 4 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 5 / 24

Freeman Chain Code � � Σ = a , a , b , b a → b ↑ a ← b ↓ S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

Freeman Chain Code � � Σ = a , a , b , b a ¯ a → b ↑ a ← b ↓ ¯ b b ¯ a ¯ ¯ b b b b ¯ a ¯ a a ¯ ¯ b b ba ba ¯ ¯ ¯ b b b b a ¯ ¯ b b b b a a w = ababbabbabbabbbbabaabbabbabb S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

Freeman Chain Code � � Σ = a , a , b , b ¯ a a → b ↑ a ← b ↓ ¯ b b a ¯ Any conjugate w ′ of w ¯ ¯ b b b b a ¯ ¯ a ¯ a codes the same polyomino. ¯ b b ba ba w and w ′ are conjugate if ¯ ¯ ¯ b b b b there exist u , v ∈ Σ ∗ such that a w = uv and w ′ = vu . ¯ ¯ b b b b a a w = ababbabbabbabbbbabaabbabbabb S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

Freeman Chain Code � � Σ = a , a , b , b ¯ a a → b ↑ a ← b ↓ ¯ b b ¯ a Any conjugate w ′ of w ¯ ¯ b b b b ¯ a a ¯ a ¯ codes the same polyomino. ¯ b b ba ba w and w ′ are conjugate if ¯ ¯ ¯ b b b b there exist u , v ∈ Σ ∗ such that a w = uv and w ′ = vu . ¯ ¯ b b b b a a w ≡ ababbabbabbabbbbabaabbabbabb S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

Freeman Chain Code � � Σ = a , a , b , b ¯ a a → b ↑ a ← b ↓ ¯ b b ¯ a Any conjugate w ′ of w ¯ ¯ b b b b ¯ a a ¯ a ¯ codes the same polyomino. ¯ b b ba ba w and w ′ are conjugate if ¯ ¯ ¯ b b b b there exist u , v ∈ Σ ∗ such that a w = uv and w ′ = vu . ¯ ¯ b b b b a a w ≡ ababbabbabbabbbbabaabbabbabb S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

Beauquier and Nivat (1991) Characterization : A polyomino P tiles the plane if and only if there exist X , Y , Z ∈ Σ ∗ such that w ≡ XYZ � X � Y � Z . ✻ X = a a b a b a b t t � ✛ X = b a b a b a a hexagon tiles square tiles ✭✭✭PPPP ✘✘✘✘✘ ✘ ❅ � Y ✘✘✘✘✘ ✘ � � X Y � ❅ ❅ ❅ X � Z � ❅ Z X Y ❅ ❅ ✘✘✘✘✘ ✘✘✘✘✘ ✘ ✘ � ✭✭✭PPPP ❅ X PPPP Y X ❅ ❅ ❅ ✭✭✭PPPP ✭ ✭ X ✘ ✘ Y ❅ ✘✘✘✘✘ ✘✘✘✘✘ � ✭ � Y Y Y ❅ � � ❅ ❅ X X � ❅ Y � ❅ Z � � X X Y ❅ ❅ ✘✘✘✘✘ ✘ � Z � ❅ X Y Z PPPP ❅ ❅ Z X ✘ X ❅ ✘✘✘✘✘ ✭ Y Y ✭ PPPP ✭ X ✭ ❅ Y ✭ ✭ S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 7 / 24

Maurits Cornelis Escher (1898-1972). Hexagonal tiling. Square tiling. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 8 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Hexagonal Tilings There are polyominoes admitting many hexagon tilings: A 1 × n rectangle tiles the plane as an hexagon in n − 1 ways and as a square in only 1 way. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 9 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

Square Tilings The pentamino has two distinct square factorizations: Conjecture (Brlek, Dulucq, F´ edou, Proven¸ cal 2007) A tile has at most 2 square factorizations. S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24