Quasicrystals are structural forms that are both ordered and nonperiodic. They form patterns that fill all the space but lack translational symmetry. Classical theory of crystals allows only 2, 3, 4, and 6fold rotational symmetries, but quasicrystals display symmetry of other orders (folds). They can be said to be in a state intermediate between crystal and glass. Just like crystals, quasicrystals produce modified Bragg diffraction, but where crystals have a simple repeating structure, quasicrystals are more complex.
Aperiodic tilings were discovered by mathematicians in the early 1960s, but some twenty years later they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography and solid state physics.^{[1]} Quasicrystals had been investigated and observed earlier^{[2]} but until the 80s they were disregarded in favor of the prevailing views about the atomic structure of matter.
Roughly, an ordering is nonperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n − 1 linearly independent directions, where n is the dimension of the space filled; i.e. the threedimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as longrange order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than 2, 3, 4, or 6. The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984.^{[3]} The distinction between quasicrystals and their corresponding mathematical models (e.g. the threedimensional version of the Penrose tiling) need not be emphasized.
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Although 20thcentury physicists were surprised by the discovery of quasicrystals, their mathematical descriptions were already well established. In 1961 Hao Wang proved that determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem, implying that aperiodic tilings exist. Two years later an example involving some 20,000 square tiles, each with different colorings, was produced. (These are now called the Wang tiles.) As the number of known aperiodic sets of tiles grew, each set seemed to contain even fewer tiles than the previous one. Then, in 1976 Roger Penrose proposed a set of just two tiles (referred to as Penrose tiles) that produced only nonperiodic tilings of the plane. These tilings displayed incidences of fivefold symmetry. In hindsight, similar patterns were observed in some decorative tilings devised by medieval Islamic architects.^{[4]}^{[5]} It was established that the Penrose tiling had a twodimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Around the same time, Robert Ammann had also discovered this solution and created a set of aperiodic tiles that produced eightfold symmetry. These two examples of mathematical quasicrystals have been shown to be derivable from a more general method which treats them as projections of a higherdimensional lattice. Just as the simple curves in the plane can be obtained as sections from a threedimensional double cone, various (aperiodic or periodic) arrangements in 2 and 3 dimensions can be obtained from postulated hyperlattices with 4 or more dimensions. This method explains both the arrangement and its ability to diffract.
The standard history of quasicrystals begins with the paper titled "Metallic Phase with LongRange Orientational Order and No Translational Symmetry" published by D. Shechtman and others in 1984. The discovery was made nearly two years before, but their work was met with resistance inside the professional community. Shechtman and coworkers demonstrated a clearcut diffraction picture with an unusual fivefold symmetry produced by samples from an AlMn alloy which has been rapidly cooled after melting. The same year Ishimasa and coauthors sent for publishing a paper titled "New ordered state between crystalline and amorphous in NiCr particles" in which a case of twelvefold symmetry was reported.^{[6]} Soon another equally challenging case presented a sample which gave a sharp eightfold diffraction picture.^{[7]} Over the years hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable. When heated, they formed regular crystals. But in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and opening the door to potential applications.
In 1972 de Wolf and van Aalst^{[8]} reported in print that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indexes but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported, but until the concept of quasicrystal came to be established they were explained away or simply denied. However at the end of the 1980s the idea became acceptable and in 1991 the International Union of Crystallography amended its definition of crystal, reducing it to the ability to produce a clearcut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic^{[9]}. Now the symmetries compatible with translations are defined as "crystallographic", leaving room for other "noncrystallographic" symmetries. Thus aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic pointgroup symmetry, to which the incommensurately modulated structures and composite structures belong, and those with noncrystallographic pointgroup symmetry, to which quasicrystal structures belong.
The term 'quasicrystal' was used for the first time in print shortly after the announcement of Shechtman's discovery in a paper by Steinhardt and Levine.^{[10]} However, the adjective 'quasicrystalline' has been loosely applied to any pattern with unusual symmetry.^{[11]}
There are several ways to mathematically define quasicrystalline patterns.
One definition, the "cut and project" construction, is that a quasicrystal consists of a slice (an intersection with one or more hyperplanes) of a higherdimensional periodic pattern. In order that the quasicrystal itself be aperiodic, this slice must not be a lattice plane of the higherdimensional lattice. (For example, Penrose tilings can be viewed as twodimensional slices of fivedimensional hypercubic structures.) Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higherdimensional lattice).^{[12]}
The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets. The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum.There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers. The aperiodic structures obtained by the cutandproject method are made diffractive by choosing a suitable orientation for the construction. This is indeed a geometric approach which has also a great appeal for physicists.
Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can by analyzed by defining an associated group (mathematics). Quasicrystals, on the other hand, are composed of more than one type of unit, so instead of lattices, quasilattices 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 construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modeling, based on the existing theories of quasicrystals, however greatly facilitated this task. Advanced programs have been developed^{[13]} allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.
Since the original discovery of Shechtman hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (AlLiCu, AlMnSi, AlNiCo, AlPdMn, AlCuFe, AlCuV, etc.), but numerous other compositions are also known (CdYb, TiZrNi, ZnMgHo, ZnMgSc, InAgYb, PdUSi, etc.).^{[14]}
In theory, there are two types in quasicrystals.^{[13]} One is called polygonal (dihedral) quasicrystals, which have one 8, 10, or 12fold axis and is periodic along this axis. They are called octagonal, decagonal, and dodecagonal quasicrystals. These structures take an ordered structure (quasiperiodic structure) in a plane normal to such a periodic axis. Another one called an icosahedral quasicrystal has no period along any directions.
Regarding thermal stability, three types of quasicrystals are distinguished:^{[15]}
Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by xray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.
The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystalforming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.
