Effective Tilings
CIMPA - Isfahan
td n◦
Self-Assembly
Exercice 1.
Stability, Transition
1.1 What is the maximal temperature at which the following patterns are stable ?
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1.2 At temperature 2, which transitions are possible from this configuration ? All tiles in the
tileset are already used in the pattern.
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Exercice 2.
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Optimal-time assembly of a square
Let S k = {[0, i[2 |i > k} be the set of all squares with edge size at least k.
A self-assembling system “assembles squares” if its seed is composed of just one tile, and its set
of final productions is S k for some k.
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Effective Tilings (td n◦ )
CIMPA - Isfahan
Recall that a production p of a self-assembling system S is ordered if there is a partial order
< p on the positions of p such that, for two positions z1 , z2 ∈ p, in any assembly sequence of p,
z1 < p z2 if and only if z1 is covered before z2 . In this case, the order < p is unique.
2.1 The system given below assembles squares. Show that it is ordered : give the order for
its “typical” productions. What is the depth of the order for the final production that is a k × k
square ?
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Let S be a self-assembling system which assembles squares. Assume all productions of S are
ordered. Define c(p) to be the function which gives the depth of the order associated with the
production p. Additionally, assume that S only has one final production pk of each size k.
2.2 Give a lower bound on c(pk ).
2.3 Show that the following order has depth c(pk ) = 2k, and that it there is a tileset which
assembles squares with this order.
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Effective Tilings (td n◦ )
Exercice 3.
CIMPA - Isfahan
Unassemblability
A set X of shapes is assemblable if there is a self-assembling system S such that the set of shapes
of final productions of S is X.
3.1 Show that the set L = {{0, . . . , n} × {0} ∪ {0, . . . , n} × {0}|n ∈ N} is not assemblable.
3.2 Show that the set H = {{0, . . . , −n} × {0, . . . , −n} ∪ {0, . . . , n} × {0, . . . n}|n ∈ N} is not
assemblable.
Let T be a self-assembling system. Let E be a cut of Z2 , that is a set of edges such that Z2 \ E
is disconnected into two parts, E + and E − . A window-movie associated on E is a sequence of
tuples (ti , ei , si ), where ti is a tile of T , ei is an edge of E, and si ∈ {+, −}.
Let R ⊂ Z2 , and B be its border, i.e. the set of edges separating R and Z2 \ R. Let s be an assembly
sequence of T , the window-movie T B (s) is the sequence (gi , ei , si ) where ti is the i-th tile added to
a position z adjacent to B, ei is the edge of B that is adjacent to z, and si is + if z ∈ R, − elsewhere.
If several edges are adjacent to z, then several copies are added, in the order N, E, S , W. Finally,
we define T B = {T B (s)|s is an assembly sequence of T }.
3.3 Let R = {(x, y) ∈ Z2 |y > 2 or x > 2}, and B its border. Give T B for the self-assembling
system of question .3.
3.4 Let R ⊂ Z2 , B its border, and τ a translation. Assume R and τ(R) do not intersect the seed
of T . Show that if τ(T B ) = T τ(B) , then for any production p, there is a production p0 such that
τ(p ∩ R) = p0 ∩ τ(R)
3.5 Show that for any s, the sets L and H of questions 1 and 2 are not assemblable. We say that
they are not assemblable at any scale.
Exercice 4.
Sierpinski
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Sierpinski’s Triangle is the set S = {(x, y)| x+y
0[2]}.
We say that a self-assembling system T weakly assembles an infinite set X at scale 1 if there is a
subset M of the tiles of T such that if p is a final production of T , p−1 (M) = X.
4.1 Give a self-assembling system which weakly assembles S .
T strictly assembles an infinite set X at scale s if for any final production of T , dom(T ) = sX.
4.2 Show that there is no self-assembling system which self-assembles S at scale 1. hint : Show
that S can be broken in two pieces using a single edge at an infinity of places.
4.3 Show for all s, that there is no self-assembling system which self-assembles S at scale s.
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