Word combinatorics, tilings and discrete geometry V. Berth e - - PowerPoint PPT Presentation

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Word combinatorics, tilings and discrete geometry

• V. Berth´

e

LIRMM-CNRS-Montpellier-France berthe@lirmm.fr http://www.lirmm.fr/˜berthe

Journ´ ees Montoises Rennes, 2006

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

From discrete surfaces to multidimensional words... ...via tilings and word combinatorics

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Some objects...

• discrete lines, planes, surfaces, rotations

...and some transformations acting on them

• substitutions

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

• flips

s

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Arithmetic discrete lines

The arithmetic discrete line D(a, b, µ, ω) is defined as D(a, b, c, µ, ω) = {(x, y) ∈ Z2 | 0 ≤ ax + by + µ < ω}

• µ is the translation parameter of D(a, b, µ, ω)
• ω is the width of D(a, b, µ, ω)
• If ω = max{|a|, |b|}, then D(a, b, µ, ω) is said naive
• If ω = |a| + |b|, then D(a, b, c, µ, ω) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Arithmetic discrete lines

The arithmetic discrete line D(a, b, µ, ω) is defined as D(a, b, c, µ, ω) = {(x, y) ∈ Z2 | 0 ≤ ax + by + µ < ω}

• µ is the translation parameter of D(a, b, µ, ω)
• ω is the width of D(a, b, µ, ω)
• If ω = max{|a|, |b|}, then D(a, b, µ, ω) is said naive
• If ω = |a| + |b|, then D(a, b, c, µ, ω) is said standard

Reveill` es’91, Fran¸ con, Andres, Debled-Renesson, Jacob-Dacol, Kiselman, Vittone, Chassery, G´ erard, Buzer, Brimkov, Barneva, Rosenfeld, Klette...

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Arithmetic discrete lines

The arithmetic discrete line D(a, b, µ, ω) is defined as D(a, b, c, µ, ω) = {(x, y) ∈ Z2 | 0 ≤ ax + by + µ < ω}

• µ is the translation parameter of D(a, b, µ, ω)
• ω is the width of D(a, b, µ, ω)
• If ω = max{|a|, |b|}, then D(a, b, µ, ω) is said naive
• If ω = |a| + |b|, then D(a, b, c, µ, ω) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete lines and Sturmian words

One can code such a discrete line (Freeman code) over the two-letter alphabet {0, 1}. One gets a Stumian word (un)n∈N ∈ {0, 1}N 0100101001001010010100100101

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete lines and Sturmian words

Let Rα : R/Z → R/Z, x → x + α mod 1.

Sturmian words [Morse-Hedlund]

Let (un)n∈N ∈ {0, 1}N be a Sturmian word. There exist α ∈ (0, 1), α ∈ Q, x ∈ R such that ∀n ∈ N, un = i ⇐ ⇒ Rn

α(x) = nα + x ∈ Ii (mod 1),

with I0 = [0, 1 − α[, I1 = [1 − α, 1[

• r

I0 =]0, 1 − α], I1 =]1 − α, 1].

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete lines and Sturmian words

Let Rα : R/Z → R/Z, x → x + α mod 1. Sturmian words are codings of irrational rotations of T1 = R/Z with respect either to the partition {I0 = [0, 1 − α[, I1 = [1 − α, 1[}

• r to

{I0 =]0, 1 − α], I1 =]1 − α, 1]}.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A key lemma

Let I0 = [0, 1 − α[, I1 = [1 − α, 1[. Let Rα : x → x + α mod 1.

Lemma

The word w = w1 · · · wn over the alphabet {0, 1} is a factor the Sturmian word u iff Iw1 ∩ R−1

α Iw2 ∩ · · · R−n+1 α

Iwn = ∅.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A key lemma

Let I0 = [0, 1 − α[, I1 = [1 − α, 1[. Let Rα : x → x + α mod 1.

Lemma

The word w = w1 · · · wn over the alphabet {0, 1} is a factor the Sturmian word u iff Iw1 ∩ R−1

α Iw2 ∩ · · · R−n+1 α

Iwn = ∅. Proof: One has ∀i ∈ N, un = i ⇐ ⇒ nα + x ∈ Ii (mod 1).

• One first notes that

ukuk+1 · · · un+k−1 = w1 · · · wn iff 8 > > < > > : kα + x ∈ Iw1 (k + 1)α + x ∈ Iw2 ... (k + n − 1)α + x ∈ Iwn

• One then applies the density of (kα)∈N in R/Z.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A key lemma

Let I0 = [0, 1 − α[, I1 = [1 − α, 1[. Let Rα : x → x + α mod 1.

Lemma

The word w = w1 · · · wn over the alphabet {0, 1} is a factor the Sturmian word u iff Iw1 ∩ R−1

α Iw2 ∩ · · · R−n+1 α

Iwn = ∅. Application: one deduces combinatorial properties on the

• number of factors of given length/ enumeration of local configurations
• densities of factors/ statistical properties of local configurations
• powers of factors, repetitions, palindromes/symmetries

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A key lemma

Let I0 = [0, 1 − α[, I1 = [1 − α, 1[. Let Rα : x → x + α mod 1.

Lemma

The word w = w1 · · · wn over the alphabet {0, 1} is a factor the Sturmian word u iff Iw1 ∩ R−1

α Iw2 ∩ · · · R−n+1 α

Iwn = ∅. Remark: The sets Iw1 ∩ R−1

α Iw2 ∩ · · · R−n+1 α

Iwn are intervals of T = R/Z. The factors of u are in one-to-one correspondence with the n + 1 intervals of T whose end-points are given by −kα mod 1, for 0 ≤ k ≤ n .

Sturmian words [Coven-Hedlund]

A word u ∈ {0, 1}N is Sturmian iff it admits eactly n + 1 factors of length n.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Tilings of the line

• By projecting the vertices of the discrete line, one gets a tiling of the line.
• This corresponds to a cut-and-project scheme in quasicrystallography.
• If one modifies the width of the selection window, then one gets a tiling of the

line by two or three tiles.

• The corresponding codings are three-interval exchange words

[Guimond-Mas´ akov´ a-Pelantov´ a].

• They have complexity n + cste, or 2n + 1.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

To summarize...

We have used

• A coding as an infinite binary word
• A dynamical system: the rotation of R/Z, Rα : x → x + α
• The key lemma: bijection between intervals and factors

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

To summarize...

We have used

• A coding as an infinite binary word
• A dynamical system: the rotation of R/Z, Rα : x → x + α
• The key lemma: bijection between intervals and factors

Dynamical system

A dynamical system (X, T) is defined as the action of a continuous and onto map T

• n a compact space X.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

To summarize...

We have used

• A coding as an infinite binary word
• A dynamical system: the rotation of R/Z, Rα : x → x + α
• The key lemma: bijection between intervals and factors

We have considered so far naive and standard lines.

• Can we work with more general widths ω?
• Can we go to higher dimensions?

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Arithmetic discrete planes

The arithmetic discrete plane P(a, b, c, µ, ω) is defined as P(a, b, c, µ, ω) = {(x, y, z) ∈ Z3 | 0 ≤ ax + by + cz + µ < ω}.

• µ is the translation parameter of P(a, b, c, µ, ω)
• ω is the width of P(a, b, c, µ, ω)
• If ω = max{|a|, |b|, |c|}, then P(a, b, c, µ, ω) is said naive
• If ω = |a| + |b| + |c|, then P(a, b, c, µ, ω) is said standard

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

From a discrete plane to a tiling by projection....

3 1 2 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 2 1 1 3 2 2 1 1 2 2 2 1 1 1 2 2 1

....and from a tiling by lozenges to a ternary coding

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete planes and two-dimensional Sturmian words

Definition

Such a coding is called a 2D Sturmian word.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete planes and two-dimensional Sturmian words

Definition

Such a coding is called a 2D Sturmian word.

Theorem [B.-Vuillon]

Let (Um,n)(m,n)∈Z2 ∈ {1, 2, 3}Z2 be a 2D Sturmian word. Then there exist x ∈ R, and α, β ∈ R such that 1, α, β are Q-linearly independent and α + β < 1 such that ∀(m, n) ∈ Z2, Um, n = i ⇐ ⇒ Rm

α Rn β(x) = x + nα + mβ ∈ Ii (mod 1),

with I1 = [0, α[, I2 = [α, α + β[, I3 = [α + β, 1[

• r

I1 =]0, α], I2 =]α, α + β], I3 =]α + β, 1].

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Combinatorial properties of 2D Sturmian words

• They key lemma still holds: rectangular factors are in one-to-one correspondence

with intervals of R/Z.

• An example of application:

Theorem [B.-Vuillon]

There exist exactly mn + m + n rectangular factors of size m × n in a 2D Sturmian word.

• One similarly gets results on densities of occurrence of factors.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Rational vs irrational arithmetic discrete planes

The arithmetic discrete plane P(a, b, c, µ, ω) is defined as P(a, b, c, µ, ω) = {(x, y, z) ∈ Z3 | 0 ≤ ax + by + cz + µ < ω}.

Remark

• Totally irrational planes: dimQ(a, b, c) = 3.
• Irrational planes (the intermediate case): dimQ(a, b, c) = 2.
• Rational planes: dimQ(a, b, c) = 1. This is a classical assumption in discrete
• geometry. One can choose a, b, c, µ, ω ∈ Z with

gcd(a, b, c) = 1 (Bezout’s Lemma). The determination of the frequencies of factors is deduced from the properties of well-distribution for the sequence ((ma + nb) mod c)(n,m)∈Z2.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Back to the discrete line case

Question

How to enlarge the width? How to project by recovering a suitable coding?

✻

− → e2

✲

− → e1

• Generalized functionality: two projections of the discrete line 0 ≤ x + 2y − 7 < 3 on

the x-axis along the directions − → e1 + − → e2 and − → e1 + 2− → e2.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Functionality of arithmetic discrete planes

Fact

Let us assume 0 ≤ a, b ≤ c. Let P := P(a, b, c, µ, c) be a naive plane. Let π be the orthogonal projection onto the plane z = 0. The set P(a, b, c, µ, c) = {(x, y, z) ∈ Z3, 0 ≤ ax + by + cz ≤ c} is in one-to-one correspondance with the set of points with integer coordinates in the plane z = 0. One associates with the point of coordinates (m, n) ∈ Z2 the point of P with coordinates (m, n, −⌊ am+bn+µ

c

⌋).

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Functionality of arithmetic discrete planes

Theorem [B.-Fiorio-Jamet-Philippe]

Let P( v, µ, ω) = { x ∈ Z3 | 0 ≤ v, x + µ < ω} be an arithmetic discrete plane, and let α ∈ Z3 be such that gcd{α1, α2, α3} = 1. Let π

α : R3 −

→ { x ∈ R3, α, x = 0} be the affine orthogonal projection map onto the plane { x ∈ R3, α, x = 0}. Then the map π

α : P(

v, µ, ω) − → π

α(Z3) is a bijection iff

| α, v| = ω.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Functionality of arithmetic discrete planes

Theorem [B.-Fiorio-Jamet-Philippe]

Let P( v, µ, ω) = { x ∈ Z3 | 0 ≤ v, x + µ < ω} be an arithmetic discrete plane, and let α ∈ Z3 be such that gcd{α1, α2, α3} = 1. Let π

α : R3 −

→ { x ∈ R3, α, x = 0} be the affine orthogonal projection map onto the plane { x ∈ R3, α, x = 0}. Then the map π

α : P(

v, µ, ω) − → π

α(Z3) is a bijection iff

| α, v| = ω. Remark: If there exists α such that α3 = 1 with | α, v| = w, then the inverse function π−1

• α

has a simple expression. We thus can apply the same strategy as previously:

• Introduce a suitable dynamical system (Z2-action by two rotations) and a suitable

coding.

• Deduce properties concerning local configurations.

For more examples of natural codings associated with discrete planes, see D. Jamet. http://www.lirmm.fr/˜jamet

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Functionality: an application to local configurations

Hypothesis

Let P = P(− → v , µ, ω) be an arithmetic discrete plane and let − → α ∈ Z3 such that gcd(− → α ) = 1 and |− → α , − → v | = ω. We assume that α3 = 1, i.e., ω ∈ v1Z + v2Z + v3. One has π

α(Z3) = Z−

→ e1 + Z− → e2.

Local configuration

By (m, n)-cube, we mean a local configuration in the discrete plane that can be

• bserved thanks to the projection π−

→ α through an (m, n)- rectangular window in the

functional lattice Z− → e1 + Z− → e2.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Height

Question

Let P = P(− → v , µ, w) be an arithmetic discrete plane with functional lattice π

α(Z3).

Given an element − → y in the lattice, how to recover the unique vector − → x ∈ P such that π−

→ α (−

→ x ) = − → y ?

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Height

Question

Let P = P(− → v , µ, w) be an arithmetic discrete plane with functional lattice π

α(Z3).

Given an element − → y in the lattice, how to recover the unique vector − → x ∈ P such that π−

→ α (−

→ x ) = − → y ?

Height

The height HP,−

→ α (−

→ y ) at − → y is defined as the third coordinate x3 of − → x = π−1

− → α (−

→ y ) ∈ P.

Case α3 = 1

If α3 = 1, then HP,−

→ α (−

→ y ) = − jv1y1 + v2y2 + µ ω k .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A coding as a two-dimensional word

Hypothesis

Let P = P(− → v , µ, w) be an arithmetic discrete plane and let − → α ∈ Z3 such that gcd(− → α ) = 1 and − → α , − → v = w. One has π

α(Z3) = Z−

→ e1 + Z− → e2. We introduce a two-dimensional word coding in a natural way the parity of the heights HP,−

→ α (−

→ y ), for − → y in the functional lattice Z− → e1 + Z− → e2. It is easy to see that, for all − → y

• HP,−

→ α

`− → y + − → e1 ´ − HP,−

→ α

`− → y ´

• HP,−

→ α

`− → y + − → e2 ´ − HP,−

→ α

`− → y ´ takes only two values. In each case, one of these values is odd, whereas the other one is even. It is now natural to introduce the following two-dimensional word of parity of heights U = (Ui1,i2)(i1,i2)∈Z2 = (HP,−

→ α (−

→ y ) mod 2)−

→ y ∈ {0, 1}Z2.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

2D Rote word [Vuillon]

The two-dimensional word U satisfies, for each (i1, i2) ∈ Z2 Ui1,i2 = 0 iff v1i1 + v2i2 + µ mod 2w ∈ [0, w[. The word U is a two-dimensional Rote word. One-dimensional Rote words are defined as the infinite words over the alphabet {0, 1} that have exactly 2n factors of length n for every positive integer n, and whose set of factors is closed under complementation, i.e., every word obtained by interchanging zeros and ones in a factor of the infinite word u is still a factor of u.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Applications of the key lemma

Complexity [B.-Fiorio-Jamet-Philippe]

Let P = P(− → v , µ, w) be an arithmetic discrete plane with w ∈ v1Z + v2 + v3. Then, P contains at most mn (m, n)-cubes. This result is well-known for naive arithmetical planes [Reveill` es, G´ erard]. One can also gets centrosymmetry/palindrome properties for local configurations:

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Discrete rotations

Definition

A discrete rotation is defined as the composition of a Euclidean rotation with the rounding operation [x] := ⌊x + 0.5⌋.

Definition

We denote by rα the Euclidean rotation of angle α: rα : R2 → R2, − → v → » cos(α) − sin(α) sin(α) cos(α) – − → v . The discrete rotation [rα] is defined as [rα] : Z2 → Z2, − → v → [rα(− → v )].

• B. Nouvel and E. R´

emila, Configurations induced by discrete rotations: periodicity and quasiperiodicity properties.

• Disc. Appl. Math. 147 (2005), 325–343.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

From discrete rotations to word combinatorics

Configurations

We associate with a dicrete rotation a two-dimensional coding word Cα called configuration.

Coding of a Z2-action

This configurations is a Z2-word obtained as a coding of the action of two rotations

• n R2/Z2 with respect to a partition into a finite number of rectangles.

Densities

We deduce results on the densities of symbols in the configurations Cα.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Codings and discrete geometry

• Standard arithmetic discrete lines and Sturmian sequences.

Rotations over T

• Standard discrete planes and 2D Sturmian sequences

Z2-actions over T

• Functionality of arithmetic disrete planes and enumeration of (m, n)-cubes

Z2-actions over T

• Discrete rotation configurations

Z2-actions over T2

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Faces

Let F1, F2, and F3 be the three following faces: F1 = {λe2 + µe3, (λ, µ) ∈ [0, 1[2} F2 = {λe1 + µe3, (λ, µ) ∈ [0, 1[2} F3 = {λe1 + µe2, (λ, µ) ∈ [0, 1[2}. We call pointed face the set (p, q, r) + Ei.

1

e2 e3 e

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Faces

Let F1, F2, and F3 be the three following faces: F1 = {λe2 + µe3, (λ, µ) ∈ [0, 1[2} F2 = {λe1 + µe3, (λ, µ) ∈ [0, 1[2} F3 = {λe1 + µe2, (λ, µ) ∈ [0, 1[2}. We call pointed face the set (p, q, r) + Ei.

1

e2 e3 e

Distinguished vertex

Let F be the set of pointed faces. Let v : F → Z3 defined by v((p, q, r) + Fi) = (p, q, r) + e1 + · · · + ei−1, for i ∈ {1, 2, 3}. The vertex v(p, q, r) is called the distinguished vertex of the face (p, q, r) + Fi.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Stepped surface

Definition

A functional discrete surface is defined as a union of pointed faces such that the

• rthogonal projection onto the diagonal plane x + y + z = 0 induces an

homeomorphism from the discrete surface onto the diagonal plane.

Projection

Let π the orthogonal projection on the diagonal plane x + y + z = 0. This projection sends the lattice Z3 onto a lattice Γ in the diagonal plane.

Fact

The restriction of π to the set of vertices of a stepped surface is a bijection on its image Γ.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Two-dimensional coding

As done for stepped planes, one provides any discrete surface with a coding as a two-dimensional word over a three-letter alphabet, by associating with any point in the lattice Γ, the type of the face having its preimage as a distinguished vertex.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Two-dimensional coding

As done for stepped planes, one provides any discrete surface with a coding as a two-dimensional word over a three-letter alphabet, by associating with any point in the lattice Γ, the type of the face having its preimage as a distinguished vertex.

3 1 3 1 3 3 3 1 1 3 3 3 3 3 3 3

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Two-dimensional coding

As done for stepped planes, one provides any discrete surface with a coding as a two-dimensional word over a three-letter alphabet, by associating with any point in the lattice Γ, the type of the face having its preimage as a distinguished vertex.

• D. Jamet.

On the Language of Discrete Planes and Surfaces. In Proceedings of the Tenth International Workshop on Combinatorial Image Analysis, pages 227-241. Springer-Verlag, 2004.

• D. Jamet and G. Paquin,

Discrete surfaces and infinite smooth words, FPSAC’05-17th Annual International Conference on Formal Power Series and Algebraic Combinatorics, June 20–25, 2005 Taormina, Italy.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Two-dimensional coding

As done for stepped planes, one provides any discrete surface with a coding as a two-dimensional word over a three-letter alphabet, by associating with any point in the lattice Γ, the type of the face having its preimage as a distinguished vertex.

1 1 1 1 1 3 1 2 1 1 2 3 2 1 1 2 2 2 2 3 1 2 3 3 3 3 2 2 3 3 1 3 3 3 n m 1 2

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Back to tilings

Lozenge tiling

A lozenge tiling of the plane x + y + z = 0 is defined as a union of lozenges π((p, q, r) + Fi) such that this union covers ∆, and furthermore, the interiors of two lozenges do not intersect.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Back to tilings

Lozenge tiling

A lozenge tiling of the plane x + y + z = 0 is defined as a union of lozenges π((p, q, r) + Fi) such that this union covers ∆, and furthermore, the interiors of two lozenges do not intersect.

Lift [Thurston]

Let T be a lozenge tiling of the plane x + y + z = 0. Then there exists a unique functional stepped surface S, up to translation by the vector e1 + e2 + e3, that projects onto T .

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Flips

We define, for s ∈ Z3 ˇ cs =

3

[

i=1

(s + Fi) and ˆ cs =

3

[

i=1

(s + ei + Fi).

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Flips

We define, for s ∈ Z3 ˇ cs =

3

[

i=1

(s + Fi) and ˆ cs =

3

[

i=1

(s + ei + Fi).

• A functional stepped surface cannot contain simultaneously ˆ

cs and ˇ cs.

• If a functional stepped surface contains one of them, then by exchanging both

unions, we still have a functional stepped surface.

s

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Flips

We define, for s ∈ Z3 ˇ cs =

3

[

i=1

(s + Fi) and ˆ cs =

3

[

i=1

(s + ei + Fi).

Definition

The flip map ϕs is defined as follows: if a union of pointed faces E contains ˆ cs (resp. ˆ cs), then ϕs(E) is obtained by replacing ˆ cs by ˇ cs (resp. ˆ cs by ˇ cs). Otherwise, ϕs(E) = E.

s

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

Flips

We define, for s ∈ Z3 ˇ cs =

3

[

i=1

(s + Fi) and ˆ cs =

3

[

i=1

(s + ei + Fi).

Projection

Let π the orthogonal projection on the diagonal plane x + y + z = 0.

Local finiteness

A sequence of flips (ϕ−

→ s n)n∈N is said to be locally finite if, for any n0 ∈ N, the set

{sn ∈ Z3 , π(sn) = π(sn0)} is bounded.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A characterization of stepped surfaces by flips

Theorem [Arnoux-B.-Fernique-Jamet]

A union of faces U is a functional stepped surface if and only if there exist a stepped plane P and a locally finite sequence of flips (ϕsn)n∈N such that U = lim

n→∞ ϕsn ◦ . . . ◦ ϕs1(P).

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A characterization of stepped surfaces by flips

Theorem [Arnoux-B.-Fernique-Jamet]

A union of faces U is a functional stepped surface if and only if there exist a stepped plane P and a locally finite sequence of flips (ϕsn)n∈N such that U = lim

n→∞ ϕsn ◦ . . . ◦ ϕs1(P).

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Corollary

Let σ be a unimodular substitution over {1, 2, 3}. The generalized substitution Σσ acts properly on every functional stepped surface. Furthermore, the image by Σσ of a functional stepped surface is a functional stepped surface.

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A characterization of stepped surfaces by flips

Theorem [Arnoux-B.-Fernique-Jamet]

A union of faces U is a functional stepped surface if and only if there exist a stepped plane P and a locally finite sequence of flips (ϕsn)n∈N such that U = lim

n→∞ ϕsn ◦ . . . ◦ ϕs1(P).

Θ

∗

Θ Θσ

∗ σ ∗

Θσ

∗ σ

Discrete line Discrete planes Functionality Discrete rotations Stepped surfaces

A characterization of flip-accessibility

Flip-accessibility [Bodini-Fernique-Remila]

There exists a CNS for two parallelogram tilings of the whole plane to be linked by a sequence of flips.

Example

Tilings by Penrose tiles.