Tilings

From CanisiusmathWiki

Table of contents

1 What Is a Tiling?

2 Quasicrystals

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What Is a Tiling?

There are various more or less rigorous and/or restrictive definitions of tilings, but for our purposes we will use the definition offered in Tilings and Patterns by Grunbaum/Shephard:

Definition: "A plane tiling is a countable family of closed sets $\mathcal{T}=\{T_{1},T_{2},...\}$ [...such that] the union of the sets $T 1, T 2,...$ (which are known as the tiles of $\mathcal{T}$ ) is to be the whole plane, and the interiors of the sets $T i$ are to be pairwise disjoint." (Grunbaum/Shephard, 16)

[One important part of this definition is that tilings are always infinite. The black and white square "tiles" in your bathroom cover the floor, but to refer to such a pattern as a tiling would be incorrect, because tilings cover the entire plane.]

To avoid unwanted configurations and tiles with bizarre shapes, we instate the following restriction:

Restriction: Let $T i$ , $T j$ be two tiles in $\mathcal{T}$ . Then (a) $T i$ and $T j$ are each homeomorphic to a closed circular disk, and thus is a bounded, connected, and simply connected set, and (b) $T_{i}\cap T_{j}=\{x\}$ is connected.

Definition: Let $T_{i}\cap T_{j}=\{x\}$ . If $x$ is a point in the plane, then $x$ is called a vertex. If $x$ is an arc, then $x$ is called an edge. We will deal only with tiles that have a finite number of vertices and edges.

Definition: Let $\mathcal{T}$ be partitioned into equivalence classes. Let $\mathcal{P}$ be a set of representatives of these classes. We say that $\mathcal {P}$ is a protoset for $\mathcal{T}$ , and each representative is a prototile. We will only consider cases with finite protosets.

Definition: If a tiling, $\mathcal{T}$ has protoset $\mathcal{P}$ , then we say that $\mathcal{P}$ admits $\mathcal{T}$ .

Definition: A patch is a finite region of a tiling. More specifically, it is the union of a finite number of tiles such that the set is simply connected and will never become disconnected if a single point is deleted.

Definition: A arrangement of tiles is locally legal if the tiles are assembled in accordance with the relevant adjacency rules.

Definition: An arrangement of tiles is globally legal if it can be extended to an infinite tiling.

Definition: The (first) corona of a tile, $T i$ is the set $\mathcal{C}(T_{i})=\{T_{j}\in \mathcal{T}\mid T_{i}\cap T_{j}\neq\empty\}$ . We will use a slightly altered definition in the work that follows. $\mathcal{C}(T_{i})=\{T_{j}\in \mathcal{T}\mid T_{i}\cap T_{j}=\{x\}$ and $\exists\,y\subset\{x\}\mid y$ is an edge $\big\}$ . With this definition, tiles that share only a vertex with $T i$ are not included in its corona.

Definition: The (first) corona atlas is the set of all (first) coronas that occur in $\mathcal{T}$ .

Definition: A reduced (first) corona atlas of $\mathcal{T}$ is a subset of the corona atlas of $\mathcal{T}$ that covers $\mathcal{T}$ .

Definition: The (first) vertex star of a vertex $v$ is the set $\mathcal{S}(v)=\{T_{j}\in\mathcal{T}\mid T_{j}\cap v \neq\empty\}$ .

Definition: The (first) vertex star atlas is the set of all (first) vertex stars that occur in $\mathcal{T}$ .

Definition: A tiling, $\mathcal{T}$ , is non-periodic if it does not have translational symmetry in more than one direction.

Definition: A set of prototiles, $\mathcal{P}$ , is aperiodic if all tilings it admits are non-periodic.

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Quasicrystals

A crystal is a solid that is composed of identical units, which may be atoms, molecules, etc., in such a way that the units completely cover $\mathbb{R}^{3}$ . Classically, a crystal is modeled by a lattice in $\mathbb{R}^{3}$ by replacing each unit of the crystal by a point located at its center of mass. If any two points are equivalent under lattice-preserving translations, then the array is called a point lattice or Bravais lattice.

However, this theory of crystals is inadequate to describe the variety of crystal-like solids found in nature. Consider a classical Bravais lattice, and let the group of isometries that act on the latice be called $G L$ . An element $R\in G_{L}$ is restricted to have order 2,3,4, or 6 (Paterson, 163). Note that the order of the element refers to the number of copies of the element that must be composed to yield the identity element, or more simply, it is the number of times an isometry must act on the lattice to exactly get back to the original configuration. (For a proof of the two dimensional case, see Patterson, 163.) One important shortcoming of this theory is that it excludes configurations with pentagonal or icosahedral symmetry. But in 1984 it was discovered that the diffraction pattern of an alloy of aluminum and manganese displayed icosahedral symmetry (Paterson, 164). Crystal-like solids such as this one have come to be called quasicrystals. The symmetry of quasicrystals is best analyzed using groupoids, not groups, and instead of point lattices, quasilattices (infinite tilings of $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ formed from a finite number of distinct tiles) are used to model quasicrystals.

(Paterson, 162-5)

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Types of Tilings

Penrose Tilings are the most widely known tilings related to quasicrystals. These tilings cover the plane with just two distinct tiles. There is also a three dimensional analogue to the rhombs version of this tiling, but not of the other versions (Senechal, 170).

Ammann Tilings cover the plane with three distinct tiles and can be derived from Penrose tilings. The work I have done with these tilings is original; only the description of their construction by Ammann (see: Grunbaum/Shephard, 548) was taken from the literature. My goal is to relate these tilings to quasicrystals by defining their related groupoid based on the method used in the Penrose case. So far, only the iteration algorithm is complete.

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Fun!

emily found a website where you can play with some tiles: [1] (http://nlvm.usu.edu/en/nav/frames_asid_171_g_3_t_2.html?open=activities). Keep in mind that these are not Penrose tiles, because they actually have hexagonal rather than pentagonal symmetry, so they do not produce an aperiodic tiling. See if you can tile the plane!

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