|
|
| Line 1: |
Line 1: |
| 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.
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Mississippi is the only location I've been residing in but I will have to move in a year or two. What me and my family adore is doing ballet but I've been taking on new issues lately. My day occupation is a travel agent.<br><br>Also visit my website ... [http://www.familysurvivalgroup.com/easy-methods-planting-looking-backyard/ tarot readings] |
| | |
| == Introduction ==
| |
| | |
| 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).
| |
| | |
| A simple example that produces a periodic tiling has only one prototile, namely a square:
| |
| | |
| <blockquote>
| |
| [[Image:subst-square.png]]
| |
| </blockquote>
| |
| | |
| 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.
| |
| | |
| <blockquote>
| |
| [[Image:subst-haus.png]]
| |
| </blockquote>
| |
| | |
| 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.
| |
| | |
| == History ==
| |
| | |
| 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.
| |
| | |
| == Mathematical definition ==
| |
| | |
| 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]].
| |
| | |
| 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'''.
| |
| | |
| A tile substitution is often loosely defined in the literature. A precise definition is as follows.<ref>
| |
| D. Frettlöh, Duality of Model Sets Generated by Substitutions, Romanian J. of Pure and Applied Math. 50, 2005</ref>
| |
| | |
| 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
| |
| : <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>
| |
| | |
| 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>
| |
| | |
| Every tiling of <math>{\mathbb R}^d</math>, where any finite part of it is congruent to a subset
| |
| of some <math>\sigma^k(T_i)</math> is called a substitution tiling (for the tile substitution <math>(Q, \sigma)</math>).
| |
| | |
| <!--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
| |
| : <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>
| |
| | |
| 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.-->
| |
| | |
| ==See also==
| |
| *[[Pinwheel tiling]]
| |
| | |
| ==References==
| |
| <references/>
| |
| * {{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 }}
| |
| | |
| == External links ==
| |
| | |
| # Dirk Frettlöh's and Edmund Harriss's [http://tilings.math.uni-bielefeld.de/tilings/index Encyclopedia of Substitution Tilings]
| |
| | |
| [[Category:Tessellation]]
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Mississippi is the only location I've been residing in but I will have to move in a year or two. What me and my family adore is doing ballet but I've been taking on new issues lately. My day occupation is a travel agent.
Also visit my website ... tarot readings