Runoff curve number: Difference between revisions
en>Danames No edit summary |
en>Otumire m fixed broken link |
||
| 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. | |||
== 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]] | |||
Revision as of 18:52, 22 January 2014
In geometry, a tile substitution is a useful method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
Introduction
A tile substitution is described by a set of prototiles (tile shapes) , an expanding map and a dissection rule showing how to dissect the expanded prototiles to form copies of some prototiles . Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Among the nonperiodic substitution tilings are some aperiodic tilings, 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:
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.
One may intuitively get an idea how this procedure yields a substitution tiling of the entire plane. A mathematically proper definition is given below. Substitution tilings are notably useful as ways of defining aperiodic tilings, 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 tilings. 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 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 that are well-behaved, in the sense that a region is a nonempty compact subset that is the closure of its interior.
We take a set of regions as prototiles. A placement of a prototile is a pair where is an isometry of . The image 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.[1]
A tile substitution with respect to the prototiles P is a pair , where is a linear map, all of whose eigenvalues are larger than one in modulus, together with a substitution rule that maps each to a tiling of . The tile substitution induces a map from any tiling T of a region W to a tiling of , defined by
Note, that the prototiles can be deduced from the tile substitution. Therefore it is not necessary to include them in the tile substitution .[2]
Every tiling of , where any finite part of it is congruent to a subset of some is called a substitution tiling (for the tile substitution ).
See also
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Dirk Frettlöh's and Edmund Harriss's Encyclopedia of Substitution Tilings