Tile substitution

In tile substitution geometry , this is a method for building mosaics . Most importantly, some tile substitutions form , that is, tilings whose do not form any mosaic with parallel transfer . The most famous of them are Penrose mosaics . Wildcard mosaics are special cases of end unit rules when geometric equality of tiles is not required.

Content

Introduction

Substitution of tiles is described by many protoplates. $T_{1},T_{2},\dots ,T_{m}$ ${\ displaystyle T_ {1}, T_ {2}, \ dots, T_ {m}}$ display extension $Q$ ${\ displaystyle Q}$ and a division rule specifying how to divide extended proto-tiles $QT_{i}$ ${\ displaystyle QT_ {i}}$ to form copies of some proto-tiles $T_{j}$ ${\ displaystyle T_ {j}}$ . Iterative tile swapping forms a mosaic on a plane called a wildcard mosaic . Some permutation mosaics are periodic , that is, they have translational symmetry . Among non-periodic permutation mosaics, some are , which means that their proto-tiles cannot be placed in the form of a periodic mosaic.

A simple example of creating periodic tiling with one tile, namely, a square:

Repeating this substitution, all large and large areas of the plane will be covered with a square grid. A more complex example of two protoflowers is shown below.

One can intuitively understand how this procedure forms a wildcard mosaic of the entire plane . The mathematical definition is given below. Wildcard mosaics are very useful as a way of defining aperiodic mosaics that are the objects of study in many areas of mathematics , including automata theory , combinatorics , combinatorial geometry , dynamical systems , group theory , harmonic analysis and number theory , not to mention the areas where these mosaics originated. crystallography and chemistry . In particular, the Penrose mosaic is an example of an aperiodic wildcard mosaic.

History

In 1973 and 1974, Roger Penrose discovered a family of aperiodic mosaics, now called Penrose mosaics . The first discovery was given in terms of “matching rules”, according to which the work with tiles went the same way as with pieces of a mosaic picture . The proof that copies of these proto-tiles can be joined together to form a plane mosaic , but this mosaic cannot form a periodic mosaic, uses a construction that can be considered as a substitution mosaic of proto-tiles. In 1977, discovered several sets of aperiodic proto-tiles, i.e. prototiles for which matching rules lead to non-periodic mosaics. In particular, he rediscovered Penrose's first example. This work affected scientists working in the field of crystallography , which ultimately led to the discovery of quasicrystals . Conversely, interest in quasicrystals led to the discovery of some well-ordered aperiodic mosaics. Many of them can easily be described as wildcard mosaics.

Mathematical Definition

Consider areas in ${\mathbb {R} }^{d}$ ${\ displaystyle {\ mathbb {R}} ^ {d}}$ which are , in the sense that a region is a non-empty compact subset, which is the closure of its interior .

Take a set of areas $\mathbf {P} =\{T_{1},T_{2},\dots ,T_{m}\}$ ${\ displaystyle \ mathbf {P} = \ {T_ {1}, T_ {2}, \ dots, T_ {m} \}}$ as protoplates. Prototype Placement $T_{i}$ ${\ displaystyle T_ {i}}$ Is a couple $(T_{i},\varphi )$ ${\ displaystyle (T_ {i}, \ varphi)}$ where $\varphi$ ${\ displaystyle \ varphi}$ is an isometry ${\mathbb {R} }^{d}$ ${\ displaystyle {\ mathbb {R}} ^ {d}}$ . Form $\varphi (T_{i})$ ${\ displaystyle \ varphi (T_ {i})}$ called the placement area. Mosaic T is a set of proto-tile placement regions in which the inner regions of proto-tiles do not have common parts. We say that a mosaic T is a mosaic on W if W is a union of placement regions from T.

Substitution of tiles in the literature is often not well defined. The exact definition is as follows ^[1] .

Substrate tile for protoplates P is a pair $(Q,\sigma )$ ${\ displaystyle (Q, \ sigma)}$ where $Q:{\mathbb {R} }^{d}\to {\mathbb {R} }^{d}$ ${\ displaystyle Q: {\ mathbb {R}} ^ {d} \ to {\ mathbb {R}} ^ {d}}$ is a linear mapping , all eigenvalues of which are greater than unity modulo, and the substitution rules $\sigma$ ${\ displaystyle \ sigma}$ display $T_{i}$ ${\ displaystyle T_ {i}}$ to the tile $QT_{i}$ ${\ displaystyle QT_ {i}}$ . Tile swap $\sigma$ ${\ displaystyle \ sigma}$ generates a map from any tile T of the region W to the tile $\sigma (\mathbf {T} )$ ${\ displaystyle \ sigma (\ mathbf {T})}$ areas of $Q_{\sigma }(\mathbf {W} )$ ${\ displaystyle Q _ {\ sigma} (\ mathbf {W})}$

\sigma (\mathbf {T} )=\bigcup _{(T_{i},\varphi )\in \mathbf {T} }\{(T_{j},Q\circ \varphi \circ Q^{-1}\circ \rho ):(T_{j},\rho )\in \sigma (T_{i})\}.

{\ displaystyle \ sigma (\ mathbf {T}) = \ bigcup _ {(T_ {i}, \ varphi) \ in \ mathbf {T}} \ {(T_ {j}, Q \ circ \ varphi \ circ Q ^ {- 1} \ circ \ rho) :( T_ {j}, \ rho) \ in \ sigma (T_ {i}) \}.}

Note that proto-tiles can be deduced from the tile substitution. Thus, there is no need to include them in tile substitutions $(Q,\sigma )$ ${\ displaystyle (Q, \ sigma)}$ ^[2] .

Any tiling ${\mathbb {R} }^{d}$ ${\ displaystyle {\ mathbb {R}} ^ {d}}$ any finite part of which is congruent to a subset of some $\sigma ^{k}(T_{i})$ ${\ displaystyle \ sigma ^ {k} (T_ {i})}$ , called wildcard mosaic (for tile swapping $(Q,\sigma )$ ${\ displaystyle (Q, \ sigma)}$ )

Notes

↑ Frettlöh, 2005 , p. 619-639.
↑ Vince, 2000 , p. 329-370.

Links

Dirk Frettlöh's and Edmund Harriss's Encyclopedia of Substitution Tilings

[_5c1ed81be8c00e41-1] Frettlöh, 2005 , p. 619-639.

[_4894bbc480a3910c-2] Vince, 2000 , p. 329-370.