A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and sixfold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance fivefold.
Aperiodic tilings were discovered by mathematicians in the early 1960s, and, 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. Quasicrystals had been investigated and observed earlier,^{[2]} but, until the 1980s, 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 two, three, four, or six. The first experimental observation of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1982 and it was reported in print two years later.^{[3]} Shechtman received the Nobel Prize in Chemistry in 2011 for his findings.^{[4]}.
In 2009, following a decade long search, a group of scientists from University of Florence in Italy reported the existence of a natural quasicrystals in mineral samples from the Koryak mountains in Russia’s far east, named icosahedrite.^{[5]}^{[6]} It was further claimed by scientists from Princeton University that icosahedrite is extraterrestrial in origin, possibly delivered to Earth by a CV3 carbonaceous chondrite asteroid.^{[7]}
240 E₈ polytope vertices using 5D orthographic_projection to 2D using 5cube (Penteract) Petrie_polygon basis_vectors overlaid on electron diffraction pattern of an Icosahedron ZnMgHo Quasicrystal.
