879:, diesel engines, and new materials that convert heat to electricity. Shechtman suggested new applications taking advantage of the low coefficient of friction and the hardness of some quasicrystalline materials, for example embedding particles in plastic to make strong, hard-wearing, low-friction plastic gears. The low heat conductivity of some quasicrystals makes them good for heat insulating coatings. One of the special properties of quasicrystals is their smooth surface, which despite the irregular atomic structure, the surface of quasicrystals can be smooth and flat.
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841:. A problem that must be resolved is the tendency for cracking due to the materials' extreme brittleness. The cracking could be suppressed by reducing sample dimensions or coating thickness. Recent studies show typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains at room temperature and sub-micrometer scales (<500 nm).
335:, was ruled out by his experiments. Therefore, Blech looked for a new structure containing cells connected to each other by defined angles and distances but without translational periodicity. He decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material, which he termed as "multiple
871:
The Nobel citation said that quasicrystals, while brittle, could reinforce steel "like armor". When
Shechtman was asked about potential applications of quasicrystals he said that a precipitation-hardened stainless steel is produced that is strengthened by small quasicrystalline particles. It does not
748:
Two types of quasicrystals are known. The first type, polygonal (dihedral) quasicrystals, have an axis of 8-, 10-, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type,
483:
in 2011 for his work on quasicrystals. "His discovery of quasicrystals revealed a new principle for packing of atoms and molecules," stated the Nobel
Committee and pointed that "this led to a paradigm shift within chemistry." In 2014, Post of Israel issued a stamp dedicated to quasicrystals and the
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587:(also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. 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
463:
in 2010. Analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid. In 2011, Bindi, Steinhardt, and a team of specialists found more icosahedrite samples from
Khatyrka. A further study of Khatyrka meteorites revealed micron-sized grains of another
156:
is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions. Symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely
821:
The quasi-ordered droplet crystals could be formed under
Dipolar forces in the Bose Einstein condensate. While the softcore Rydberg dressing interaction has forms triangular droplet-crystals, adding a Gaussian peak to the plateau type interaction would form multiple roton unstable points in the
798:
The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the
330:
The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked
Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, a type of
745:, hundreds of quasicrystals have been reported and confirmed. Quasicrystals are found most often in aluminium alloys (Al–Li–Cu, Al–Mn–Si, Al–Ni–Co, Al–Pd–Mn, Al–Cu–Fe, Al–Cu–V, etc.), but numerous other compositions are also known (Cd–Yb, Ti–Zr–Ni, Zn–Mg–Ho, Zn–Mg–Sc, In–Ag–Yb, Pd–U–Si, etc.).
649:
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 allowing one to construct,
882:
Other potential applications include selective solar absorbers for power conversion, broad-wavelength reflectors, and bone repair and prostheses applications where biocompatibility, low friction and corrosion resistance are required. Magnetron sputtering can be readily applied to other stable
511:
In 2018, chemists from Brown
University announced the successful creation of a self-constructing lattice structure based on a strangely shaped quantum dot. While single-component quasicrystal lattices have been previously predicted mathematically and in computer simulations, they had not been
376:
reported twelvefold symmetry in Ni-Cr particles. Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys. Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were
352:. Meanwhile, on seeing the draft of the paper, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move". This paper, published in the
813:
Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick properties. Many metallic quasicrystalline substances are impractical for most applications due to their
3772:
429:
In 2001, Steinhardt hypothesized that quasicrystals could exist in nature and developed a method of recognition, inviting all the mineralogical collections of the world to identify any badly cataloged crystals. In 2007 Steinhardt received a reply by
650:
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
503:
of uniformly shaped, nano-sized molecular units at an air-liquid interface. It was demonstrated that these units can be both inorganic and organic. Additionally in the 2010s, two-dimensional molecular quasicrystals were discovered, driven by
868:; and the pan could withstand temperatures of 1,000 °C (1,800 °F) without harm. However, after an initial introduction the pans were a chrome steel, probably because of the difficulty of controlling thin films of the quasicrystal.
749:
icosahedral quasicrystals, are aperiodic in all directions. Icosahedral quasicrystals have a three dimensional quasiperiodic structure and possess fifteen 2-fold, ten 3-fold and six 5-fold axes in accordance with their icosahedral symmetry.
903:
Applications in macroscopic engineering have been suggested, building quasi-crystal-like large scale engineering structures, which could have interesting physical properties. Also, aperiodic tiling lattice structures may be used instead of
132:
Quasicrystals had been investigated and observed earlier, but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding,
176:
alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the scientific community's reaction, it took him two years to publish the results for which he was awarded the
476:. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock.
4424:
633:
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
377:
thermodynamically unstable: when heated, they formed regular crystals. However, in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and applications.
326:
of the NIST, who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something, and I challenge you to find out what it is".
794:
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.
108:
were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a
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Mineralogical
Collection. The crystal samples were sent to Princeton University for other tests, and in late 2009, Steinhardt confirmed its quasicrystalline character. This quasicrystal, with a composition of
205:
Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.). A subdivision rule to construct perfect quasi-crystalline tilings has been
2610:
Talapin, Dmitri V.; Shevchenko, Elena V.; Bodnarchuk, Maryna I.; Ye, Xingchen; Chen, Jun; Murray, Christopher B. (2009). "Quasicrystalline order in self-assembled binary nanoparticle superlattices".
822:
Bogoliubov spectrum. Therefore, the excitation around the roton instabilities would grow exponentially and form multiple allowed lattice constants leading to quasi-ordered periodic droplet crystals.
591:), 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.
499:
systems. Soft quasicrystal structures have been found in supramolecular dendrimer liquids and ABC Star
Polymers in 2004 and 2007. In 2009, it was found that thin-film quasicrystals can be formed by
1268:
2320:
Bindi, L.; Yao, N.; Lin, C.; Hollister, L.S.; Andronicos, C.L.; Distler, V.V.; Eddy, M.P.; Kostin, A.; Kryachko, V.; MacPherson, G.J.; Steinhardt, W.M.; Yudovskaya, M.; Steinhardt, P.J. (2015).
384:
altered its definition of a crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.
677:
as temperature tends to zero. It is suggested that the electronic system of some quasicrystals is located at a quantum critical point without tuning, while quasicrystals exhibit the typical
1043:
3255:
Deguchi, Kazuhiko; Matsukawa, Shuya; Sato, Noriaki K.; Hattori, Taisuke; Ishida, Kenji; Takakura, Hiroyuki; Ishimasa, Tsutomu (2012). "Quantum critical state in a magnetic quasicrystal".
322:(later NIST). Shechtman related his observation to Ilan Blech, who responded that such diffractions had been seen before. Around that time, Shechtman also related his finding to
144:, 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
3881:
161:. 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. In 1982,
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corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material.
2094:
2867:
Engel, Michael; Damasceno, Pablo F.; Phillips, Carolyn L.; Glotzer, Sharon C. (Dec 8, 2014). "Computational self-assembly of a one-component icosahedral quasicrystal".
2720:
Wasio, Natalie A.; Quardokus, Rebecca C.; Forrest, Ryan P.; Lent, Craig S.; Corcelli, Steven A.; Christie, John A.; Henderson, Kenneth W.; Kandel, S. Alex (Mar 2014).
3943:
Shaginyan, V. R.; Msezane, A. Z.; Popov, K. G.; Japaridze, G. S.; Khodel, V. A. (2013). "Common quantum phase transition in quasicrystals and heavy-fermion metals".
342:
Shechtman accepted Blech's discovery of a new type of material and chose to publish his observation in a paper entitled "The
Microstructure of Rapidly Solidified Al
3083:
264:) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974
1587:
339:", and found a ten-fold structure similar to what was observed. The multiple polyhedral structure was termed later by many researchers as icosahedral glass.
3583:
2469:
256:(hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student
3811:
2206:
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230:, a glass-like substance formed from fused sand and copper transmission lines. Identified in 2021, they are the oldest known anthropogenic quasicrystals.
3848:
638:. 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,
575:
There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of
2023:
610:
structures; similarly, icosahedral quasicrystals in three dimensions are projected from a six-dimensional hypercubic lattice, as first described by
3667:
2794:
Paßens, M.; Caciuc, V.; Atodiresei, N.; Feuerbacher, M.; Moors, M.; Dunin-Borkowski, R. E.; Blügel, S.; Waser, R.; Karthäuser, S. (May 22, 2017).
818:; the Al–Cu–Fe ternary system and the Al–Cu–Fe–Cr and Al–Co–Fe–Cr quaternary systems, thermally stable up to 700 °C, are notable exceptions.
2589:
189:
were awarded the Aspen
Institute 2018 Prize for collaboration and scientific research between Italy and the United States, after they discovered
1347:
226:
nuclear bomb test produced icosahedral quasicrystals. They went unnoticed at the time of the test but were later identified in samples of red
299:. Other puzzling cases have been reported, but until the concept of quasicrystal came to be established, they were explained away or denied.
4119:
1907:
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864:(PFOA); the surface was very hard, claimed to be ten times harder than stainless steel, and not harmed by metal utensils or cleaning in a
706:
423:
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1008:"Nucleation and growth of Ag islands on fivefold Al-Pd-Mn quasicrystal surfaces: Dependence of island density on temperature and flux"
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or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it
1957:
381:
272:, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later
1857:
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3113:
3077:
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1172:"Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals"
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cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in
3691:
Zou, Yu; Kuczera, Pawel; Sologubenko, Alla; Sumigawa, Takashi; Kitamura, Takayuki; Steurer, Walter; Spolenak, Ralph (2016).
2377:
Bindi, Luca; John M. Eiler; Yunbin Guan; Lincoln S. Hollister; Glenn MacPherson; Paul J. Steinhardt; Nan Yao (Jan 3, 2012).
697:
Tiling of a plane by regular pentagons is impossible but can be realized on a sphere in the form of pentagonal dodecahedron.
87:
2123:
Ishimasa, T.; Nissen, H.-U.; Fukano, Y. (1985). "New ordered state between crystalline and amorphous in Ni-Cr particles".
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2773:
2531:
1802:
1498:
1123:
1729:
Pauling, L (Jan 26, 1987). "So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal".
249:
2296:
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Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by
4479:
4085:
2017:
4443:
4112:
4035:
1007:
810:
A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals.
408:
1579:
3359:
C, Cui; M, Shimoda; AP, Tsai (2014). "Studies on icosahedral Ag-In-Yb: A prototype for Tsai-type quasicrystals".
2466:
361:
Originally, the new form of matter was dubbed "Shechtmanite". The term "quasicrystal" was first used in print by
33:
2167:
1978:
4091:
3833:
2486:
Zeng, Xiangbing; Ungar, Goran; Liu, Yongsong; Percec, Virgil; Dulcey, Andrés E.; Hobbs, Jamie K. (Mar 2004).
257:
17:
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348:
319:
75:
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1997:
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Kramer, P.; Neri, R. (1984). "On periodic and non-periodic space fillings of E obtained by projection".
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291:
In 1972, R. M. de Wolf and W. van Aalst reported that the diffraction pattern produced by a crystal of
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891:
599:
1979:"NIST and the Nobel (September 30, 2016, Updated November 17, 2019) The Nobel Moment: Dan Shechtman"
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480:
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178:
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2912:"Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule"
2663:"Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule"
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556:
529:
354:
1929:
1879:
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showed theoretically that the diffraction pattern from the Penrose tiling had a two-dimensional
4489:
3530:"Rydberg noisy dressing and applications in making soliton molecules and droplet quasicrystals"
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861:
761:
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658:
439:
223:
141:
117:. In crystallography the quasicrystals were predicted in 1981 by a five-fold symmetry study of
83:
4013:
3396:"Icosahedral clusters, icosaheral order and stability of quasicrystals – a view of metallurgy"
1822:"Multiply Twinned Particles at Earlier Stages of Gold Film Formation on Alkalihalide Crystals"
4484:
210:
The first representations of perfect quasicrystalline patterns can be found in several early
91:
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281:
4069:"Indiana Steinhardt and the Quest for Quasicrystals – A Conversation with Paul Steinhardt"
2436:
1773:
1580:"Newly discovered quasicrystal was created by the first nuclear explosion at Trinity Site"
358:, repeated Shechtman's observation and used the same illustrations as the original paper.
8:
4453:
4413:
4197:
3922:
3010:
de Bruijn, N. (1981). "Algebraic theory of Penrose's non-periodic tilings of the plane".
2553:
1521:"Accidental synthesis of a previously unknown quasicrystal in the first atomic bomb test"
909:
635:
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245:
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149:
118:
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3708:
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Louzguine-Luzgin, D. V.; Inoue, A. (2008). "Formation and Properties of Quasicrystals".
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Hayashida, Kenichi; Dotera, Tomonari; Takano, Atsushi; Matsushita, Yushu (May 8, 2007).
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129:, the possibility of identifying quasiperiodic order in a material through diffraction.
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4064:
Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart
3978:
3952:
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3907:
3725:
3693:"Superior room-temperature ductility of typically brittle quasicrystals at small sizes"
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2009:
1555:
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1413:
1244:
1035:
984: – Structure that repeats in time; a novel type or phase of non-equilibrium matter
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627:
404:
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4425:
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
3563:
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An application was the use of low-friction Al–Cu–Fe–Cr quasicrystals as a coating for
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2581:
2554:"Polymeric Quasicrystal: Mesoscopic Quasicrystalline Tiling in $ ABC$ Star Polymers"
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2013:
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82:. A quasicrystalline pattern can continuously fill all available space, but it lacks
3750:
Al-Cu-Fe quasicrystalline coatings and composites studied by mechanical spectroscopy
3605:
2853:
2796:"Interface-driven formation of a two-dimensional dodecagonal fullerene quasicrystal"
1417:
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quasicrystal have revealed a quantum critical point defining the divergence of the
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332:
292:
158:
105:
60:
4051:
4041:
2577:
1304:"Metallic Phase with Long-Range Orientational Order and No Translational Symmetry"
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Bindi, L.; Steinhardt, P. J.; Yao, N.; Lu, P. J. (2009). "Natural Quasicrystals".
890:
Rendering of a quasicrystalline structure, created using an open-source model for
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3069:
Quasicrystals: An Introduction to Structure, Physical Properties and Applications
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3752:(Thesis). École polytechnique fédérale de Lausanne EPFL, Thesis n° 2707 (2002).
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MacIá, Enrique (2006). "The role of aperiodic order in science and technology".
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Shechtman, Dan; I. A. Blech (1985). "The Microstructure of Rapidly Solidified Al
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natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al
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Study of quasicrystals may shed light on the most basic notions related to the
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asked whether determining if a set of tiles admits a tiling of the plane is an
238:
126:
110:
3052:
2721:
2487:
1806:
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654:. However, spectra of quasicrystals can still be computed with error control.
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In order that the quasicrystal itself be aperiodic, this slice must avoid any
201:
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2827:
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de Wolf, R.M. & van Aalst, W. (1972). "The four dimensional group of γ-Na
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3834:"An Overview of Quasicrystals, Their Types, Preparation Methods, Properties"
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for the stable quasicrystals, whereas the icosahedral order prevails in the
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Mn", which was written around June 1984 and published in a 1985 edition of
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134:
1370:"Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture"
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Quasicrystalline substances have potential applications in several forms.
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3777:. Evanston, Illinois, USA: Northwestern University. pp. Appendix A.
3606:"Supersolid Vortex Crystals in Rydberg-Dressed Bose-Einstein Condensates"
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1880:"The structure and orientation of crystals in deposits of metals on mica"
1845:
1269:"Tecnion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery"
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284:' peaks arranged in a fivefold symmetric pattern. Around the same time,
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3874:"Could centuries-old islamic patterns be the key to hypersonic flight?"
3604:
Henkel, N.; Cinti, F.; Jain, P.; Pupillo, G.; Pohl, T. (Jun 26, 2012).
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865:
845:
701:
693:
607:
588:
580:
431:
366:
261:
182:
2910:
Chen, Ou; Eggert, Dennis; Zhu, Hua; Nagaoka, Yasutaka (Dec 21, 2018).
2345:
2255:
415:
4448:
4351:
4097:
3286:
2880:
1109:
853:
779:
752:
Quasicrystals fall into three groups of different thermal stability:
670:
547:
311:
307:
227:
173:
169:
71:
52:
48:
44:
2379:"Evidence for the extraterrestrial origin of a natural quasicrystal"
520:
4177:
4142:
3546:
2441:
1444:"Octagon-Based Quasicrystalline Formations in Islamic Architecture"
957:
639:
496:
288:
created a set of aperiodic tiles that produced eightfold symmetry.
99:
3957:
3912:
3622:
3269:
387:
4303:
4298:
4286:
4264:
4249:
4157:
2793:
2232:"Binary Quasicrystals Discovered That Are Stable and Icosahedral"
2168:"Two-dimensional quasicrystal with eightfold rotational symmetry"
905:
856:
and easy to clean; heat transfer and durability were better than
63:
4092:
Quasicrystal Blocks: Description and Cut & Fold Instructions
3494:"Sputtering technique forms versatile quasicrystalline coatings"
2722:"Self-assembly of hydrogen-bonded two-dimensional quasicrystals"
1608:
Mackay, A.L. (1982). "Crystallography and the Penrose Pattern".
942: – A state similar to a liquid and a crystal in properties.
618:
of such a quasicrystal is nonzero only at a dense set of points
39:
2551:
2095:"Impossible' Form of Matter Takes Spotlight In Study of Solids"
969:
875:
Quasicrystals were also being used to develop heat insulation,
726:
552:
525:
137:, offered evidence for the existence of natural quasicrystals.
3942:
2661:
Nagaoka, Yasutaka; Zhu, Hua; Eggert, Dennis; Chen, Ou (2018).
1446:. In Schmid, Siegbert; Withers, Ray L.; Lifshitz, Ron (eds.).
733:
4336:
4063:
3690:
2866:
2609:
685:
and belong to the well-known family of heavy fermion metals.
3774:
Quasicrystalline thin films: growth, structure and interface
966: – Structure found in atomic clusters and nanoparticles
729:, the quasicrystal has faces that are true regular pentagons
606:
can be viewed as two-dimensional slices of five-dimensional
4214:
4152:
4147:
3254:
1156:
Alan L. Mackay, "Crystallography and the Penrose Pattern",
1006:; A.R. Ross; T.A. Lograsso; J.W. Evans; P.A. Thiel (2007).
857:
666:
491:
components, later on quasicrystals were also discovered in
3134:"Software package for structure analysis of quasicrystals"
3106:
Groupoids, inverse semigroups, and their operator algebras
1642:
912:. None of these seem to have been put to use in practice.
487:
While the first quasicrystals discovered were made out of
260:
constructed a set of some 20,000 square tiles (now called
55:(Al–Pd–Mn) quasicrystal surface. Similar to Fig. 6 in Ref.
3189:
Colbrook, Matthew; Roman, Bogdan; Hansen, Anders (2019).
2719:
1301:
1108:
Lifshitz, Ron; Schmid, Siegbert; Withers, Ray L. (2013).
876:
306:
patterns in 1982, while conducting a routine study of an
3906:
Kramer, Peter (2010). "Gateways towards quasicrystals".
3066:
Suck, Jens-Boie; Schreiber, M.; Häussler, Peter (2002).
2004:. Aperiodicity and Order. Vol. 3. pp. 37–104.
1302:
Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. (1984).
1669:
953:
Pages displaying short descriptions of redirect targets
268:
discovered a set of just two tiles, now referred to as
121:,—that also brought in 1982, with the crystallographic
3603:
3072:. Springer Science & Business Media. pp. 1–.
646:, is the appropriate tool for studying quasicrystals.
391:
Atomic image of a micron-sized grain of the natural Al
98:
pattern of quasicrystals shows sharp peaks with other
4036:
A Partial Bibliography of Literature on Quasicrystals
3450:
2289:"Quasicrystal – Online Dictionary of Crystallography"
1107:
1002:Ünal, B; V. Fournée; K.J. Schnitzenbaumer; C. Ghosh;
833:
Metallic quasicrystalline coatings can be applied by
90:, can possess only two-, three-, four-, and six-fold
43:
Potential energy surface for silver depositing on an
2909:
2660:
2319:
2122:
2043:
1210:
972: – Collective excitation in aperiodic materials
944:
Pages displaying wikidata descriptions as a fallback
930: – Polyhedra in which all vertices are the same
898:
27:
Ordered chemical structure with no repeating pattern
3188:
1491:"Islamic Quasicrystal Tilings | Paul J. Steinhardt"
1450:. Dordrecht: Springer Netherlands. pp. 49–57.
222:. On July 16, 1945, in Alamogordo, New Mexico, the
193:, the first quasicrystal known to occur naturally.
2485:
1774:"Israeli Scientist Wins Nobel Prize for Chemistry"
1368:Lu, Peter J.; Steinhardt, Paul J. (Feb 23, 2007).
1069:"Quasicrystals: A New Class of Ordered Structures"
960: – Five crystals arranged round a common axis
1675:"Lattice Textures in Cholesteric Liquid Crystals"
951: – Binary sequence from Fibonacci recurrence
140:Roughly, an ordering is non-periodic if it lacks
4466:
1066:
3801:
2488:"Supramolecular dendritic liquid quasicrystals"
2383:Proceedings of the National Academy of Sciences
1877:
1525:Proceedings of the National Academy of Sciences
642:, the mathematical generalization of groups in
369:shortly after Shechtman's paper was published.
3990:Thiel, P. A. (2008). "Quasicrystal Surfaces".
2322:"Natural quasicrystal with decagonal symmetry"
936: – Scientific study of crystal structures
756:Stable quasicrystals grown by slow cooling or
4113:
4042:Quasicrystals: an introduction by R. Lifshitz
3686:
3684:
1771:
860:non-stick cookware and the pan was free from
626:, which are the projections of the primitive
434:, who found a quasicrystalline specimen from
86:. While crystals, according to the classical
3400:Science and Technology of Advanced Materials
3138:Science and Technology of Advanced Materials
1571:
1367:
725:shape of some cubic-system crystals such as
614:and Roberto Neri in 1984. Equivalently, the
3841:Journal of Environmental Friendly Materials
3191:"How to Compute Spectra with Error Control"
2979:"Zur Theorie fastperiodischer Funktionen I"
2165:
1262:
1260:
1258:
924: – Crystal type lacking 3D periodicity
630:vectors of the higher-dimensional lattice.
567:. This is used to understand the aperiodic
214:such as the Gunbad-i-Kabud tomb tower, the
4120:
4106:
4047:Quasicrystals: an introduction by S. Weber
3747:
3681:
3028:
1441:
1169:
883:quasicrystalline alloys such as Al–Pd–Mn.
4014:10.1146/annurev.physchem.59.032607.093736
3956:
3911:
3797:
3795:
3724:
3621:
3545:
3509:
3427:
3268:
3165:
3009:
2994:
2945:
2935:
2835:
2696:
2686:
2412:
2402:
2353:
1554:
1544:
1327:
1195:
1092:
848:. Food did not stick to it as much as to
403:quasicrystal (shown in the inset) from a
3920:
3182:
3131:
3103:
1995:
1826:Journal of the Physical Society of Japan
1819:
1663:
1255:
1143:Alan L. Mackay, "De Nive Quinquangula",
978: – Tiling of a plane in mathematics
885:
732:
700:
692:
665:metals. Experimental measurements on an
622:by integer multiples of a finite set of
546:
519:
414:
386:
232:
200:
38:
3770:
3528:Khazali, Mohammadsadegh (Aug 5, 2021).
3527:
3475:10.1146/annurev.matsci.38.060407.130318
3127:
3125:
2315:
2313:
1930:"Structure of small metallic particles"
1728:
1577:
774:Metastable quasicrystals formed by the
461:International Mineralogical Association
14:
4467:
4127:
3905:
3871:
3792:
3488:
3486:
3484:
3059:
2092:
1927:
1878:Allpress, J.G.; Sanders, J.V. (1967).
1607:
1266:
1067:Levine, Dov; Steinhardt, Paul (1984).
539:overlaid on the diffractogram from an
382:International Union of Crystallography
302:Dan Shechtman first observed ten-fold
4101:
3989:
3831:
3523:
3521:
3315:
1518:
1336:
767:Metastable quasicrystals prepared by
737:TiMn quasicrystal approximant lattice
212:Islamic works of art and architecture
3393:
3358:
3122:
2976:
2310:
2166:Wang, N.; Chen, H.; Kuo, K. (1987).
1971:
1062:
1060:
688:
88:crystallographic restriction theorem
4406:The Chemical Basis of Morphogenesis
3993:Annual Review of Physical Chemistry
3481:
3454:Annual Review of Materials Research
2229:
1344:"The Nobel Prize in Chemistry 2011"
1297:
1295:
598:of the higher-dimensional lattice.
24:
3899:
3518:
2159:
2116:
2093:Browne, Malcolm W. (Sep 5, 1989).
2010:10.1016/B978-0-12-040603-6.50007-6
1795:
1765:
1722:
1636:
1601:
1512:
807:for the metastable quasicrystals.
707:Ho–Mg–Zn dodecahedral quasicrystal
250:algorithmically unsolvable problem
25:
4501:
4088:showing pictures of Quasicrystals
4029:
3564:10.1103/physrevresearch.3.l032033
2467:Crystallography matters ... more!
2437:"Nobel win for crystal discovery"
1820:Ino, Shozo; Ogawa, Shiro (1967).
1204:
1163:
1150:
1057:
899:Non-material science applications
4228:
3802:Kalman, Matthew (Oct 12, 2011).
1292:
741:Since the original discovery by
102:orders—for instance, five-fold.
3884:from the original on 2023-03-20
3878:Josefine Lissner and Lin Kayser
3865:
3854:from the original on 2021-10-31
3825:
3814:from the original on 2020-07-28
3781:from the original on 2024-09-18
3764:
3741:
3670:from the original on 2024-09-18
3597:
3586:from the original on 2024-09-18
3444:
3387:
3352:
3309:
3248:
3237:from the original on 2021-01-22
3097:
3086:from the original on 2024-09-18
3022:
3003:
2970:
2903:
2860:
2787:
2776:from the original on 2024-09-18
2713:
2654:
2603:
2592:from the original on 2024-09-18
2545:
2534:from the original on 2022-11-06
2479:
2460:
2449:from the original on 2011-10-05
2429:
2370:
2299:from the original on 2024-04-04
2281:
2270:from the original on 2021-11-03
2223:
2212:from the original on 2018-07-21
2105:from the original on 2017-12-01
2086:
2037:
2026:from the original on 2021-05-07
2002:Extended Icosahedral Structures
1989:
1960:from the original on 2024-09-18
1921:
1910:from the original on 2023-12-03
1871:
1860:from the original on 2023-03-24
1813:
1784:from the original on 2017-02-19
1711:from the original on 2020-04-26
1590:from the original on 2021-06-21
1578:Mullane, Laura (May 18, 2021).
1501:from the original on 2023-05-29
1483:
1472:from the original on 2023-05-30
1435:
1424:from the original on 2024-09-18
1361:
1350:from the original on 2017-10-05
1126:from the original on 2024-09-18
1046:from the original on 2020-07-28
825:
34:Quasi-crystals (supramolecular)
3640:10.1103/physrevlett.108.265301
3318:Reports on Progress in Physics
3215:10.1103/PhysRevLett.122.250201
1267:Gerlin, Andrea (Oct 5, 2011).
1137:
1101:
995:
515:
13:
1:
3104:Paterson, Alan L. T. (1999).
2578:10.1103/PhysRevLett.98.195502
1998:"The Icosahedral Glass Model"
1772:Kenneth Chang (Oct 5, 2011).
988:
181:in 2011. On 25 October 2018,
3923:"What is... a Quasicrystal?"
3872:Kayser, Lin (Mar 20, 2023).
3420:10.1088/1468-6996/9/1/013008
3158:10.1088/1468-6996/9/3/013001
3012:Nederl. Akad. Wetensch. Proc
2230:Day, Charles (Feb 1, 2001).
1954:10.1016/0039-6028(77)90375-2
1904:10.1016/0039-6028(67)90062-3
1630:10.1016/0378-4371(82)90359-4
1197:10.1524/zkri.219.7.391.35643
512:demonstrated prior to this.
508:and interface-interactions.
420:Electron diffraction pattern
407:fragment. The corresponding
349:Metallurgical Transactions A
320:National Bureau of Standards
7:
3921:Senechal, Marjorie (2006).
3804:"The Quasicrystal Laureate"
2195:10.1103/PhysRevLett.59.1010
1996:Stephens, Peter W. (1989).
1519:Bindi, Luca (Jun 1, 2021).
1456:10.1007/978-94-007-6431-6_7
1329:10.1103/PhysRevLett.53.1951
1094:10.1103/PhysRevLett.53.2477
915:
571:structure of quasicrystals.
506:intermolecular interactions
459:and it was approved by the
411:reveal a ten-fold symmetry.
10:
4506:
3975:10.1103/PhysRevB.87.245122
3338:10.1088/0034-4885/69/2/R03
2145:10.1103/PhysRevLett.55.511
1751:10.1103/PhysRevLett.58.365
1147:, Vol. 26, 910–919 (1981).
1032:10.1103/PhysRevB.75.064205
940:Disordered hyperuniformity
852:making the pan moderately
579:(mathematician brother of
479:Shechtman was awarded the
372:Also in 1985, T. Ishimasa
196:
31:
4436:
4386:D'Arcy Wentworth Thompson
4329:
4237:
4226:
4135:
3108:. Springer. p. 164.
3053:10.1107/S0108767384001203
2947:21.11116/0000-0002-B8DF-4
2698:21.11116/0000-0002-B8DF-4
1442:Al Ajlouni, Rima (2013).
892:Computational Engineering
4480:Condensed matter physics
3758:10.5075/epfl-thesis-2707
3534:Physical Review Research
3132:Yamamoto, Akiji (2008).
3032:Acta Crystallographica A
1702:10.1002/prop.19810290503
805:undercooled liquid state
683:thermodynamic properties
585:almost periodic function
481:Nobel Prize in Chemistry
179:Nobel Prize in Chemistry
32:Not to be confused with
3610:Physical Review Letters
3195:Physical Review Letters
2937:10.1126/science.aav0790
2688:10.1126/science.aav0790
2558:Physical Review Letters
2404:10.1073/pnas.1111115109
2175:Physical Review Letters
2125:Physical Review Letters
1731:Physical Review Letters
1682:Fortschritte der Physik
1546:10.1073/pnas.2101350118
1394:10.1126/science.1135491
1308:Physical Review Letters
1233:10.1126/science.1170827
1073:Physical Review Letters
709:formed as a pentagonal
675:magnetic susceptibility
557:rhombic triacontahedron
530:orthographic projection
355:Physical Review Letters
3394:Tsai, An Pang (2008).
949:Fibonacci quasicrystal
895:
862:perfluorooctanoic acid
738:
730:
698:
659:quantum critical point
572:
544:
440:University of Florence
426:
412:
241:
207:
168:observed that certain
142:translational symmetry
84:translational symmetry
56:
4220:Widmanstätten pattern
4052:Steinhardt's proposal
3808:MIT Technology Review
3771:Widjaja, Edy (2004).
3697:Nature Communications
2800:Nature Communications
889:
736:
721:. Unlike the similar
704:
696:
583:). The concept of an
550:
543:Ho–Mg–Zn quasicrystal
523:
424:Ho–Mg–Zn quasicrystal
418:
390:
280:consisting of sharp '
236:
204:
92:rotational symmetries
42:
4094:Space-filling models
3511:10.1557/mrs.2011.190
2051:Metall Mater Trans A
1846:10.1143/JPSJ.22.1365
1673:and Maki K. (1981).
839:magnetron sputtering
792:electron diffraction
409:diffraction patterns
304:electron diffraction
150:linearly independent
4454:Mathematics and art
4444:Pattern recognition
4414:Aristid Lindenmayer
4006:2008ARPC...59..129T
3967:2013PhRvB..87x5122S
3748:Fikar, Jan (2003).
3717:10.1038/ncomms12261
3709:2016NatCo...712261Z
3632:2012PhRvL.108z5301H
3556:2021PhRvR...3c2033K
3467:2008AnRMS..38..403L
3412:2008STAdM...9a3008T
3373:2014RSCAd...446907C
3367:(87): 46907–46921.
3330:2006RPPh...69..397M
3279:2012NatMa..11.1013D
3207:2019PhRvL.122y0201C
3150:2008STAdM...9a3001Y
3045:1984AcCrA..40..580K
2928:2018Sci...362.1396N
2922:(6421): 1396–1400.
2820:10.1038/ncomms15367
2812:2017NatCo...815367P
2746:10.1038/nature12993
2738:2014Natur.507...86W
2679:2018Sci...362.1396N
2673:(6421): 1396–1400.
2632:10.1038/nature08439
2624:2009Natur.461..964T
2570:2007PhRvL..98s5502H
2504:10.1038/nature02368
2395:2012PNAS..109.1396B
2338:2015NatSR...5E9111B
2293:dictionary.iucr.org
2248:2001PhT....54b..17D
2187:1987PhRvL..59.1010W
2137:1985PhRvL..55..511I
2064:1985MTA....16.1005S
1946:1977SurSc..67..139G
1896:1967SurSc...7....1A
1838:1967JPSJ...22.1365I
1743:1987PhRvL..58..365P
1694:1981ForPh..29..219K
1653:Acta Crystallogr. A
1622:1982PhyA..114..609M
1537:2021PNAS..11801350B
1531:(22): e2101350118.
1386:2007Sci...315.1106L
1380:(5815): 1106–1110.
1320:1984PhRvL..53.1951S
1225:2009Sci...324.1306B
1188:2004ZK....219..391S
1182:(7–2004): 391–446.
1170:Steurer W. (2004).
1085:1984PhRvL..53.2477L
1024:2007PhRvB..75f4205U
816:thermal instability
555:projected into the
220:Al-Attarine Madrasa
163:materials scientist
119:Alan Lindsay Mackay
4392:On Growth and Form
4292:Logarithmic spiral
4129:Patterns in nature
4074:2016-11-04 at the
4057:2016-10-18 at the
3931:Notices of the AMS
3381:10.1039/C4RA07980A
2996:10.1007/BF02395468
2472:2018-12-21 at the
2326:Scientific Reports
2072:10.1007/BF02811670
1928:Gillet, M (1977).
1495:paulsteinhardt.org
1448:Aperiodic Crystals
1346:. Nobelprize.org.
1160:114 A, 609 (1982).
1111:Aperiodic crystals
910:honeycomb patterns
896:
739:
731:
699:
628:reciprocal lattice
573:
545:
484:2011 Nobel Prize.
427:
422:of an icosahedral
413:
405:Khatyrka meteorite
242:
208:
152:directions, where
57:
4462:
4461:
4419:Benoît Mandelbrot
4319:Self-organization
4255:Natural selection
4245:Pattern formation
3945:Physical Review B
3115:978-0-8176-4051-4
3079:978-3-540-64224-4
2977:Bohr, H. (1925).
2618:(7266): 964–967.
2498:(6979): 157–160.
2346:10.1038/srep09111
2256:10.1063/1.1359699
1465:978-94-007-6431-6
1314:(20): 1951–1953.
1079:(26): 2477–2480.
1012:Physical Review B
964:Icosahedral twins
928:Archimedean solid
922:Aperiodic crystal
689:Materials science
679:scaling behaviour
616:Fourier transform
278:Fourier transform
123:Fourier transform
106:Aperiodic tilings
96:Bragg diffraction
16:(Redirected from
4497:
4270:Sexual selection
4232:
4122:
4115:
4108:
4099:
4098:
4025:
3986:
3960:
3939:
3927:
3917:
3915:
3893:
3892:
3890:
3889:
3869:
3863:
3862:
3860:
3859:
3853:
3838:
3829:
3823:
3822:
3820:
3819:
3799:
3790:
3789:
3787:
3786:
3768:
3762:
3761:
3745:
3739:
3738:
3728:
3688:
3679:
3678:
3676:
3675:
3625:
3601:
3595:
3594:
3592:
3591:
3549:
3525:
3516:
3515:
3513:
3504:(8): 581. 2011.
3490:
3479:
3478:
3448:
3442:
3441:
3431:
3391:
3385:
3384:
3356:
3350:
3349:
3313:
3307:
3306:
3287:10.1038/nmat3432
3272:
3257:Nature Materials
3252:
3246:
3245:
3243:
3242:
3186:
3180:
3179:
3169:
3129:
3120:
3119:
3101:
3095:
3094:
3092:
3091:
3063:
3057:
3056:
3026:
3020:
3019:
3007:
3001:
3000:
2998:
2983:Acta Mathematica
2974:
2968:
2967:
2949:
2939:
2907:
2901:
2900:
2881:10.1038/nmat4152
2869:Nature Materials
2864:
2858:
2857:
2839:
2791:
2785:
2784:
2782:
2781:
2717:
2711:
2710:
2700:
2690:
2658:
2652:
2651:
2607:
2601:
2600:
2598:
2597:
2549:
2543:
2542:
2540:
2539:
2483:
2477:
2464:
2458:
2457:
2455:
2454:
2433:
2427:
2426:
2416:
2406:
2389:(5): 1396–1401.
2374:
2368:
2367:
2357:
2317:
2308:
2307:
2305:
2304:
2285:
2279:
2278:
2276:
2275:
2227:
2221:
2220:
2218:
2217:
2211:
2181:(9): 1010–1013.
2172:
2163:
2157:
2156:
2120:
2114:
2113:
2111:
2110:
2090:
2084:
2083:
2058:(6): 1005–1012.
2041:
2035:
2034:
2032:
2031:
1993:
1987:
1986:
1975:
1969:
1968:
1966:
1965:
1925:
1919:
1918:
1916:
1915:
1875:
1869:
1868:
1866:
1865:
1832:(6): 1365–1374.
1817:
1811:
1810:
1805:. Archived from
1799:
1793:
1792:
1790:
1789:
1769:
1763:
1762:
1726:
1720:
1719:
1717:
1716:
1710:
1679:
1667:
1661:
1660:
1640:
1634:
1633:
1605:
1599:
1598:
1596:
1595:
1575:
1569:
1568:
1558:
1548:
1516:
1510:
1509:
1507:
1506:
1487:
1481:
1480:
1478:
1477:
1439:
1433:
1432:
1430:
1429:
1365:
1359:
1358:
1356:
1355:
1340:
1334:
1333:
1331:
1299:
1290:
1289:
1287:
1286:
1277:. Archived from
1264:
1253:
1252:
1219:(5932): 1306–9.
1208:
1202:
1201:
1199:
1167:
1161:
1154:
1148:
1145:Krystallografiya
1141:
1135:
1134:
1132:
1131:
1105:
1099:
1098:
1096:
1064:
1055:
1054:
1052:
1051:
999:
954:
945:
835:Thermal spraying
760:with subsequent
333:crystal twinning
297:reciprocal space
293:sodium carbonate
159:long-range order
113:in the field of
21:
4505:
4504:
4500:
4499:
4498:
4496:
4495:
4494:
4475:Crystallography
4465:
4464:
4463:
4458:
4432:
4325:
4233:
4224:
4131:
4126:
4076:Wayback Machine
4059:Wayback Machine
4032:
3925:
3902:
3900:Further reading
3897:
3896:
3887:
3885:
3870:
3866:
3857:
3855:
3851:
3836:
3830:
3826:
3817:
3815:
3800:
3793:
3784:
3782:
3769:
3765:
3746:
3742:
3689:
3682:
3673:
3671:
3602:
3598:
3589:
3587:
3526:
3519:
3492:
3491:
3482:
3449:
3445:
3392:
3388:
3357:
3353:
3314:
3310:
3253:
3249:
3240:
3238:
3187:
3183:
3130:
3123:
3116:
3102:
3098:
3089:
3087:
3080:
3064:
3060:
3027:
3023:
3008:
3004:
2975:
2971:
2908:
2904:
2865:
2861:
2792:
2788:
2779:
2777:
2732:(7490): 86–89.
2718:
2714:
2659:
2655:
2608:
2604:
2595:
2593:
2550:
2546:
2537:
2535:
2484:
2480:
2474:Wayback Machine
2465:
2461:
2452:
2450:
2445:. Oct 5, 2011.
2435:
2434:
2430:
2375:
2371:
2318:
2311:
2302:
2300:
2287:
2286:
2282:
2273:
2271:
2228:
2224:
2215:
2213:
2209:
2170:
2164:
2160:
2121:
2117:
2108:
2106:
2091:
2087:
2047:
2042:
2038:
2029:
2027:
2020:
1994:
1990:
1985:. Sep 30, 2016.
1977:
1976:
1972:
1963:
1961:
1934:Surface Science
1926:
1922:
1913:
1911:
1884:Surface Science
1876:
1872:
1863:
1861:
1818:
1814:
1801:
1800:
1796:
1787:
1785:
1770:
1766:
1727:
1723:
1714:
1712:
1708:
1677:
1668:
1664:
1650:
1646:
1641:
1637:
1606:
1602:
1593:
1591:
1576:
1572:
1517:
1513:
1504:
1502:
1489:
1488:
1484:
1475:
1473:
1466:
1440:
1436:
1427:
1425:
1366:
1362:
1353:
1351:
1342:
1341:
1337:
1300:
1293:
1284:
1282:
1265:
1256:
1209:
1205:
1168:
1164:
1155:
1151:
1142:
1138:
1129:
1127:
1106:
1102:
1065:
1058:
1049:
1047:
1000:
996:
991:
952:
943:
934:Crystallography
918:
901:
850:stainless steel
828:
776:crystallization
691:
652:Bloch's theorem
644:category theory
604:Penrose tilings
518:
475:
471:
467:
454:
450:
446:
402:
398:
394:
363:Paul Steinhardt
345:
317:
218:shrine and the
199:
187:Paul Steinhardt
148: – 1
115:crystallography
37:
28:
23:
22:
15:
12:
11:
5:
4503:
4493:
4492:
4487:
4482:
4477:
4460:
4459:
4457:
4456:
4451:
4446:
4440:
4438:
4434:
4433:
4431:
4430:
4429:
4428:
4416:
4411:
4410:
4409:
4397:
4396:
4395:
4383:
4381:Wilson Bentley
4378:
4376:Joseph Plateau
4373:
4368:
4363:
4362:
4361:
4349:
4344:
4339:
4333:
4331:
4327:
4326:
4324:
4323:
4322:
4321:
4316:
4314:Plateau's laws
4311:
4309:Fluid dynamics
4306:
4296:
4295:
4294:
4289:
4284:
4274:
4273:
4272:
4267:
4262:
4257:
4247:
4241:
4239:
4235:
4234:
4227:
4225:
4223:
4222:
4217:
4212:
4207:
4202:
4201:
4200:
4195:
4190:
4185:
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4139:
4137:
4133:
4132:
4125:
4124:
4117:
4110:
4102:
4096:
4095:
4089:
4083:
4080:Ideas Roadshow
4066:
4061:
4049:
4044:
4039:
4031:
4030:External links
4028:
4027:
4026:
3987:
3940:
3918:
3901:
3898:
3895:
3894:
3864:
3832:Bakhtiari, H.
3824:
3791:
3763:
3740:
3680:
3616:(26): 265301.
3596:
3517:
3480:
3443:
3386:
3351:
3324:(2): 397–441.
3308:
3263:(12): 1013–6.
3247:
3201:(25): 250201.
3181:
3121:
3114:
3096:
3078:
3058:
3039:(5): 580–587.
3021:
3002:
2969:
2902:
2875:(1): 109–116.
2859:
2786:
2712:
2653:
2602:
2564:(19): 195502.
2544:
2478:
2459:
2428:
2369:
2309:
2280:
2222:
2158:
2131:(5): 511–513.
2115:
2099:New York Times
2085:
2045:
2036:
2018:
1988:
1970:
1940:(1): 139–157.
1920:
1870:
1812:
1809:on 2011-10-07.
1794:
1764:
1737:(4): 365–368.
1721:
1688:(5): 219–259.
1662:
1648:
1644:
1635:
1616:(1): 609–613.
1600:
1570:
1511:
1482:
1464:
1434:
1360:
1335:
1291:
1254:
1203:
1176:Z. Kristallogr
1162:
1149:
1136:
1100:
1056:
993:
992:
990:
987:
986:
985:
979:
973:
967:
961:
955:
946:
937:
931:
925:
917:
914:
900:
897:
827:
824:
784:
783:
772:
765:
690:
687:
534:Petrie polygon
532:into 2D using
517:
514:
473:
469:
465:
452:
448:
444:
400:
396:
392:
343:
318:Mn, at the US
315:
239:Penrose tiling
198:
195:
127:Penrose tiling
111:paradigm shift
26:
9:
6:
4:
3:
2:
4502:
4491:
4490:Quasicrystals
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4472:
4470:
4455:
4452:
4450:
4447:
4445:
4442:
4441:
4439:
4435:
4427:
4426:
4422:
4421:
4420:
4417:
4415:
4412:
4408:
4407:
4403:
4402:
4401:
4398:
4394:
4393:
4389:
4388:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4371:Ernst Haeckel
4369:
4367:
4366:Adolf Zeising
4364:
4360:
4359:
4355:
4354:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4334:
4332:
4328:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4301:
4300:
4297:
4293:
4290:
4288:
4285:
4283:
4280:
4279:
4278:
4275:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4252:
4251:
4248:
4246:
4243:
4242:
4240:
4236:
4231:
4221:
4218:
4216:
4213:
4211:
4210:Vortex street
4208:
4206:
4203:
4199:
4196:
4194:
4191:
4189:
4188:Quasicrystals
4186:
4184:
4181:
4180:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4140:
4138:
4134:
4130:
4123:
4118:
4116:
4111:
4109:
4104:
4103:
4100:
4093:
4090:
4087:
4084:
4081:
4077:
4073:
4070:
4067:
4065:
4062:
4060:
4056:
4053:
4050:
4048:
4045:
4043:
4040:
4037:
4034:
4033:
4023:
4019:
4015:
4011:
4007:
4003:
3999:
3995:
3994:
3988:
3984:
3980:
3976:
3972:
3968:
3964:
3959:
3954:
3951:(24) 245122.
3950:
3946:
3941:
3938:(8): 886–887.
3937:
3933:
3932:
3924:
3919:
3914:
3909:
3904:
3903:
3883:
3879:
3875:
3868:
3850:
3846:
3842:
3835:
3828:
3813:
3809:
3805:
3798:
3796:
3780:
3776:
3775:
3767:
3759:
3755:
3751:
3744:
3736:
3732:
3727:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3687:
3685:
3669:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3633:
3629:
3624:
3619:
3615:
3611:
3607:
3600:
3585:
3581:
3577:
3573:
3569:
3565:
3561:
3557:
3553:
3548:
3543:
3540:(3): 032033.
3539:
3535:
3531:
3524:
3522:
3512:
3507:
3503:
3499:
3495:
3489:
3487:
3485:
3476:
3472:
3468:
3464:
3460:
3456:
3455:
3447:
3439:
3435:
3430:
3425:
3421:
3417:
3413:
3409:
3406:(1): 013008.
3405:
3401:
3397:
3390:
3382:
3378:
3374:
3370:
3366:
3362:
3355:
3347:
3343:
3339:
3335:
3331:
3327:
3323:
3319:
3312:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3271:
3266:
3262:
3258:
3251:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3192:
3185:
3177:
3173:
3168:
3163:
3159:
3155:
3151:
3147:
3144:(1): 013001.
3143:
3139:
3135:
3128:
3126:
3117:
3111:
3107:
3100:
3085:
3081:
3075:
3071:
3070:
3062:
3054:
3050:
3046:
3042:
3038:
3034:
3033:
3025:
3017:
3013:
3006:
2997:
2992:
2988:
2984:
2980:
2973:
2965:
2961:
2957:
2953:
2948:
2943:
2938:
2933:
2929:
2925:
2921:
2917:
2913:
2906:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2863:
2855:
2851:
2847:
2843:
2838:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2790:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2735:
2731:
2727:
2723:
2716:
2708:
2704:
2699:
2694:
2689:
2684:
2680:
2676:
2672:
2668:
2664:
2657:
2649:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2606:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2559:
2555:
2548:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2482:
2475:
2471:
2468:
2463:
2448:
2444:
2443:
2438:
2432:
2424:
2420:
2415:
2410:
2405:
2400:
2396:
2392:
2388:
2384:
2380:
2373:
2365:
2361:
2356:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2316:
2314:
2298:
2294:
2290:
2284:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2236:Physics Today
2233:
2226:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2169:
2162:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2119:
2104:
2100:
2096:
2089:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2052:
2040:
2025:
2021:
2019:9780120406036
2015:
2011:
2007:
2003:
1999:
1992:
1984:
1980:
1974:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1924:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1874:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1816:
1808:
1804:
1803:"QC Hot News"
1798:
1783:
1779:
1775:
1768:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1725:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1676:
1672:
1666:
1658:
1654:
1639:
1631:
1627:
1623:
1619:
1615:
1611:
1604:
1589:
1585:
1581:
1574:
1566:
1562:
1557:
1552:
1547:
1542:
1538:
1534:
1530:
1526:
1522:
1515:
1500:
1496:
1492:
1486:
1471:
1467:
1461:
1457:
1453:
1449:
1445:
1438:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1364:
1349:
1345:
1339:
1330:
1325:
1321:
1317:
1313:
1309:
1305:
1298:
1296:
1281:on 2014-12-05
1280:
1276:
1275:
1270:
1263:
1261:
1259:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1207:
1198:
1193:
1189:
1185:
1181:
1177:
1173:
1166:
1159:
1153:
1146:
1140:
1125:
1121:
1117:
1113:
1112:
1104:
1095:
1090:
1086:
1082:
1078:
1074:
1070:
1063:
1061:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1018:(6): 064205.
1017:
1013:
1009:
1005:
998:
994:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
950:
947:
941:
938:
935:
932:
929:
926:
923:
920:
919:
913:
911:
907:
893:
888:
884:
880:
878:
873:
869:
867:
863:
859:
855:
851:
847:
842:
840:
836:
831:
823:
819:
817:
811:
808:
806:
802:
796:
793:
789:
781:
777:
773:
770:
769:melt spinning
766:
763:
759:
755:
754:
753:
750:
746:
744:
743:Dan Shechtman
735:
728:
724:
720:
716:
712:
708:
703:
695:
686:
684:
680:
676:
672:
668:
664:
663:heavy fermion
660:
655:
653:
647:
645:
641:
637:
631:
629:
625:
624:basis vectors
621:
617:
613:
609:
605:
601:
597:
596:lattice plane
592:
590:
586:
582:
578:
570:
566:
565:basis vectors
562:
558:
554:
549:
542:
538:
537:basis vectors
535:
531:
527:
522:
513:
509:
507:
502:
501:self-assembly
498:
494:
490:
489:intermetallic
485:
482:
477:
462:
458:
441:
437:
433:
425:
421:
417:
410:
406:
389:
385:
383:
380:In 1992, the
378:
375:
370:
368:
364:
359:
357:
356:
351:
350:
340:
338:
334:
328:
325:
321:
313:
309:
305:
300:
298:
294:
289:
287:
286:Robert Ammann
283:
279:
275:
271:
270:Penrose tiles
267:
266:Roger Penrose
263:
259:
258:Robert Berger
255:
251:
247:
240:
235:
231:
229:
225:
221:
217:
213:
203:
194:
192:
188:
184:
180:
175:
171:
167:
166:Dan Shechtman
164:
160:
157:described as
155:
151:
147:
143:
138:
136:
130:
128:
124:
120:
116:
112:
107:
103:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
62:
61:quasiperiodic
54:
50:
46:
41:
35:
30:
19:
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