In General > s.a. cohomology; Penrose Tiling; Phonon; Root Lattice.
* Idea: A new atomic structure, neither crystalline (in the sense of periodic) nor glassy, with long-range translational and orientational order; Theoretically, the inspiration came from the Penrose tiling; The name "quasicrystal" was coined by P J Steinhardt.
* Structure: The structure function does not factorize into an intrinsic part and a geometric part, as for regular lattices, and the definition of the unit cell is not arbitrary as in the regular case; Roughly, they are made of "clusters of clusters" of atoms, where individual clusters (typically with a magic number of atoms, like 13 or 55) are more tightly packed than in crystals.
* Properties: Extremely low electrical and thermal resistivity (worse than glass); Harder than steel; Friction and stickiness lower than Teflon (and can be still lowered); The only drawback is that they are brittle – ok if used as thin films on substrates; All of this probably is a consequence of their tight-cluster structure.
@ General references: Levine & Steinhardt PRL(84); Mermin & Troian PRL(85) [mean-field theory]; Janssen PRP(88); Maddox Nat(89)jul, DiVincenzo Nat(89)aug; Stephens & Goldman SA(91)apr; Goldman et al AS(96); Pelantová & Masáková mp/06-proc [mathematical models]; news SA(09)oct [more normal than assumed]; Allouche & Meyer CRP(14)-a1401, Adiceam a1604 [mathematical properties]; Grimm & Kramer a1906 [rev].
@ Books: Amann et al ed-88; Jaric ed-88; Senechal 95; Janot 97; Di Vincenzo & Steinhardt ed-99; Barber 08 [r CP(10)]; Baake & Grimm 13, 17; Steinhardt 19.
@ Related topics, types: Lifshitz RMP(97) [colored]; Fisher & Rabson mp/01 [classification with group cohomology]; Gouliaev cm/01/ACA [analytic]; Lifshitz FP(03) [without forbidden symmetries]; Cornwell PS(04) [icosahedral]; Masáková et al JPA(05) [Voronoi and Delaunay tiles]; Au-Yang & Perk mp/06-conf [projections from 5D]; Böröczky et al JGP(06) [combinatorial properties]; Fujita ACA(09)-a0906 [decagonal, inflation rules]; Kraus et al PRL(12) + Quandt Phy(12) [and topological insulators]; Gambaudo & Vignolo NJP(14)-a1309 [Brillouin zone labelling].

Examples in Nature, Experiments, Applications > s.a. bose-einstein condensates; Dimer Models.
* In nature: A meteorite fragment kept in a Florence Museum holds a grain of quasicrystal; P J Steinhardt and others found more where that one came from in Russia.
* Experiment: 3D non-periodic tilings with icosahedral structure have been seen first by Dan Shechtman of NBS in 1983 [@ Shechtman et al PRL(84)], from diffraction patterns produced by some alloys of Al and Mn, then made by An-Pang Tsai in Japan in 1987, and others; The golden mean appears all over the place in the patterns; We now know that they are not a rare exception, and they can be found in many alloy systems.
@ In nature: Steinhardt & Bindi RPP(12) + news nat(12)aug, pw(12)aug [samples from meteorite]; news sn(16)dec [in Russian meteorite].
@ And physics: Albuquerque & Cottam PRP(03) [elementary excitations]; Monreal et al IJMPA(08)-a0804 [quantum particle and effective non-commutative geometry]; Colli & Mariano PLA(11) [corrections to the standard formulation of quasicrystal linear elasticity].
@ Experiments: news pw(11)nov [new type made of hard triangular bipyramids]; news PhysOrg(13)oct [new form of 12-sided quasicrystal accidentally discovered].
@ Uses, applications: news pw(07)feb [in Islamic art]; news pw(07)mar [as filter for terahertz light]; news pw(12)jan [in ancient Islamic architecture].

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