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| In geometry, a '''tile substitution''' is a useful method for constructing highly ordered [[Tessellation|tiling]]s. Most importantly, some tile substitutions generate [[aperiodic tiling]]s, which are tilings whose prototiles do not admit any tiling with [[translational symmetry]]. The most famous of these are the [[Penrose tiling]]s. Substitution tilings are special cases of [[finite subdivision rules]], which do not require the tiles to be geometrically rigid.
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| == Introduction ==
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| A tile substitution is described by a [[Set (mathematics)|set]] of '''prototiles''' (tile shapes) <math>T_1,T_2,\dots, T_m</math>, an '''expanding map''' <math>Q</math> and a '''dissection rule''' showing how to dissect the expanded prototiles <math>Q T_i</math> to form copies of some prototiles <math>T_j</math>. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a '''substitution tiling'''. Some substitution tilings are [[Periodic function|periodic]], defined as having [[translational symmetry]]. Among the nonperiodic substitution tilings are some [[aperiodic tiling]]s, those whose prototiles cannot be rearranged to form a periodic tiling (usually if one requires in addition some matching rules).
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| A simple example that produces a periodic tiling has only one prototile, namely a square:
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| <blockquote>
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| [[Image:subst-square.png]]
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| </blockquote>
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| By iterating this tile substitution, larger and larger regions of the plane are covered with a square grid. A more sophisticated example with two prototiles is shown below, with the two steps of blowing up and dissecting are merged into one step in the figure.
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| <blockquote>
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| [[Image:subst-haus.png]]
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| </blockquote>
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| One may intuitively get an idea how this procedure yields a substitution tiling of the entire [[Plane (mathematics)|plane]]. A mathematically proper definition is given below. Substitution tilings are notably useful as ways of defining [[aperiodic tiling]]s, which are objects of interest in many fields of [[mathematics]], including [[automata theory]], [[combinatorics]], [[discrete geometry]], [[dynamical systems]], [[group theory]], [[harmonic analysis]] and [[number theory]], not to mention the impact which were induced by those tilings in [[crystallography]] and [[chemistry]]. In particular, the celebrated [[Penrose tiling]] is an example of an aperiodic substitution tiling.
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| == History ==
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| In 1973 and 1974, [[Roger Penrose]] discovered a family of aperiodic tilings, now called [[Penrose tiling]]s. The first description was given in terms of 'matching rules' treating the prototiles as [[jigsaw puzzle]] pieces. The proof that copies of these prototiles can be put together to form a [[tessellation|tiling]] of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 [[Robert Ammann]] discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings; in particular, he rediscovered Penrose's first example. This work gave an impact to scientists working in [[crystallography]], eventually leading to the discovery of [[quasicrystals]]. In turn, the interest in quasicrystals led to the discovery of several well-ordered aperiodic tilings. Many of them can be easily described as substitution tilings.
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| == Mathematical definition ==
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| We will consider '''regions''' in <math>{\mathbb R}^d</math> that are [[well-behaved]], in the sense that a region is a nonempty compact subset that is the [[closure (topology)|closure]] of its [[Interior (topology)|interior]].
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| We take a set of regions <math>\bold{P} = \{ T_1, T_2,\dots, T_m \}</math> as prototiles. A '''placement''' of a prototile <math>T_i</math> is a pair <math>( T_i, \varphi )</math> where <math>\varphi</math>is an [[isometry]] of <math>{\mathbb R}^d</math>. The image <math>\varphi(T_i)</math> is called the placement's region. A '''tiling T''' is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling '''T''' is a '''tiling of W''' where '''W''' is the union of the regions of the placements in '''T'''.
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| A tile substitution is often loosely defined in the literature. A precise definition is as follows.<ref>
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| D. Frettlöh, Duality of Model Sets Generated by Substitutions, Romanian J. of Pure and Applied Math. 50, 2005</ref>
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| A '''tile substitution''' with respect to the prototiles '''P''' is a pair <math>(Q, \sigma)</math>, where <math>Q: {\mathbb R}^d \to {\mathbb R}^d</math> is a [[linear map]], all of whose [[eigenvalues]] are larger than one in modulus, together with a '''substitution rule''' <math>\sigma</math> that maps each <math>T_i</math> to a tiling of <math>Q T_i</math>. The tile substitution <math>\sigma</math> induces a map from any tiling '''T''' of a region '''W''' to a tiling <math>\sigma(\bold{T})</math> of <math>Q_\sigma(\bold{W})</math>, defined by
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| : <math>\sigma(\bold{T}) = \bigcup_{(T_i,\varphi) \in \bold{T}} \{ ( T_j, Q \circ \varphi \circ Q^{-1} \circ \rho ) : (T_j, \rho) \in \sigma(T_i) \} .</math>
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| Note, that the prototiles can be deduced from the tile substitution. Therefore it is not necessary to include them in the tile substitution <math>(Q,\sigma)</math>.<ref>A. Vince, Digit Tiling of Euclidean Space, in: Directions in Mathematical Quasicrystals, eds: M. Baake, R.V. Moody, AMS, 2000</ref>
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| Every tiling of <math>{\mathbb R}^d</math>, where any finite part of it is congruent to a subset
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| of some <math>\sigma^k(T_i)</math> is called a substitution tiling (for the tile substitution <math>(Q, \sigma)</math>).
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| <!--Consider a tiling <math>\bold{T}_0 = \{ (T_j, \varphi_0) \}</math>, and suppose that ''k'' iterations of an expanding map <math>\sigma</math> map <math>\bold{T}_0</math> to a tiling that contains a placement <math>(T_j,\rho_0)</math> whose region is interior to the region of <math>\sigma^k(\bold{T_0})</math>. This allows us to define sequence of tilings <math>\bold{T}_1, \bold{T}_2, \bold{T}_3, \cdots</math> where
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| : <math>\bold{T}_{i+1} = \{ (T_j, \varphi_0 \circ \rho_0^{-1} \circ \rho ) : (T_j, \rho) \in \sigma^k(\bold{T}_i) \} .</math>
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| Then each <math>\bold{T}_i</math> is a subset of <math>\bold{T}_{i+1}</math>, and <math>\bold{T} = \bigcup_i \bold{T}_i</math> is a tiling of <math>{\mathbb R}^d</math>. The tiling <math>\bold{T}</math> is called a substitution tiling.-->
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| ==See also==
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| *[[Pinwheel tiling]]
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| ==References==
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| <references/>
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| * {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| == External links ==
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| # Dirk Frettlöh's and Edmund Harriss's [http://tilings.math.uni-bielefeld.de/tilings/index Encyclopedia of Substitution Tilings]
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| [[Category:Tessellation]]
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