# Quasicrystal

**quasicrystalsquasi-crystalquasicrystallineIcosahedral Phasequasiperiodic crystalsAperiodic order and disorderquasi-crystallinequasi-crystalline tilingquasicrystalline materialsquasiperiodic**

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.wikipedia

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### Crystallographic restriction theorem

**crystallographic restrictioncompatiblecrystallographic**

While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders — for instance, five-fold.

However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

### Crystal

**crystallinecrystalscrystalline solid**

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.

(Quasicrystals are an exception, see below).

### Icosahedrite

In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals.

Icosahedrite is the first known naturally occurring quasicrystal phase.

### Aperiodic tiling

**aperiodicaperiodicallycut-and-project method**

Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals.

quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011.

### Paul Steinhardt

**Paul J. SteinhardtProf. Paul J. Steinhardt**

On 25 October 2018, Luca Bindi and Paul Steinhardt were awarded the Aspen Institute 2018 Prize for collaboration and scientific research between Italy and the United States.

He is also well known for his exploration of a new form of matter, known as quasicrystals.

### Dan Shechtman

**Daniel ShechtmanProf. Dan ShechtmanDan Schetman**

In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures.

On April 8, 1982, while on sabbatical at the U.S. National Bureau of Standards in Washington, D.C., Shechtman discovered the icosahedral phase, which opened the new field of quasiperiodic crystals.

### Aluminium

**aluminumAlall-metal**

In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures.

Additionally, one of the main motifs of boron chemistry is regular icosahedral structures, and aluminium forms an important part of many icosahedral quasicrystal alloys, including the Al–Zn–Mg class.

### Luca Bindi

On 25 October 2018, Luca Bindi and Paul Steinhardt were awarded the Aspen Institute 2018 Prize for collaboration and scientific research between Italy and the United States.

He discovered natural quasicrystals in 2009, showing that quasicrystals can form spontaneously in nature and remain stable for geological times.

### Robert Ammann

**Ammann, Robert**

Around the same time, Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.

Robert Ammann (October 1, 1946 – May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings.

### Order and disorder

**long-range orderorderedquenched disorder**

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.

During much of the 20th century, the converse was also taken for granted – until the discovery of quasicrystals in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity.

### Roger Penrose

**Sir Roger PenrosePenrosePenrose, Roger**

In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane.

In 1984, such patterns were observed in the arrangement of atoms in quasicrystals.

### Alan Lindsay Mackay

**Alan MackayA. L. MacKayAlan L. Mackay**

One year later Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern.

He is a pioneer in the introduction of five-fold symmetry in materials and in 1981 predicted quasicrystals in a paper (in Russian) entitled "De Nive Quinquangula" in which he used a Penrose tiling in two and three dimensions to predict a new kind of ordered structures not allowed by traditional crystallography.

### Wang tile

**Domino ProblemWang tilesWang domino**

Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called "Wang tiles") that can tile the plane but not in a periodic fashion.

This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal.

### Penrose tiling

**Penrose tilesPenrose tilingsPenrose tile**

In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane.

It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

### Delone set

**covering radiusuniformly discretedensity**

The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets.

These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals.

### Crystallography

**crystallographercrystallographiccrystallographically**

The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography.

### National Institute of Standards and Technology

**NISTNational Bureau of StandardsBureau of Standards**

The observation was made during a routine investigation, by electron microscopy, of a rapidly cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards (later NIST).

In 2011, Dan Shechtman was awarded the Nobel in chemistry for his work on quasicrystals in the Metallurgy Division from 1982 to 1984.

### Peter Kramer (physicist)

**Peter Kramer**

Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984.

In the early eighties he and his student Roberto Neri developed a mathematical model for quasiperiodic tesselations of three-dimensional space.

### Meyer set

**Meyer**

The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets.

Nowadays Meyer sets are best known as mathematical model for quasicrystals.

### Girih tiles

**girihdecorative patternsGirih tilings**

For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.

In 2007, the physicists Peter J. Lu and Paul J. Steinhardt suggested that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings, predating them by five centuries.

### Self-assembly

**self-assembledself assemblyself-assemble**

Earlier in 2009, it was found that thin-film quasicrystals can be formed by self-assembly of uniformly shaped, nano-sized molecular units at an air-liquid interface.

Another remarkable example of macroscopic self-assembly is the formation of thin quasicrystals at an air-liquid interface, which can be built up not only by inorganic, but also by organic molecular units.

### Fibonacci quasicrystal

**quasicrystal**

Both names are acceptable as a 'Fibonacci crystal' denotes a quasicrystal and a 'Fibonacci' quasicrystal is a specific type of quasicrystal.

### Substitution tiling

**substitutionsubstitutions**

Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.

This work gave an impact to scientists working in crystallography, eventually leading to the discovery of quasicrystals.

### Lattice plane

**crystallographic planesplaneplanes**

In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice.

Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).

### Phason

Phason is a quasiparticle existing in quasicrystals due to their specific, quasiperiodic lattice structure.