Symmetric Aperiodicity in Quasicrystals
As we delve into the intriguing world of aperiodic tilings, quasicrystals, lattices, crystallographic point sets, and cut and project sets in Euclidean space. Our aim is to establish a connection and provide an explanation for these complex structures. By investigating rotational symmetries, the crystallographic restriction theorem, diffraction patterns, and the intricate patterns observed in patches, we contribute to the understanding of these fascinating phenomena. Furthermore, we explore the relationship between Euclidean Space and Rotational Symmetries, considering the role of group sets and ordered sets. This research sheds light on the rich interplay between solid-state physics, discrete geometry, and the theory of quasicrystals.
To develop some of these ideas, one has to start with a valid idealization of the physical structures one has in mind. Since we are interested in solids of some relevant size here, it is reasonable to replace their atomic arrangements with suitable infinite point sets. These should be uniformly discrete (i.e., there should be a uniform minimal distance between the points) and, usually, they should be relatively dense (i.e., there is a maximal hole). Sets with this property are called Delone sets and are widely used for this purpose.
In analogy to ordinary crystallography, many people prefer to think in terms of cells or tiles. Here, one may start from a (usually finite) number of proto-tiles that fit together to tile space without gaps or overlaps. If we now decorate the tiles by finitely many points (giving the atomic positions, say), we return to a Delone set.
Wang Tiles
In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane. The idea of constraining adjacent tiles to match each other occurs in the game of dominoes, so Wang tiles are also known as Wang dominoes. The algorithmic problem of determining whether a tile set can tile the plane became known as the domino problem.
This is one of the major problems and concepts in solid states, tesselations not only help us understand the structure, but they help us understand the physical properties.
The question that pondered the minds of these people was very simple.
Is there an algorithm which, when given any finite collection of Wang tiles, can decide whether or not it can tile the plane?
Wang proposed in 1961: There is an algorithm which can determine whether or not a finite collection of Wang tiles can tile the plane periodically.
The Domino Problem
deals with the class of all domino sets. It consists of deciding, for each domino set, whether or not it is solvable. We say that the Domino Problem is decidable or undecidable according to whether there exists or does not exist an algorithm which, given the specifications of an arbitrary domino set, will decide whether or not the set is solvable.
In 1966, Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt\footnote{}. The undecidability of the halting problem then implies the undecidability of Wang’s tiling problem.
Combining Berger’s undecidability result with Wang’s observation shows that there must exist a finite set of Wang tiles that tiles the plane, but only aperiodically.
Aperiodic Tiling Structures
In 1964 Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable. This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles. An even smaller set of six aperiodic tiles (based on Wang tiles) was discovered by Raphael M. Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977.
The aperiodic Penrose tilings can be generated not only by an aperiodic set of prototiles but also by a substitution and by a cut-and-project method. After the discovery of quasicrystals, aperiodic tilings become studied intensively by physicists and mathematicians. The cut-and-project method of N.G. de Bruijn for Penrose tilings eventually turned out to be an instance of the theory of Meyer sets. ( Yves Meyer and harmonic analysis)
Euclidian Space
One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections.
In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a “mathematical” space is a number, not something expressed in inches or metres.
The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts, the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows for defining lines, planes, subspaces, dimensions, and parallelism. The inner product allows for defining distance and angles.
Cut and Project Methods
There are several ways to mathematically define quasicrystalline patterns. One definition, the “cut and project” construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including the work of Bohl and Escanglon. He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. These functions are not exactly periodic, but they are arbitrarily close in some sense, as well as being a projection of an exactly periodic function.
The classical theory of crystals reduces crystals to point lattices where each point is the centre of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasi-lattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals.
Using mathematics for the construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modelling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns. The aperiodic nature of quasicrystals can also make theoretical studies of physical properties, such as electronic structure, difficult due to the inapplicability of Bloch’s theorem. However, spectra of quasicrystals can still be computed with error control.
Blochs Theorem
In condensed matter physics, Bloch’s theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. And hence these quasicrystals, break periodicity.
Conclusion
Periodicity as a Pseudo Function — We observe frequently in the case of such tesselation that aperiodicity arises quite frequently in Quasicrystals, It is expected because of the unique projected lattice structures.
We do observe mathematical accuracy in such cases, but it is worth mentioning that these crystals don't exist in the traditional sense as stable so there might be some inaccuracies in the projections.
Substitution tiling systems — provide a rich source of aperiodic tilings. A set of tiles that forces a substitution structure to emerge is said to enforce the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non-periodic, in precisely the same manner as described above, but the chair tile itself is not aperiodic — it is easy to find periodic tilings by unmarked chair tiles.
Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic.
Aperiodic tilings were considered mathematical artefacts until 1984 when physicist Dan Shechtman announced the discovery of a phase of an aluminium-manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry — so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction to a three-dimensional icosahedral equivalent.
Although the mathematics behind the concept is solid and pure in an abstract sense there exists, an amount of discomfort in bringing this to the physical sense.
There had been much debate about the works of Bekker and Schectman, as there had been much debate with the works of Pauling, taunting upon the existence of these crystals.
But with diffraction patterns clearly projecting the existence of these such structures, question our basic understanding of atoms to the core.
It's again important to understand that these exist as diffraction patterns but not at physical tangible objects per se.
After this little excursion into the world of aperiodic order, one should realize that the displayed material was not only a short (and possibly insufficient) presentation of the same.