- Substitution tiling
A tile substitution is a useful method for constructing highly ordered tilings. Most importantly, some tile substitutions generate
aperiodic tiling s, which are tilings whose prototiles do not admit any tiling withtranslational symmetry . The most famous of these are thePenrose tiling s.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 someaperiodic 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:
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 tiling s, which are objects of interest in many fields ofmathematics , includingautomata theory ,combinatorics ,discrete geometry ,dynamical systems ,group theory ,harmonic analysis andnumber theory , not to mention the impact which were induced by those tilings incrystallography andchemistry . In particular, the celebratedPenrose tiling is an example of an aperiodic substitution tiling.History
In 1973 and 1974,
Roger Penrose discovered a family of aperiodic tilings, now calledPenrose tiling s. The first description was given in terms of 'matching rules' treating the prototiles asjigsaw 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 1977Robert 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 incrystallography , eventually leading to the discovery ofquasicrystals . 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 itsinterior .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 [D. Frettlöh, Duality of Model Sets Generated by Substitutions, Romanian J. of Pure and Applied Math. 50, 2005] .
A tile substitution with respect to the prototiles P is a pair , where is a
linear map , all of whoseeigenvalues 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 [A. Vince, Digit Tiling of Euclidean Space, in: Directions in Mathematical Quasicrystals, eds: M. Baake, R.V. Moody, AMS, 2000] .
Every tiling of , where any finite part of it is congruent to a subsetof some is called a substitution tiling (for the tile substitution ).
References
External links
# Dirk Frettlöh's and Edmund Harriss's [http://tilings.math.uni-bielefeld.de/tilings/index Encyclopedia of Substitution Tilings]
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