Local Rules for Planar Tilings
Nicolas B´edaride (I2M) Thomas Fernique (LIPN)
Outline
Planar tilings
Local rules
Coincidences
Subperiods
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Outline
Planar tilings
Local rules
Coincidences
Subperiods
n → d tilings
Let~v1, . . . , ~vn be vectors inRd that define the tiles:
Ti1,...,id :=
d
X
j=1
λij~vij
0≤λij ≤1
.
Definition (n →d tiling)
Ann →d tiling is a “face-to-face” tiling of Rd by such tiles.
Such a tiling naturally lifts onto ad-dim. surface of Rn via~vi 7→~ei.
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n → d tilings
Let~v1, . . . , ~vn be vectors inRd that define the tiles:
Ti1,...,id :=
d
X
j=1
λij~vij
0≤λij ≤1
.
Definition (n →d tiling)
Ann →d tiling is a “face-to-face” tiling of Rd by such tiles.
Such a tiling naturally lifts onto ad-dim. surface of Rn via~vi 7→~ei.
Examples
A rhombille tiling in Marmoutier Abbey (Alsace).
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Examples
Examples
A Penrose parketry (Paris).
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Planarity
Definition (Planar tiling)
Ann→d tiling is said to beplanarif it lifts into a tubeE+ [0,1]n, whereE is an affine d-plane ofRn called theslopeof the tiling.
n d example
2 1 Sturmian words 3 1 Billiard words 3 2 Discrete planes
4 2 Ammann-Beenker tilings 5 2 Penrose tilings
6 3 Icosahedral tilings ... ... ...
Quasiperiodicity
Planar tilings arequasiperiodic (oruniformly recurrent):
any pattern reoccurs within bounded distance from any point.
They are often use to modelquasicrystals.
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Quasiperiodicity
Planar tilings arequasiperiodic (oruniformly recurrent):
any pattern reoccurs within bounded distance from any point.
They are often use to modelquasicrystals.
Outline
Planar tilings
Local rules
Coincidences
Subperiods
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Local rules
Apattern is a finite union of tiles:
Definition (Local rules)
A tilingT admits local rulesif there is a finite setF of patterns so that any tiling without pattern inF has the same patterns as T. In other words: the local appearance enforces the global ones.
Local rules
Apattern is a finite union of tiles:
Definition (Local rules)
A tilingT admits local rulesif there is a finite setF of patterns so that any tiling without pattern inF has the same patterns as T.
In other words: the local appearance enforces the global ones.
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Local rules
Apattern is a finite union of tiles:
Definition (Local rules)
A tilingT admits local rulesif there is a finite setF of patterns so that any tiling without pattern inF has the same patterns as T. In other words: the local appearance enforces the global ones.
Local rules for planar tilings
Question
Which planarn→d tilings admit local rules?
I Every rational slopes (periodic tiling).
I Not every irrational slopes (cardinality).
Theorem (Penrose, 1978 & De Bruijn, 1981) The Penrose tilings (quadratic slope) admit local rules.
Theorem (Burkov, 1988 & B´edaride-Fernique, 2015) Not the Ammann-Beenker tilings (quadratic slope as well).
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Local rules for planar tilings
Question
Which planarn→d tilings admit local rules?
I Every rational slopes (periodic tiling).
I Not every irrational slopes (cardinality).
Theorem (Penrose, 1978 & De Bruijn, 1981) The Penrose tilings (quadratic slope) admit local rules. Theorem (Burkov, 1988 & B´edaride-Fernique, 2015) Not the Ammann-Beenker tilings (quadratic slope as well).
Local rules for planar tilings
Question
Which planarn→d tilings admit local rules?
I Every rational slopes (periodic tiling).
I Not every irrational slopes (cardinality).
Theorem (Penrose, 1978 & De Bruijn, 1981) The Penrose tilings (quadratic slope) admit local rules.
Theorem (Burkov, 1988 & B´edaride-Fernique, 2015) Not the Ammann-Beenker tilings (quadratic slope as well).
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Local rules for planar tilings
Question
Which planarn→d tilings admit local rules?
I Every rational slopes (periodic tiling).
I Not every irrational slopes (cardinality).
Theorem (Penrose, 1978 & De Bruijn, 1981) The Penrose tilings (quadratic slope) admit local rules.
Theorem (Burkov, 1988 & B´edaride-Fernique, 2015) Not the Ammann-Beenker tilings (quadratic slope as well).
A great tool
Definition (Window)
Thewindow W of a planarn→d tiling of slopeE is the orthogonal projection of [0,1]n ontoE⊥:
W :=π0([0,1]n).
Proposition
To anypointedpattern P corresponds a subregion R of the window in which project the vertices which point this pattern in the tiling:
R := \
π~y∈V(P)
W −π0(~y−~x) .
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Examples
Examples
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Outline
Planar tilings
Local rules
Coincidences
Subperiods
Exceptional pattern
Definition (Coincidence)
Acoincidenceof a planarn→d tiling is a set ofn−d+ 1 unit faces ofZn of dim. n−d−1 with a common intersection inW.
This is the smallest region corresponding to a pointed pattern. This patterns appears at most once in the tiling (if aperiodic).
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Exceptional pattern
Definition (Coincidence)
Acoincidenceof a planarn→d tiling is a set ofn−d+ 1 unit faces ofZn of dim. n−d−1 with a common intersection inW. This is the smallest region corresponding to a pointed pattern.
This patterns appears at most once in the tiling (if aperiodic).
Coincidences and local rules
Theorem (B´edaride-Fernique, 2020)
If a planar tiling has local rules, then its slope ischaracterized by finitely many coincidences.
The converse holds. . . up to planarity! Proof sketch.
(⇒) Otherwise, we could change the slope without modifying patterns of a given size (easy, wait the next slide).
(⇐) We show that any coincidence is ensured in any planar tiling by patterns of a sufficiently large size (4 lemmas, read the paper).
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Coincidences and local rules
Theorem (B´edaride-Fernique, 2020)
If a planar tiling has local rules, then its slope ischaracterized by finitely many coincidences. The converse holds. . . up to planarity!
Proof sketch.
(⇒) Otherwise, we could change the slope without modifying patterns of a given size (easy, wait the next slide).
(⇐) We show that any coincidence is ensured in any planar tiling by patterns of a sufficiently large size (4 lemmas, read the paper).
Coincidences and local rules
Theorem (B´edaride-Fernique, 2020)
If a planar tiling has local rules, then its slope ischaracterized by finitely many coincidences. The converse holds. . . up to planarity!
Proof sketch.
(⇒) Otherwise, we could change the slope without modifying patterns of a given size (easy, wait the next slide).
(⇐) We show that any coincidence is ensured in any planar tiling by patterns of a sufficiently large size (4 lemmas, read the paper).
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Example
Any finite set of coicidences of Ammann-Beenker tilings can be preserved by modifying suitably its slope (left).
Example
Any finite set of coicidences of Ammann-Beenker tilings can be preserved by modifying suitably its slope (left).
Small patterns are preserved, only theirfrequencies change (right).
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Example
Any finite set of coicidences of Ammann-Beenker tilings can be preserved by modifying suitably its slope (left).
Example
Any finite set of coicidences of Ammann-Beenker tilings can be preserved by modifying suitably its slope (left).
Small patterns are preserved, only theirfrequencies change (right).
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Example
Any finite set of coicidences of Ammann-Beenker tilings can be preserved by modifying suitably its slope (left).
In equations
Proposition (B´edaride-Fernique, 2020)
Each coincidence of a planar tiling corresponds to an algebraic equation on theGrassmann coordinates of its slope.
For ann→d tiling, this equation is homogeneous of degreen−d. Corollary (Le, 1995)
If a slope is charaterized by forbidden patterns, then it is algebraic.
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Outline
Planar tilings
Local rules
Coincidences
Subperiods
Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
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Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
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Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
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Hidden periods
Definition (Subperiod)
Asubperiodof a planarn →d tiling is a vector in its slope with d+ 1 integer entries.
In the window
Proposition
A planar n→d tiling has a subperiod iff there are infinitely many points ofZn which project into a facet of its (shifted) window.
Proof sketch.
A subperiodp~∈E can be written~p =~q+~x with
I ~q ∈Zn (the ”floor part”),
I ~x in ann−d −1 dim. unit face ofZn(the ”fractional part”).
The integer vector~q thus projects into a facet of the window:
π0p~= 0 =π0~q+π0~x.
Ify∈Zn projects into this facet, theny+~q as well (up to±~ei).
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In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
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In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
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In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
In pictures
When the window is shifted, points inπ0Zn enter/exit in groups.
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A consequence
Proposition
A planar tiling without any subperiod does not admit local rules.
A consequence
Proposition
A planar tiling without any subperiod does not admit local rules.
A slope shift(in a suitable direction)does not change the patterns.
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A consequence
Proposition
A planar tiling without any subperiod does not admit local rules.
A consequence
Proposition
A planar tiling without any subperiod does not admit local rules.
A partial change breaks planarity but leaves the patterns intact.
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Further results
Theorem (Levitov, 1988)
If a planar tiling has local rules, then it has alltypes of subperiods.
The converse may not hold (e.g., Ammann-Beenker).
Theorem (B´edaride-Fernique, 2015, 2017)
A planar4→2tiling has local rules iff its slope is characterized by its subperiods.
Open question
Does the same holds for planarn→d tilings? Remark
Subperiods may force a continuum of planar tilings (e.g., Beenker).
Further results
Theorem (Levitov, 1988)
If a planar tiling has local rules, then it has alltypes of subperiods.
The converse may not hold (e.g., Ammann-Beenker).
Theorem (B´edaride-Fernique, 2015, 2017)
A planar4→2tiling has local rules iff its slope is characterized by its subperiods.
Open question
Does the same holds for planarn→d tilings? Remark
Subperiods may force a continuum of planar tilings (e.g., Beenker).
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Further results
Theorem (Levitov, 1988)
If a planar tiling has local rules, then it has alltypes of subperiods.
The converse may not hold (e.g., Ammann-Beenker).
Theorem (B´edaride-Fernique, 2015, 2017)
A planar4→2tiling has local rules iff its slope is characterized by its subperiods.
Open question
Does the same holds for planarn→d tilings?
Remark
Subperiods may force a continuum of planar tilings (e.g., Beenker).
Further results
Theorem (Levitov, 1988)
If a planar tiling has local rules, then it has alltypes of subperiods.
The converse may not hold (e.g., Ammann-Beenker).
Theorem (B´edaride-Fernique, 2015, 2017)
A planar4→2tiling has local rules iff its slope is characterized by its subperiods.
Open question
Does the same holds for planarn→d tilings?
Remark
Subperiods may force a continuum of planar tilings (e.g., Beenker).
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Outline
Planar tilings
Local rules
Coincidences
Subperiods
Selected references
N. G. De Bruijn,Algebraic theory of Penrose’s nonperiodic tilings of the plane, Indag. Math.43 (1981)
L. S. Levitov,Local rules for quasicrystals, Comm. Math.
Phys.119 (1988)
J. E. S. Socolar Weak matching rules for quasicrystals, Comm.
Math. Phys.129(1990)
T. Q. T. Le,Local rules for quasiperiodic tilings, in NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci.489(1995)
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Our papers
The Ammann-Beenker tilings revisited, in Aperiodic Crystals (2013)
When periodicities enforce aperiodicity, Comm. Math. Phys.
335(2015)
No weak local rules for the4p-fold tilings, Disc. Comput.
Geom.54(2015)
Weak local rules for octagonal tilings, Israel J. Math.222 (2017)
Canonical projection tilings defined by patterns, Geom. Dedic.
(2020)
