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

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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 30th, 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

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Outline

1

The Tiling by Translation Problem Discrete Figures Tilings Beauquier and Nivat Hexagonal and Square Tilings

2

Double Square Tiles Christoffel Tiles Fibonacci Tiles

3

Conclusion

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 2 / 24

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Discrete Figures and Polyominoes

Discrete plane : Z2
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

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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

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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

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

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

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

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

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

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

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

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

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

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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

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Freeman Chain Code

Σ =

a, a, b, b
a →

b ↑ a ← b ↓ a ba ¯ b ¯ b a b ba ¯ b ¯ b a b b b b ¯ a ¯ b ¯ a ¯ a b b ¯ a ¯ b ¯ b ¯ a ¯ b ¯ b w = ababbabbabbabbbbabaabbabbabb

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

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Freeman Chain Code

Σ =

a, a, b, b
a →

b ↑ a ← b ↓ a ba ¯ b ¯ b a b ba ¯ b ¯ b a b b b b ¯ a ¯ b ¯ a ¯ a b b ¯ a ¯ b ¯ b ¯ a ¯ b ¯ b w = ababbabbabbabbbbabaabbabbabb Any conjugate w′ of w codes the same polyomino. w and w′ are conjugate if there exist u, v ∈ Σ∗ such that w = uv and w′ = vu.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

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Freeman Chain Code

Σ =

a, a, b, b
a →

b ↑ a ← b ↓ Any conjugate w′ of w codes the same polyomino. w and w′ are conjugate if there exist u, v ∈ Σ∗ such that w = uv and w′ = vu. a ba ¯ b ¯ b a b ba ¯ b ¯ b a b b b b ¯ a ¯ b ¯ a ¯ a b b ¯ a ¯ b ¯ b ¯ a ¯ b ¯ b w ≡ ababbabbabbabbbbabaabbabbabb

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

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Freeman Chain Code

Σ =

a, a, b, b
a →

b ↑ a ← b ↓ Any conjugate w′ of w codes the same polyomino. w and w′ are conjugate if there exist u, v ∈ Σ∗ such that w = uv and w′ = vu. a ba ¯ b ¯ b a b ba ¯ b ¯ b a b b b b ¯ a ¯ b ¯ a ¯ a b b ¯ a ¯ b ¯ b ¯ a ¯ b ¯ b w ≡ ababbabbabbabbbbabaabbabbabb

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 6 / 24

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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 ✻

X = b a b a b a a

t ✛

hexagon tiles square tiles

✭✭✭PPPP ✭ ✭ ✭ PPPP X Y Z

X
Y
Z

✭✭✭PPPP ✭ ✭ ✭ PPPP X Y Z

X
Y
Z

✭✭✭PPPP ✭ ✭ ✭ PPPP X Y Z

X
Y
Z

✘✘✘✘✘ ✘ ❅ ❅ ❅ ❅ ❅ ❅ ✘✘✘✘✘ ✘ X Y

X
Y

✘✘✘✘✘ ✘ ❅ ❅ ❅ ❅ ❅ ❅ ✘✘✘✘✘ ✘ X Y

X
Y

✘✘✘✘✘ ✘ ❅ ❅ ❅ ❅ ❅ ❅ ✘✘✘✘✘ ✘ X Y

X
Y

✘✘✘✘✘ ✘ ❅ ❅ ❅ ❅ ❅ ❅ ✘✘✘✘✘ ✘ X Y

X
Y

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 7 / 24

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Maurits Cornelis Escher (1898-1972). Hexagonal tiling. Square tiling.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 8 / 24

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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

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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

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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

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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

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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

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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

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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

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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Square Tilings

The pentamino has two distinct square factorizations:

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 10 / 24

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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

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 11 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 11 / 24

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Prime Tiles

Let P be a polyomino and S be a square tile. Then the composition P ◦ S is the polyomino defined by replacing each unit cell of P by S.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 12 / 24

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Prime Tiles

Let P be a polyomino and S be a square tile. Then the composition P ◦ S is the polyomino defined by replacing each unit cell of P by S. = Note: This is not commutative.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 12 / 24

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Prime Tiles

Let P be a polyomino and S be a square tile. Then the composition P ◦ S is the polyomino defined by replacing each unit cell of P by S. = Note: This is not commutative.

Definition

A polyomino Q is prime if Q = P ◦ S implies that P or S is the 1 × 1 unit square.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 12 / 24

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Prime Double Squares

Conjecture (X. Proven¸ cal and L. Vuillon, 2008)

If XY X Y describes the contour of a prime double square, then both X and Y are palindromes. Note: a palindrome is a word that reads the same forward as it does backward. a b a b ¯ a b ¯ a ¯ b ¯ a ¯ b a ¯ b a b a b ¯ a b ¯ a ¯ b ¯ a ¯ b a ¯ b

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 13 / 24

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Prime Double Squares

Conjecture (X. Proven¸ cal and L. Vuillon, 2008)

If XY X Y describes the contour of a prime double square, then both X and Y are palindromes. Note: a palindrome is a word that reads the same forward as it does backward. a b a b ¯ a b ¯ a ¯ b ¯ a ¯ b a ¯ b a ba ¯ a ¯ b ¯ a ¯ b a¯ b b ¯ a b

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 13 / 24

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Prime Double Squares

Conjecture (X. Proven¸ cal and L. Vuillon, 2008)

If XY X Y describes the contour of a prime double square, then both X and Y are palindromes. Note: a palindrome is a word that reads the same forward as it does backward. a ba ¯ a ¯ b ¯ a ¯ b a¯ b b ¯ a b ba b ¯ b ¯ a ¯ b a¯ b a ¯ a b ¯ a

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 13 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 14 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 14 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ ∋ΛςΜΩΞΣÿΙΠ8ΜΠΙΩ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 14 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below that may be factorized in , , and :

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below that may be factorized in , , and :

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below that may be factorized in , , and : w w

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below that may be factorized in , , and : w w w w w w

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

There are prime double squares like the one below that may be factorized in , , and : w w w w w w Those three can be obtained from smaller words ww via the morphism λ : a → , b → , ¯ a → , ¯ b → .

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 15 / 24

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Christoffel Tiles

λ : a → , b → , ¯ a → , ¯ b → .

Theorem (Blondin Mass´ e, Brlek, Garon, L.)

Let w = apb where a and b are letters. (i) If p is a palindrome, then λ(ww) is a square tile. (ii) λ(ww) is a double square if and only if w is a Christoffel word. Christoffel words are discretization

f finite segments.

(0, 0) (8, 5) S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 16 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ ∋ΛςΜΩΞΣÿΙΠ8ΜΠΙΩ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 17 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ ∋ΛςΜΩΞΣÿΙΠ8ΜΠΙΩ ∗ΜΦΣΡΕΓΓΜ8ΜΠΙΩ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 17 / 24

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Fibonacci Tiles

We define a sequence in {r, l}∗ by q0 = ε, q1 = r and qn =

qn−1qn−2

if n ≡ 2 mod 3, qn−1qn−2 if n ≡ 0, 1 mod 3. The first terms are

q0 = ε q3 = rl q6 = rllrllrr q1 = r q4 = rll q7 = rllrllrrlrrlr q2 = r q5 = rllrl q8 = rllrllrrlrrlrrllrllrr

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 18 / 24

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Fibonacci Tiles

q2 q3 q4 q5 q6 q7 q8 q9 q10

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 19 / 24

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Fibonacci Tiles

q4 q7 q10

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 19 / 24

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Fibonacci Tiles

q4 q7 q10 (q4)4 (q7)4 (q10)4

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 19 / 24

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Fibonacci Tiles

(q4)4 (q7)4 (q10)4

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 19 / 24

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Lemma (Blondin Mass´ e, Brlek, L., Mend` es France)

(i) The path qn is self-avoiding. (ii) The path (q3n+1)4 codes the boundary of a polyomino.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 20 / 24

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Lemma (Blondin Mass´ e, Brlek, L., Mend` es France)

(i) The path qn is self-avoiding. (ii) The path (q3n+1)4 codes the boundary of a polyomino.

Figure: Fibonacci tiles of order n = 0, 1, 2, 3, 4.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 20 / 24

SLIDE 76

Lemma (Blondin Mass´ e, Brlek, L., Mend` es France)

(i) The path qn is self-avoiding. (ii) The path (q3n+1)4 codes the boundary of a polyomino.

Figure: Fibonacci tiles of order n = 0, 1, 2, 3, 4.

Theorem (Blondin Mass´ e, Brlek, Garon, L.)

(i) Fibonacci tiles of order n ≥ 1 are double squares. (ii) If AB ˆ Aˆ B is a BN-factorisation of a Fibonacci tile, then A and B are palindromes.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 20 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96):

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ ∋ΛςΜΩΞΣÿΙΠ8ΜΠΙΩ ∗ΜΦΣΡΕΓΓΜ8ΜΠΙΩ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 21 / 24

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Definition

A double square is a tile having two distinct square factorizations. Table of the first double squares (Proven¸ cal’s Thesis, p.96): Cookies are crenelated versions

f Fibonacci tiles.

96 P´ erim` etre Premiers Compos´ es 12 16 18 20 24 28 30 32 Tableau 4.1 Les doubles pseudo-carr´ es de p´ erim` etre inf´ erieur ou ´ egal ` a 32.

2ΣΞΤςΜΘΙ ∋ΛςΜΩΞΣÿΙΠ8ΜΠΙΩ ∗ΜΦΣΡΕΓΓΜ8ΜΠΙΩ ∋ΣΣΟΜΙΩ

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 21 / 24

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Problem

Does the Christoffel tiles and (generalized) Fibonacci tiles describe all the prime double square tiles?

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 22 / 24

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Problem

Does the Christoffel tiles and (generalized) Fibonacci tiles describe all the prime double square tiles? No!!!!

Figure: Some double squares not in the Christoffel tiles nor in the Fibonacci tiles families.

S´ ebastien Labb´ e (LaCIM) Christoffel and Fibonacci Tiles DGCI 2009 22 / 24

SLIDE 81

Useful Software

This research was driven by computer exploration using the open-source mathematical software Sage [1] and its algebraic combinatorics features developed by the Sage-Combinat community [2], and in particular, F. Saliola, A. Bergeron and S. Labb´ e. The pictures have been produced using Sage, pgf/tikz and Xournal.

W. A. Stein et al., Sage Mathematics Software (Version 4.1.1), The