Editing Quasicrystal (section)

==Mathematics==
[[File:Ho-Mg-Zn E8-5Cube.jpg|thumb|right |A [[5-cube]] as an [[orthographic projection]] into 2D using [[Petrie polygon]] [[basis vector]]s overlaid on the diffractogram from an [[icosahedron|icosahedral]] Ho–Mg–Zn quasicrystal]]
[[File:6Cube-QuasiCrystal.png|thumb|A [[6-cube]] projected into the [[rhombic triacontahedron]] using the [[golden ratio]] in the [[basis vector]]s. This is used to understand the aperiodic [[icosahedron|icosahedral]] structure of quasicrystals.]]
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 work of Bohl and Escanglon.<ref>{{cite journal|doi=10.1007/BF02395468|title=Zur Theorie fastperiodischer Funktionen I|year=1925|last1=Bohr|first1=H.|journal=Acta Mathematica|volume=45|page=580|doi-access=free}}</ref>
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 [[hyperplane]]s), 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.

In order that the quasicrystal itself be aperiodic, this slice must avoid any [[lattice plane]] of the higher-dimensional lattice. [[Nicolaas Govert de Bruijn|De Bruijn]] showed that [[Penrose tilings]] can be viewed as two-dimensional slices of five-dimensional [[hypercubic]] structures;<ref>{{cite journal|title=Algebraic theory of Penrose's non-periodic tilings of the plane|year=1981|last1=de Bruijn|first1=N.|journal=Nederl. Akad. Wetensch. Proc|volume=A84|page=39}}</ref> similarly, icosahedral quasicrystals in three dimensions are projected from a six-dimensional hypercubic lattice, as first described by [[Peter Kramer (physicist)|Peter Kramer]] and Roberto Neri in 1984.<ref>{{cite journal|doi=10.1107/S0108767384001203|title=On periodic and non-periodic space fillings of E<sup>m</sup> obtained by projection|year=1984|last1=Kramer|first1=P.|last2=Neri|first2=R.|journal=[[Acta Crystallographica A]] |volume=40|pages=580–587|issue=5|bibcode=1984AcCrA..40..580K }}</ref> Equivalently, the [[Fourier transform]] of such a quasicrystal is nonzero only at a dense set of points [[linear span|spanned]] by integer multiples of a finite set of [[basis vectors]], which are the projections of the primitive [[reciprocal lattice]] vectors of the higher-dimensional lattice.<ref name="Suck04" />

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 be analyzed by defining an associated [[group (mathematics)|group]]. 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, [[groupoid]]s, the mathematical generalization of groups in [[category theory]], is the appropriate tool for studying quasicrystals.<ref>{{cite book|last=Paterson|first=Alan L. T.|title=Groupoids, inverse semigroups, and their operator algebras|year=1999|publisher=Springer|page=164|isbn=978-0-8176-4051-4}}</ref>

Using mathematics for construction and analysis of quasicrystal structures is a difficult task. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed<ref name="comp" /> 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.<ref name="speccomp" />

Study of quasicrystals may shed light on the most basic notions related to the [[quantum critical point]] observed in [[heavy fermion]] metals. Experimental measurements on an [[gold|Au]]–Al–[[ytterbium|Yb]] quasicrystal have revealed a quantum critical point defining the divergence of the [[magnetic susceptibility]] as temperature tends to zero.<ref>{{Cite journal|last1=Deguchi|first1=Kazuhiko|last2=Matsukawa|first2=Shuya|last3=Sato|first3=Noriaki K.|last4=Hattori|first4=Taisuke|last5=Ishida|first5=Kenji|last6=Takakura|first6=Hiroyuki|last7=Ishimasa|first7=Tsutomu|title=Quantum critical state in a magnetic quasicrystal|journal=Nature Materials|doi=10.1038/nmat3432|year=2012|pmid=23042414|volume=11|issue=12|pages=1013–6|arxiv=1210.3160|bibcode=2012NatMa..11.1013D|s2cid=7686382}}</ref> It is suggested that the electronic system of some quasicrystals is located at a quantum critical point without tuning, while quasicrystals exhibit the typical [[scaling behaviour]] of their [[thermodynamic properties]] and belong to the well-known family of heavy fermion metals.