Flip Dynamics,
Structure of Tiling Spaces.
Eric Rémila
Université de Lyon
GATE LSE (umr 5824 CNRS)
Université Jean Monnet Saint-Etienne.
3 september 2015
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Flips
Figure: a domino flip and an lozenge fllip
Flip : local transformation of a tiling involving a few tiles.
In this lecture, we will work with domino tilings, but lozenge tilings
can be treated in a similar way.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Domain
Domain : finite simply connected (i.e. with no hole) union of cells
of the square lattice.
Figure: Left : a domain
Right : a non simply connected region
We study the set of tilings of a fixed domain D.
Conventions : the origin O = (0, 0) is on the boundary of D.
In the coloring of cells of Z2 , as a checkerboard, the cell whose
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lower left corner is O is white.
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Tiling space
Tiling space of D : the undirected graph
whose vertex set is the set of tilings of D,
the pair (T , T 0 ) is an edge if one can pass from T to T 0 by a
single flip.
Figure: A tiling space
Question : what about the structure of tiling spaces ?
Eric Rémila (U. Lyon)
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Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
A tool : path value
First direct edges of the square lattice, according to cell colorings.
An edge is well directed when it is directed in such a way.
Definition
δh (v , v 0 ) = 1 if (v , v 0 ) is well directed,
δh (v , v 0 ) = −1 otherwise (i.e. if δh (v 0 , v ) = 1).
By extension, for each path P of Z2 , starting in O :
X
δh (P) =
δh (v , v 0 )
(v ,v 0 ) is an edge of P
Figure: Computation of δh (P) (first way)
Eric Rémila (U. Lyon)
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Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Path value, definition with no need of orientation
Compute the word wP , in the alphabet {v , h}, induced by the
path P.
Color edges of Z, alternatively with v and h, with the
convention : color ([0, 1]) = h iff the origin of P is the lower
left corner of a white cell.
Compute the path P 0 of Z starting from 0 encoded by wP .
δh (P) = endpoint of P 0
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Figure: Computation of δh (P) (second way).
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Cycle value
Remark : The value δh (C ) of a cycle C around a single domino is
null.
Proposition : Let C be an elementary cycle takes place in a tiled
domain, and cuts no tile. We have
δh (C ) = 0.
Two proofs
by the “camel principle”,
by induction on the number of dominoes included in the cycle.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Height function of a tiling : definition
Corollary : if P and P 0 are paths with the same endpoints, and cut
no tile, then δh (P) = δh (P 0 ).
This allows the consistence of the following definition :
Definition : For each tiling T of a domain D, and each vertex v ,
hT (v ) = δ(P(v ,T ) )
where P(v ,T ) denotes any path, from O of the boundary of D to v ,
which cuts no tile in T .
-2
0
1
0
-1
-3
-1
-2
-1
0
-2
-3
-2
1
0
-3
0
-1
1
2
-1
-2
-2
0
-3
-2
-1
-2
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Figure: From a tiling to its height function
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Height function of a tiling : directing the flips.
Remark : If v is on the boundary of D, then the value hT (v ) does
not depend on the tiling T .
Remark : if T and T 0 only differ by a single flip located in v , then
hT (v 0 ) = hT 0 (v 0 ) for v 0 6= v ,
|hT (v ) − hT 0 (v )| = 4.
Figure: upwards flips
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This allows to give an orientation to flips.
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Height function of a tiling : directed tiling space
The tiling space becomes a directed acyclic graph.
Figure: Tiling space with edges directed by height functions university-logo
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Height function of a tiling : flip interpretation
Proposition : let T and T 0 be two tilings of D. The following
conditions are equivalent :
1
2
for each vertex v of D, hT (v ) ≤ hT 0 (v )
there exists a finite sequence (T = T0 , T1 , ..., Tp = T 0 ) such
that, for each i < p, one can pass from Ti to Ti+1 by a single
upward flip.
(2) =⇒ (1) is obvious,
(1) =⇒ (2) ....
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Height function of a tiling : local characterisation
Proposition : (local characterization) Let h be a function
V → Z. there exists a tiling T such that h = hT if and only if :
f (O) = 0,
for each (well) directed edge (v , v 0 )
either h(v 0 ) = h(v ) + 1 or h(v 0 ) = h(v ) − 3,
for each (well) directed edge (v , v 0 ) such that [v , v 0 ] is on the
boundary of D, h(v 0 ) = h(v ) + 1 .
=⇒ is obvious,
⇐= ....
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Lattice structure
Applying the proposition of local characterization, one gets the
following :
Proposition : (Lattice structure) let T and T 0 be two tilings of
D. There exists
a tiling Tmin such that hTmin = min(hT , hT 0 ),
a tiling Tmax such that hTmax = max(hT , hT 0 ).
...
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Summary of the previous part
Tilings ⇐⇒ Height functions
Therefore tilings can be ordered in such a way that the tiling
space becomes a (distributive) lattice.
The order can be interpreted with flips.
The order confers to the tiling space a structure of
(distributive) lattice.
Now, we can turn towards applications.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Flip connectivity
From the lattice structure, the space tiling admits a global
minimal tiling T0 .
From the geometrical interpretation, for any tiling T , there
exists a sequence of upward flips to pass from T0 to T .
Thus :
Proposition : The tiling space is connected : for any pair (T , T 0 )
of tilings, one can pass from T to T ” be a sequence of flips.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Tiling algorithm (preliminaries)
Question : given a domain D, how to compute a tiling of D (or
claim that there is no tiling) ?
Idea : compute the minimal tiling T0 .
Lemma : (convexity lemma) let M0 = max{hT0 (v ), v ∈ D}.
If hT0 (v ) < M0 , then there exists an edge (v , v 0 ) such that
hT0 (v 0 ) = hT0 (v ) + 1,
If M0 = hT0 (v ), then v is the boundary of D.
Proof : in two cases, according to the position of v :
if v is in the interior of D, then...
Thus M0 > hT0 (v ).
il v is in the boundary of D, then it is obvious.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Tiling algorithm (realization)
The minimal tiling T0 can be constructed from the top to the
bottom, “slice by slice”.
Initialisation : construct hT0 on the boundary of D.
Loop : for M ≤ M0 , let VM = {v 0 , hT0 (v 0 ) ≥ M}.
Assume that hT0 is constructed on VM . We have hT0 (v ) = M − 1
/ VM and there exists a well directed edge
if and only if hT0 (v ) ∈
(v , v 0 ) such that hT0 (v 0 ) = M. This allows to construct VM−1 .
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Figure: first steps of the algorithm : successive constructions of V2 , V0 ,
V−1 and V−2 .
Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Tiling algorithm (a failure case)
There is no tiling when the local characterization is not satisfied.
Figure: A case when the algorithm detects an impossibility.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Distance between two tilings
Proposition : let d(T , T 0 ) be the minimal number of flips to pass
from T to T 0 . We have :
1X
d(T , T 0 ) =
|hT (v ) − hT 0 (v )|
4 v
The inequality : d(T , T 0 ) ≥
1
4
For the inequality d(T , T 0 ) ≤
P
|hT (v ) − hT 0 (v )| is obvious.
v
1
4
P
v
|hT (v ) − hT 0 (v )| :
d(T , T 0 ) ≤ d(T , Tmin ) + d(Tmin , T 0 )
...
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))
Conclusion
Height functions allows to get some structural properties of tilings
From these properties, we have the following applications :
connectivity by flips,
tiling algorithm,
flip distance between tilings,
random sampling of tilings
(lecture from Olivier Bodini, tomorrow)
Thank you for all.
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Eric Rémila (U. Lyon)
Flip Dynamics (Tilings and Tessellations (Isfahan, 2015))