View source for Quasicrystal

{{Short description|Ordered chemical structure with no repeating pattern}}
{{use mdy dates|date=September 2024 |cs1-dates=sy }}
{{Distinguish|Quasi-crystals (supramolecular)}}
[[File:Quasicrystal1.jpg|thumb|Potential energy surface for silver depositing on an [[aluminium]]–[[palladium]]–[[manganese]] (Al–Pd–Mn) quasicrystal surface. Similar to Fig. 6 in Ref.<ref>{{cite journal|last=Ünal|first=B|author2=V. Fournée|author3=K.J. Schnitzenbaumer|author4=C. Ghosh|author5=C.J. Jenks|author5-link=Cynthia Jenks|author6=A.R. Ross|author7=T.A. Lograsso|author8=J.W. Evans|author9=P.A. Thiel|title=Nucleation and growth of Ag islands on fivefold Al-Pd-Mn quasicrystal surfaces: Dependence of island density on temperature and flux|journal=Physical Review B|year=2007|volume=75|pages=064205|doi=10.1103/PhysRevB.75.064205|bibcode=2007PhRvB..75f4205U|issue=6|s2cid=53382207|url=http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1034&context=chem_pubs|access-date=2018-12-21|archive-date=2020-07-28|archive-url=https://web.archive.org/web/20200728104031/https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1034&context=chem_pubs|url-status=live}}</ref>]]

A [[quasiperiodicity|quasiperiodic]] [[crystal]], or '''quasicrystal''', is a [[structure]] that is [[Order and disorder (physics)|ordered]] but not [[Bravais lattice|periodic]]. A quasicrystalline pattern can continuously fill all available space, but it lacks [[translational symmetry]].<ref name="r9" /> 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.<ref>{{Cite book|last1=Lifshitz|first1=Ron|last2=Schmid|first2=Siegbert|last3=Withers|first3=Ray L.|url=http://worldcat.org/oclc/847002667|title=Aperiodic crystals|date=2013|publisher=Springer|oclc=847002667|access-date=2022-12-13|archive-date=2024-09-18|archive-url=https://web.archive.org/web/20240918002535/https://search.worldcat.org/title/847002667|url-status=live}}</ref>

[[Aperiodic tiling]]s 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 [[paradigm shift]] in the field of [[crystallography]]. In crystallography the quasicrystals were predicted in 1981 by a five-fold symmetry study of [[Alan Lindsay Mackay]],<ref>Alan L. Mackay, "De Nive Quinquangula", ''Krystallografiya'', Vol. 26, 910–919 (1981).</ref>—that also brought in 1982, with the crystallographic [[Fourier transform]] of a [[Penrose tiling]],<ref name=":1">Alan L. Mackay, "Crystallography and the Penrose Pattern", ''Physica'' 114 A, 609 (1982).</ref> the possibility of identifying quasiperiodic order in a material through diffraction.

Quasicrystals had been investigated and observed earlier,<ref name=":2">{{cite journal|author=Steurer W.|journal= Z. Kristallogr. |volume=219 |year=2004|pages= 391–446|doi=10.1524/zkri.219.7.391.35643|title=Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals|issue=7–2004|bibcode = 2004ZK....219..391S |doi-access=free}}</ref> 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, [[icosahedrite]], offered evidence for the existence of natural quasicrystals.<ref name="r5" />

Roughly, an ordering is non-periodic 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''&nbsp;–&nbsp;1 [[linear independence|linearly independent]] directions, where ''n'' 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 described as [[long-range 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. 

In 1982, [[materials science|materials scientist]] [[Dan Shechtman]] observed that certain [[aluminium]]–[[manganese]] alloys produced 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<ref name="bloomberg" /><ref name="s" />. Shechtman's discovery challenged the long-held belief that all crystals are periodic. Observed in a rapidly solidified Al-Mn alloy, quasicrystals exhibited [[icosahedral symmetry]], which was previously thought impossible in crystallography<ref>{{Cite journal |last=Blech |first=Ilan A. |last2=Cahn |first2=John W. |last3=Gratias |first3=Denis |date=2012-10-01 |title=Reminiscences About a Chemistry Nobel Prize Won with Metallurgy: Comments on D. Shechtman and I. A. Blech; Metall. Trans. A, 1985, vol. 16A, pp. 1005–12 |url=https://link.springer.com/article/10.1007/s11661-012-1323-1 |journal=Metallurgical and Materials Transactions A |language=en |volume=43 |issue=10 |pages=3411–3422 |doi=10.1007/s11661-012-1323-1 |issn=1543-1940}}</ref>. This breakthrough, supported by theoretical models and experimental evidence, led to a paradigm shift in the understanding of solid-state matter. Despite initial skepticism, the discovery gained widespread acceptance, prompting the [[International Union of Crystallography]] to redefine the term "[[crystal]]."<ref>{{Cite web |title=articles |url=https://www.iucr.org/news/newsletter/etc/articles?issue=151351&result_138339_result_page=17 |access-date=2024-11-22 |website=www.iucr.org}}</ref> The work ultimately earned Shechtman the 2011 [[Nobel Prize in Chemistry]] <ref name="nobel" /> and inspired significant advancements in materials science and mathematics.

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, after they discovered [[icosahedrite]], the first quasicrystal known to occur naturally.

==History==
[[File:Darb-i Imam shrine spandrel.JPG|thumb|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 identified<ref>{{Cite journal |last1=Lu |first1=Peter J. |last2=Steinhardt |first2=Paul J. |date=2007-02-23 |title=Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture |url=https://www.science.org/doi/10.1126/science.1135491 |journal=Science |language=en |volume=315 |issue=5815 |pages=1106–1110 |doi=10.1126/science.1135491 |pmid=17322056 |bibcode=2007Sci...315.1106L |s2cid=10374218 |issn=0036-8075 |access-date=2023-05-30 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002534/https://www.science.org/doi/10.1126/science.1135491 |url-status=live }}</ref>]]
The first representations of perfect quasicrystalline patterns can be found in several early [[Islamic art|Islamic works of art and architecture]] such as the Gunbad-i-Kabud tomb tower, the [[Darb-e Imam]] shrine and the [[Al-Attarine Madrasa]].<ref>{{Cite book |last=Al Ajlouni |first=Rima |title=Aperiodic Crystals |chapter=Octagon-Based Quasicrystalline Formations in Islamic Architecture |date=2013 |editor-last=Schmid |editor-first=Siegbert |editor2-last=Withers |editor2-first=Ray L. |editor3-last=Lifshitz |editor3-first=Ron |chapter-url=https://link.springer.com/chapter/10.1007/978-94-007-6431-6_7 |language=en |location=Dordrecht |publisher=Springer Netherlands |pages=49–57 |doi=10.1007/978-94-007-6431-6_7 |isbn=978-94-007-6431-6 |access-date=2023-05-30 |archive-date=2023-05-30 |archive-url=https://web.archive.org/web/20230530081621/https://link.springer.com/chapter/10.1007/978-94-007-6431-6_7 |url-status=live }}</ref><ref>{{Cite web |title=Islamic Quasicrystal Tilings {{!}} Paul J. Steinhardt |url=https://paulsteinhardt.org/islamic-tilings/ |access-date=2023-05-29 |website=paulsteinhardt.org |language=en-US |archive-date=2023-05-29 |archive-url=https://web.archive.org/web/20230529082126/https://paulsteinhardt.org/islamic-tilings/ |url-status=live }}</ref> On July 16, 1945, in Alamogordo, New Mexico, the [[Trinity (nuclear test)|Trinity]] nuclear bomb test produced icosahedral quasicrystals. They went unnoticed at the time of the test but were later identified in samples of red [[trinitite]], a glass-like substance formed from fused sand and copper transmission lines. Identified in 2021, they are the oldest known anthropogenic quasicrystals.<ref name=":3">{{cite journal |last1=Bindi |first1=Luca |title=Accidental synthesis of a previously unknown quasicrystal in the first atomic bomb test |journal=Proceedings of the National Academy of Sciences |date=2021-06-01 |volume=118 |issue=22 |pages=e2101350118 |doi=10.1073/pnas.2101350118 |pmid=34001665 |pmc=8179242 |bibcode=2021PNAS..11801350B |doi-access=free }}</ref><ref name="redtrinitite">{{cite news |last=Mullane |first=Laura |date=May 18, 2021 |title=Newly discovered quasicrystal was created by the first nuclear explosion at Trinity Site |url=https://phys.org/news/2021-05-newly-quasicrystal-nuclear-explosion-trinity.html |work=Phys.org |location= |access-date=May 21, 2021 |archive-date=June 21, 2021 |archive-url=https://web.archive.org/web/20210621144831/https://phys.org/news/2021-05-newly-quasicrystal-nuclear-explosion-trinity.html |url-status=live }}<!--Note that the author is from the same university as the lead researcher and co-author of the paper, so this is effectively a primary source.--></ref>

[[File:Penrose Tiling (Rhombi).svg|thumb|A [[Penrose tiling]]]]
In 1961, [[Hao Wang (academic)|Hao Wang]] asked whether determining if a set of tiles admits a tiling of the plane is an [[algorithmically unsolvable problem]] 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 ''periodically'' (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student [[Robert Berger (mathematician)|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. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974 [[Roger Penrose]] discovered a set of just two tiles, now referred to as [[Penrose tiling|Penrose tiles]], that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later [[Alan Lindsay Mackay|Alan Mackay]] showed theoretically that the diffraction pattern from the Penrose tiling had a two-dimensional [[Fourier transform]] consisting of sharp '[[Dirac delta function|delta]]' peaks arranged in a fivefold symmetric pattern.<ref name="r2" /> Around the same time, [[Robert Ammann]] created a set of aperiodic tiles that produced eightfold symmetry.

In 1972, R. M. de Wolf and W. van Aalst<ref name="r6" /> reported that the diffraction pattern produced by a crystal of [[sodium carbonate]] cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in [[Reciprocal lattice|reciprocal space]]. Other puzzling cases have been reported,<ref name="Kleinert" /> but until the concept of quasicrystal came to be established, they were explained away or denied.<ref name="r7" /><ref name="r8" />

Dan Shechtman first observed ten-fold [[electron diffraction]] patterns in 1982, while conducting a routine study of an [[aluminium]]–[[manganese]] alloy, Al<sub>6</sub>Mn, at the US [[National Bureau of Standards]] (later NIST).<ref name=":4">{{Cite web | url=http://www.quasi.iastate.edu/discovery.html | archive-url=https://web.archive.org/web/20111007225727/http://www.quasi.iastate.edu/discovery.html | url-status=dead | archive-date=2011-10-07 | title=QC Hot News}}</ref> Shechtman related his observation to Ilan Blech, who responded that such diffractions had been seen before.<ref>{{Cite journal |last1=Ino |first1=Shozo |last2=Ogawa |first2=Shiro |date=1967 |title=Multiply Twinned Particles at Earlier Stages of Gold Film Formation on Alkalihalide Crystals |url=https://journals.jps.jp/doi/10.1143/JPSJ.22.1365 |journal=Journal of the Physical Society of Japan |language=en |volume=22 |issue=6 |pages=1365–1374 |doi=10.1143/JPSJ.22.1365 |bibcode=1967JPSJ...22.1365I |issn=0031-9015 |access-date=2023-03-24 |archive-date=2023-03-24 |archive-url=https://web.archive.org/web/20230324203315/https://journals.jps.jp/doi/10.1143/JPSJ.22.1365 |url-status=live }}</ref><ref>{{Cite journal |last1=Allpress |first1=J.G. |last2=Sanders |first2=J.V. |date=1967 |title=The structure and orientation of crystals in deposits of metals on mica |url=https://linkinghub.elsevier.com/retrieve/pii/0039602867900623 |journal=Surface Science |language=en |volume=7 |issue=1 |pages=1–25 |doi=10.1016/0039-6028(67)90062-3 |bibcode=1967SurSc...7....1A |access-date=2023-03-24 |archive-date=2023-12-03 |archive-url=https://web.archive.org/web/20231203003246/https://linkinghub.elsevier.com/retrieve/pii/0039602867900623 |url-status=live }}</ref><ref>{{Cite journal |last=Gillet |first=M |date=1977 |title=Structure of small metallic particles |url=https://linkinghub.elsevier.com/retrieve/pii/0039602877903752 |journal=Surface Science |language=en |volume=67 |issue=1 |pages=139–157 |doi=10.1016/0039-6028(77)90375-2 |bibcode=1977SurSc..67..139G |access-date=2023-03-24 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002514/https://www.sciencedirect.com/unsupported_browser |url-status=live }}</ref> Around that time, Shechtman also related his finding to [[John W. Cahn]] 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".<ref name= cahn>{{cite journal| url = https://www.nist.gov/nist-and-nobel/dan-shechtman/nobel-moment-dan-shechtman| title = NIST and the Nobel (September 30, 2016, Updated November 17, 2019) The Nobel Moment: Dan Shechtman| journal = NIST| date = 30 September 2016}}</ref>

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 [[crystal twinning]], 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 [[polyhedral symmetry|polyhedral]]", and found a ten-fold structure similar to what was observed. The multiple polyhedral structure was termed later by many researchers as icosahedral glass.<ref>{{cite book | chapter-url=https://www.sciencedirect.com/science/article/pii/B9780120406036500076 | doi=10.1016/B978-0-12-040603-6.50007-6 | chapter=The Icosahedral Glass Model | title=Extended Icosahedral Structures | series=Aperiodicity and Order | year=1989 | last1=Stephens | first1=Peter W. | volume=3 | pages=37–104 | isbn=9780120406036 | access-date=2021-05-07 | archive-date=2021-05-07 | archive-url=https://web.archive.org/web/20210507114808/https://www.sciencedirect.com/science/article/pii/B9780120406036500076 | url-status=live }}</ref>

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<sub>6</sub>Mn", which was written around June 1984 and published in a 1985 edition of ''[[Metallurgical Transactions A]]''.<ref>{{cite journal|last=Shechtman|first=Dan|author2=I. A. Blech |title=The Microstructure of Rapidly Solidified Al<sub>6</sub>Mn|journal=[[Metall Mater Trans A]]|year=1985|volume=16A|pages=1005–1012|doi=10.1007/BF02811670|issue=6|bibcode = 1985MTA....16.1005S |s2cid=136733193}}</ref> 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 ''[[Physical Review Letters]]'',<ref name="s" /> repeated Shechtman's observation and used the same illustrations as the original paper.

Originally, the new form of matter was dubbed "Shechtmanite".<ref>{{cite news|title=Impossible' Form of Matter Takes Spotlight In Study of Solids|newspaper=New York Times|url=https://www.nytimes.com/1989/09/05/science/impossible-form-of-matter-takes-spotlight-in-study-of-solids.html?pagewanted=all&src=pm|first=Malcolm W.|last=Browne|date=1989-09-05|access-date=2017-02-12|archive-date=2017-12-01|archive-url=https://web.archive.org/web/20171201135113/http://www.nytimes.com/1989/09/05/science/impossible-form-of-matter-takes-spotlight-in-study-of-solids.html?pagewanted=all&src=pm|url-status=live}}</ref> The term "quasicrystal" was first used in print by [[Paul Steinhardt]] and [[Dov Levine]]<ref name="r9" /> shortly after Shechtman's paper was published.

Also in 1985, T. Ishimasa ''et al.'' reported twelvefold symmetry in Ni-Cr particles.<ref name="r3" /> Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys.<ref name="r4" /> Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were 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.<ref>{{Cite journal|last=Day|first=Charles|date=2001-02-01|title=Binary Quasicrystals Discovered That Are Stable and Icosahedral|url=https://physicstoday.scitation.org/doi/10.1063/1.1359699|journal=Physics Today|volume=54|issue=2|pages=17–18|doi=10.1063/1.1359699|bibcode=2001PhT....54b..17D|issn=0031-9228|access-date=2021-01-10|archive-date=2021-11-03|archive-url=https://web.archive.org/web/20211103024025/https://physicstoday.scitation.org/doi/10.1063/1.1359699|url-status=live}}</ref>

In 1992, the [[International Union of Crystallography]] 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.<ref name="bloomberg" /><ref>{{Cite web |title=Quasicrystal – Online Dictionary of Crystallography |url=https://dictionary.iucr.org/Quasicrystal |access-date=2024-04-04 |website=dictionary.iucr.org |archive-date=2024-04-04 |archive-url=https://web.archive.org/web/20240404024622/https://dictionary.iucr.org/Quasicrystal |url-status=live }}</ref>

[[File:Al71Ni24Fe5 TEM.jpg|thumb|Atomic image of a micron-sized grain of the natural Al<sub>71</sub>Ni<sub>24</sub>Fe<sub>5</sub> quasicrystal (shown in the inset) from a [[Khatyrka meteorite]] fragment. The corresponding [[:File:Al71Ni24Fe5 diffraction.jpg|diffraction patterns]] reveal a ten-fold symmetry.<ref name="natq" />]]
[[File:Zn-Mg-HoDiffraction.JPG|thumb|[[Electron crystallography|Electron diffraction pattern]] of an icosahedral [[holmium–magnesium–zinc quasicrystal|Ho–Mg–Zn quasicrystal]]]]

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 [[Luca Bindi]], who found a quasicrystalline specimen from [[Khatyrka (river)|Khatyrka]] in the [[University of Florence]] 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 Al<sub>63</sub>Cu<sub>24</sub>Fe<sub>13</sub>, was named [[icosahedrite]] and it was approved by the [[International Mineralogical Association]] 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.<ref>{{Cite journal | doi = 10.1073/pnas.1111115109 | pmid = 22215583 | last = Bindi | first = Luca |author2=John M. Eiler |author3=Yunbin Guan |author4=Lincoln S. Hollister |author5=Glenn MacPherson |author6=Paul J. Steinhardt |author7=Nan Yao | title = Evidence for the extraterrestrial origin of a natural quasicrystal | journal = Proceedings of the National Academy of Sciences | date = 2012-01-03 |bibcode = 2012PNAS..109.1396B | volume = 109 | issue = 5 | pages = 1396–1401 |pmc=3277151| doi-access = free }}</ref> A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al<sub>71</sub>Ni<sub>24</sub>Fe<sub>5</sub>. 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.<ref name="natq" />

Shechtman was awarded the [[Nobel Prize in Chemistry]] 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."<ref name="bloomberg" /><ref>{{cite news|url=https://www.bbc.co.uk/news/science-environment-15181187|title=Nobel win for crystal discovery|newspaper=[[BBC News]]|access-date=2011-10-05|date=2011-10-05|archive-date=2011-10-05|archive-url=https://web.archive.org/web/20111005104716/http://www.bbc.co.uk/news/science-environment-15181187|url-status=live}}</ref> In 2014, Post of Israel issued a stamp dedicated to quasicrystals and the 2011 Nobel Prize.<ref>[https://www.iycr2014.org/countries/events-by-country/israel/iycr-philately-day Crystallography matters ... more!] {{Webarchive|url=https://web.archive.org/web/20181221134716/https://www.iycr2014.org/countries/events-by-country/israel/iycr-philately-day |date=2018-12-21 }} iycr2014.org</ref>

While the first quasicrystals discovered were made out of [[intermetallic]] components, later on quasicrystals were also discovered in [[Soft matter|soft-matter]] and [[Molecule|molecular]] systems. Soft quasicrystal structures have been found in supramolecular dendrimer liquids<ref>{{Cite journal |last1=Zeng |first1=Xiangbing |last2=Ungar |first2=Goran |last3=Liu |first3=Yongsong |last4=Percec |first4=Virgil |last5=Dulcey |first5=Andrés E. |last6=Hobbs |first6=Jamie K. |date=March 2004 |title=Supramolecular dendritic liquid quasicrystals |url=https://www.nature.com/articles/nature02368 |journal=Nature |language=en |volume=428 |issue=6979 |pages=157–160 |doi=10.1038/nature02368 |pmid=15014524 |s2cid=4429689 |issn=1476-4687 |access-date=2022-11-06 |archive-date=2022-11-06 |archive-url=https://web.archive.org/web/20221106132759/https://www.nature.com/articles/nature02368 |url-status=live }}</ref> and ABC Star Polymers<ref>{{Cite journal |last1=Hayashida |first1=Kenichi |last2=Dotera |first2=Tomonari |last3=Takano |first3=Atsushi |last4=Matsushita |first4=Yushu |date=2007-05-08 |title=Polymeric Quasicrystal: Mesoscopic Quasicrystalline Tiling in $ABC$ Star Polymers |url=https://link.aps.org/doi/10.1103/PhysRevLett.98.195502 |journal=Physical Review Letters |volume=98 |issue=19 |pages=195502 |doi=10.1103/PhysRevLett.98.195502 |pmid=17677627 |bibcode=2007PhRvL..98s5502H |access-date=2022-11-06 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002513/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.195502 |url-status=live }}</ref> in 2004 and 2007. 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.<ref>{{cite journal|doi=10.1038/nature08439|pmid=19829378|title=Quasicrystalline order in self-assembled binary nanoparticle superlattices|journal=Nature|volume=461|issue=7266|pages=964–967|year=2009|last1=Talapin|first1=Dmitri V.|last2=Shevchenko|first2=Elena V.|last3=Bodnarchuk|first3=Maryna I.|last4=Ye|first4=Xingchen|last5=Chen|first5=Jun|last6=Murray|first6=Christopher B.|bibcode=2009Natur.461..964T|s2cid=4344953}}</ref> It was demonstrated that these units can be both inorganic and organic.<ref>{{cite journal|doi=10.1126/science.aav0790 |pmid=30573624 |title=Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule |journal=Science |volume=362 |issue=6421 |pages=1396–1400 |year=2018 |last1=Nagaoka |first1=Yasutaka |last2=Zhu |first2=Hua |last3=Eggert |first3=Dennis |last4=Chen |first4=Ou |bibcode=2018Sci...362.1396N |doi-access=free|hdl=21.11116/0000-0002-B8DF-4 |hdl-access=free }}</ref> Additionally in the 2010s, two-dimensional molecular quasicrystals were discovered, driven by [[Intermolecular force|intermolecular interactions]]<ref>{{Cite journal |last1=Wasio |first1=Natalie A. |last2=Quardokus |first2=Rebecca C. |last3=Forrest |first3=Ryan P. |last4=Lent |first4=Craig S. |last5=Corcelli |first5=Steven A. |last6=Christie |first6=John A. |last7=Henderson |first7=Kenneth W. |last8=Kandel |first8=S. Alex |date=March 2014 |title=Self-assembly of hydrogen-bonded two-dimensional quasicrystals |url=https://www.nature.com/articles/nature12993 |journal=Nature |language=en |volume=507 |issue=7490 |pages=86–89 |doi=10.1038/nature12993 |pmid=24598637 |bibcode=2014Natur.507...86W |s2cid=4401013 |issn=1476-4687 |access-date=2022-11-06 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002622/https://www.nature.com/articles/nature12993 |url-status=live }}</ref> and interface-interactions.<ref>{{Cite journal |last1=Paßens |first1=M. |last2=Caciuc |first2=V. |last3=Atodiresei |first3=N. |last4=Feuerbacher |first4=M. |last5=Moors |first5=M. |last6=Dunin-Borkowski |first6=R. E. |last7=Blügel |first7=S. |last8=Waser |first8=R. |last9=Karthäuser |first9=S. |date=2017-05-22 |title=Interface-driven formation of a two-dimensional dodecagonal fullerene quasicrystal |journal=Nature Communications |language=en |volume=8 |issue=1 |article-number=15367 |doi=10.1038/ncomms15367 |pmid=28530242 |pmc=5458153 |bibcode=2017NatCo...815367P |s2cid=22736155 |issn=2041-1723}}</ref>

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,<ref>{{Cite journal|last1=Engel|first1=Michael|last2=Damasceno|first2=Pablo F.|last3=Phillips|first3=Carolyn L.|last4=Glotzer|first4=Sharon C.|date=2014-12-08|title=Computational self-assembly of a one-component icosahedral quasicrystal|journal=Nature Materials|language=en|volume=14|issue=1|pages=109–116|doi=10.1038/nmat4152|issn=1476-4660|pmid=25485986}}</ref> they had not been demonstrated prior to this.<ref>{{Cite journal|last1=Chen|first1=Ou|last2=Eggert|first2=Dennis|last3=Zhu|first3=Hua|last4=Nagaoka|first4=Yasutaka|date=2018-12-21|title=Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule|journal=Science|language=en|volume=362|issue=6421|pages=1396–1400|doi=10.1126/science.aav0790|issn=0036-8075|pmid=30573624|bibcode=2018Sci...362.1396N|doi-access=free|hdl=21.11116/0000-0002-B8DF-4|hdl-access=free}}</ref>

==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 for most experimentalists. 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.

==Materials science==
[[Image:frustrationPentagons.jpg|thumb|Tiling of a plane by regular pentagons is impossible but can be realized on a sphere in the form of pentagonal dodecahedron.]]
[[File:Ho-Mg-ZnQuasicrystal.jpg|thumb|A [[holmium–magnesium–zinc quasicrystal|Ho–Mg–Zn dodecahedral quasicrystal]] formed as a pentagonal [[dodecahedron]], the [[dual polyhedron|dual]] of the [[icosahedron]]. Unlike the similar [[pyritohedron]] shape of some cubic-system crystals such as [[pyrite]], the quasicrystal has faces that are true regular pentagons]]
[[File:TiMnImage2y.png|right|thumb|TiMn quasicrystal approximant lattice]]

Since the original discovery by [[Dan Shechtman]], 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.).<ref>{{cite journal|doi=10.1088/0034-4885/69/2/R03|title=The role of aperiodic order in science and technology|year=2006|last1=MacIá|first1=Enrique|journal=Reports on Progress in Physics|volume=69|pages=397–441|bibcode = 2006RPPh...69..397M|issue=2|s2cid=120125675 }}</ref>

Two types of quasicrystals are known.<ref name="comp" /> 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, 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.<ref>{{cite journal |last1=C |first1=Cui |last2=M |first2=Shimoda |last3=AP |first3=Tsai |title=Studies on icosahedral Ag-In-Yb: A prototype for Tsai-type quasicrystals |journal=RSC Advances |date=2014 |volume=4 |issue=87 |pages=46907–46921 |doi=10.1039/C4RA07980A|bibcode=2014RSCAd...446907C }}</ref>

Quasicrystals fall into three groups of different thermal stability:<ref>{{cite journal|doi=10.1088/1468-6996/9/1/013008|pmc=5099795|title=Icosahedral clusters, icosaheral order and stability of quasicrystals – a view of metallurgy|year=2008|last1=Tsai|first1=An Pang|journal=Science and Technology of Advanced Materials|volume=9|issue=1|page=013008|bibcode = 2008STAdM...9a3008T|pmid=27877926}}</ref>
* Stable quasicrystals grown by slow cooling or [[casting (metalworking)|casting]] with subsequent [[annealing (metallurgy)|annealing]],
* Metastable quasicrystals prepared by [[melt spinning]], and
* Metastable quasicrystals formed by the [[crystallization]] of the [[amorphous]] phase.

Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by [[X-ray diffraction|X-ray]] and [[electron diffraction]] 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.

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 ''liquid state'' for the stable quasicrystals, whereas the icosahedral order prevails in the ''undercooled liquid state'' for the metastable quasicrystals.

A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals.<ref>{{Cite journal | doi = 10.1146/annurev.matsci.38.060407.130318| title = Formation and Properties of Quasicrystals| journal = [[Annual Review of Materials Research]]| volume = 38| pages = 403–423| year = 2008| last1 = Louzguine-Luzgin | first1 = D. V. | last2 = Inoue | first2 = A.|bibcode = 2008AnRMS..38..403L}}</ref>

Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick
properties.<ref name="mrs" /> Many metallic quasicrystalline substances are impractical for most applications due to their [[Thermal stability|thermal instability]]; the Al–Cu–Fe ternary system and the Al–Cu–Fe–Cr and Al–Co–Fe–Cr quaternary systems, thermally stable up to 700&nbsp;°C, are notable exceptions.

The quasi-ordered droplet crystals could be formed under Dipolar forces in the Bose Einstein condensate.<ref name=":0">{{Cite journal|last=Khazali|first=Mohammadsadegh|date=2021-08-05|title=Rydberg noisy dressing and applications in making soliton molecules and droplet quasicrystals|url=http://dx.doi.org/10.1103/physrevresearch.3.l032033|journal=Physical Review Research|volume=3|issue=3|page=032033|doi=10.1103/physrevresearch.3.l032033|issn=2643-1564|arxiv=2007.01039|bibcode=2021PhRvR...3c2033K|s2cid=220301701|access-date=2021-09-30|archive-date=2024-09-18|archive-url=https://web.archive.org/web/20240918002539/https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.L032033|url-status=live}}</ref> While the softcore Rydberg dressing interaction has forms triangular droplet-crystals,<ref>{{Cite journal|last1=Henkel|first1=N.|last2=Cinti|first2=F.|last3=Jain|first3=P.|last4=Pupillo|first4=G.|last5=Pohl|first5=T.|date=2012-06-26|title=Supersolid Vortex Crystals in Rydberg-Dressed Bose-Einstein Condensates|url=http://dx.doi.org/10.1103/physrevlett.108.265301|journal=Physical Review Letters|volume=108|issue=26|page=265301|doi=10.1103/physrevlett.108.265301|pmid=23004994|issn=0031-9007|arxiv=1111.5761|bibcode=2012PhRvL.108z5301H|s2cid=1782501|access-date=2021-09-30|archive-date=2024-09-18|archive-url=https://web.archive.org/web/20240918002514/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.265301|url-status=live}}</ref> adding a Gaussian peak to the plateau type interaction would form multiple roton unstable points in the 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.<ref name=":0" />

===Applications===
Quasicrystalline substances have potential applications in several forms.

Metallic quasicrystalline coatings can be applied by [[Thermal spraying]] or [[magnetron sputtering]]. A problem that must be resolved is the tendency for cracking due to the materials' extreme brittleness.<ref name="mrs" /> The cracking could be suppressed by reducing sample dimensions or coating thickness.<ref name="nature.com" /> Recent studies show typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains at room temperature and sub-micrometer scales (<500&nbsp;nm).<ref name="nature.com" />

An application was the use of low-friction Al–Cu–Fe–Cr quasicrystals<ref>{{cite thesis|doi=10.5075/epfl-thesis-2707|author=Fikar, Jan |year=2003|title=Al-Cu-Fe quasicrystalline coatings and composites studied by mechanical spectroscopy|publisher=École polytechnique fédérale de Lausanne EPFL, Thesis n° 2707 (2002)}}</ref> as a coating for [[frying pan]]s. Food did not stick to it as much as to [[stainless steel]] making the pan moderately [[non-stick]] and easy to clean; heat transfer and durability were better than [[PTFE]] non-stick cookware and the pan was free from [[perfluorooctanoic acid]] (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 [[dishwasher]]; and the pan could withstand temperatures of {{Convert|1000|C|sigfig=2}} 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.<ref>{{Cite book |last=Widjaja |first=Edy |url=https://www.proquest.com/openview/44d6ab76271620818381a74684e9945a/1?pq-origsite=gscholar&cbl=18750&diss=y&casa_token=9d5DGvy1Q3EAAAAA:InGBLgm8r0b-OQ2GMmc4bosumNpFp5ucdJQzXDqKzTe3fp2Uz5-XVpa2qV5FVNOIBVBb4ZfnQg |title=Quasicrystalline thin films: growth, structure and interface |publisher=Northwestern University |year=2004 |location=Evanston, Illinois, USA |pages=Appendix A |access-date=2023-12-02 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002538/https://www.proquest.com/openview/44d6ab76271620818381a74684e9945a/1?pq-origsite=gscholar&cbl=18750&diss=y&casa_token=9d5DGvy1Q3EAAAAA:InGBLgm8r0b-OQ2GMmc4bosumNpFp5ucdJQzXDqKzTe3fp2Uz5-XVpa2qV5FVNOIBVBb4ZfnQg |url-status=live }}</ref>

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 corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material.<ref name="mitrev" />

Quasicrystals were also being used to develop heat insulation, [[LED]]s, 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.<ref name="mitrev" /> 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.<ref>{{cite journal |last1=Bakhtiari |first1=H. |title=An Overview of Quasicrystals, Their Types, Preparation Methods, Properties |journal=Journal of Environmental Friendly Materials |volume=5 |pages=69–76 |url=http://jefm.kiau.ac.ir/article_682540_041d8d272bb5706da2472b22eace5d2c.pdf |access-date=2021-10-31 |archive-date=2021-10-31 |archive-url=https://web.archive.org/web/20211031204626/http://jefm.kiau.ac.ir/article_682540_041d8d272bb5706da2472b22eace5d2c.pdf |url-status=live }}</ref>

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 quasicrystalline alloys such as Al–Pd–Mn.<ref name="mrs" />
[[File:Rendering of quasicrystal structure.jpg|thumb|Rendering of a quasicrystalline structure, created using an open-source model for [[Computational engineering|Computational Engineering]].]]

==Non-material science applications==
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 [[isogrid]] or [[Honeycomb structure|honeycomb patterns]]. None of these seem to have been put to use in practice.<ref>{{Cite web |last=Kayser |first=Lin |date=2023-03-20 |title=Could centuries-old islamic patterns be the key to hypersonic flight? |url=https://jlk.ae/2023/03/20/could-centuries-old-islamic-patterns-be-the-key-to-hypersonic-flight/ |access-date=2023-03-20 |website=Josefine Lissner and Lin Kayser |language=en-US |archive-date=2023-03-20 |archive-url=https://web.archive.org/web/20230320095514/https://jlk.ae/2023/03/20/could-centuries-old-islamic-patterns-be-the-key-to-hypersonic-flight/ |url-status=live }}</ref>

==See also==
* {{Annotated link|Aperiodic crystal}}
* {{Annotated link|Archimedean solid}}
* {{Annotated link|Crystallography}}
* {{Annotated link|Disordered hyperuniformity}}
* {{Annotated link|Fibonacci quasicrystal}}
* {{Annotated link|Fiveling}}
* {{Annotated link|Icosahedral twins}}
* {{Annotated link|Phason}}
* {{Annotated link|Tessellation}}
* {{Annotated link|Time crystal}}

==References==
{{reflist|refs=
<ref name="Kleinert">{{cite journal | title = Lattice Textures in Cholesteric Liquid Crystals | author = [[Hagen Kleinert|Kleinert H.]] and Maki K. | journal = Fortschritte der Physik | volume = 29 | issue = 5 | pages = 219–259 | year = 1981 | doi = 10.1002/prop.19810290503 | url = http://www.physik.fu-berlin.de/~kleinert/75/75.pdf | bibcode = 1981ForPh..29..219K | access-date = 2011-10-07 | archive-date = 2020-04-26 | archive-url = https://web.archive.org/web/20200426005943/http://users.physik.fu-berlin.de/~kleinert/75/75.pdf | url-status = live }}</ref>

<ref name="r2">{{cite journal|doi=10.1016/0378-4371(82)90359-4|title=Crystallography and the Penrose Pattern|year=1982|last1=Mackay|first1=A.L.|journal=Physica A|volume=114|issue=1|pages=609–613|bibcode = 1982PhyA..114..609M}}</ref>

<ref name="r3">{{cite journal|doi=10.1103/PhysRevLett.55.511|pmid=10032372|title=New ordered state between crystalline and amorphous in Ni-Cr particles|year=1985|last1=Ishimasa|first1=T.|last2=Nissen|first2=H.-U.|last3=Fukano|first3=Y.|journal=Physical Review Letters|volume=55|issue=5|pages=511–513|bibcode=1985PhRvL..55..511I}}</ref>

<ref name="r4">{{cite journal|doi=10.1103/PhysRevLett.59.1010|pmid=10035936|title=Two-dimensional quasicrystal with eightfold rotational symmetry|year=1987|last1=Wang|first1=N.|last2=Chen|first2=H.|last3=Kuo|first3=K.|journal=Physical Review Letters|volume=59|issue=9|pages=1010–1013|bibcode=1987PhRvL..59.1010W|url=http://repository.ust.hk/ir/bitstream/1783.1-50899/1/PhysRevLett.59.1010.pdf|access-date=2018-12-21|archive-date=2018-07-21|archive-url=https://web.archive.org/web/20180721024645/http://repository.ust.hk/ir/bitstream/1783.1-50899/1/PhysRevLett.59.1010.pdf|url-status=live}}</ref>

<ref name="r5">{{cite journal|doi=10.1126/science.1170827|pmid=19498165|title=Natural Quasicrystals|year=2009|last1=Bindi|first1=L.|last2=Steinhardt|first2=P. J.|last3=Yao|first3=N.|last4=Lu|first4=P. J.|journal=Science|volume=324|issue=5932|pages=1306–9|bibcode = 2009Sci...324.1306B |s2cid=14512017}}</ref>

<ref name="r6">{{cite journal|author1=de Wolf, R.M. |author2=van Aalst, W. |name-list-style=amp |title= The four dimensional group of γ-Na<sub>2</sub>CO<sub>3</sub>|journal=Acta Crystallogr. A |volume=28 |year=1972|page= S111}}</ref>

<ref name="r7">{{cite journal | title=So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal |journal=Physical Review Letters |date=1987-01-26|volume=58|issue=4|pages=365–368| pmid=10034915 | doi=10.1103/PhysRevLett.58.365 | last1=Pauling | first1=L|bibcode = 1987PhRvL..58..365P}}</ref>

<ref name="r8">{{cite news | newspaper=NY Times | url=https://www.nytimes.com/2011/10/06/science/06nobel.html | title=Israeli Scientist Wins Nobel Prize for Chemistry | author=Kenneth Chang | date=October 5, 2011 | access-date=February 12, 2017 | archive-date=February 19, 2017 | archive-url=https://web.archive.org/web/20170219052709/http://www.nytimes.com/2011/10/06/science/06nobel.html | url-status=live }}</ref>

<ref name="r9">{{cite journal|doi=10.1103/PhysRevLett.53.2477|title=Quasicrystals: A New Class of Ordered Structures|year=1984|last1=Levine|first1=Dov|last2=Steinhardt|first2=Paul|journal=Physical Review Letters|volume=53|pages=2477–2480|bibcode=1984PhRvL..53.2477L|issue=26|doi-access=free}}</ref>

<ref name="Suck04">{{cite book|author1=Suck, Jens-Boie|author2=Schreiber, M.|author3=Häussler, Peter|title=Quasicrystals: An Introduction to Structure, Physical Properties and Applications|url=https://books.google.com/books?id=a0_F4CiHevkC&pg=PP1|date=2002|publisher=Springer Science & Business Media|isbn=978-3-540-64224-4|pages=1–|access-date=2016-10-29|archive-date=2024-09-18|archive-url=https://web.archive.org/web/20240918003014/https://books.google.com/books?id=a0_F4CiHevkC&pg=PP1#v=onepage&q&f=false|url-status=live}}</ref>

<ref name="comp">{{cite journal |last1=Yamamoto |first1=Akiji |year=2008 |title=Software package for structure analysis of quasicrystals |journal=Science and Technology of Advanced Materials |volume=9 |issue=1 |page=013001 |pmc=5099788|doi=10.1088/1468-6996/9/3/013001 |bibcode=2008STAdM...9a3001Y |pmid=27877919}}</ref>

<ref name="s">{{cite journal |last1=Shechtman |first1=D. |last2=Blech |first2=I. |last3=Gratias |first3=D. |last4=Cahn |first4=J. |year=1984 |title=Metallic Phase with Long-Range Orientational Order and No Translational Symmetry|journal=Physical Review Letters |volume=53 |issue=20 |pages=1951–1953 |doi=10.1103/PhysRevLett.53.1951 |bibcode=1984PhRvL..53.1951S|doi-access=free}}</ref>

<ref name="nobel">{{cite web |url=https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/ |title=The Nobel Prize in Chemistry 2011 |publisher=Nobelprize.org |access-date=2011-10-06 |archive-date=2017-10-05 |archive-url=https://web.archive.org/web/20171005013742/https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/ |url-status=live }}</ref>

<ref name="speccomp">{{cite journal|first1=Matthew|last1=Colbrook|first2=Bogdan|last2=Roman|first3=Anders|last3=Hansen|title=How to Compute Spectra with Error Control|journal=Physical Review Letters|volume=122|issue=25|pages=250201|year=2019|doi=10.1103/PhysRevLett.122.250201|pmid=31347861|bibcode=2019PhRvL.122y0201C|s2cid=198463498|url=https://www.repository.cam.ac.uk/handle/1810/295873|access-date=2020-09-02|archive-date=2021-01-22|archive-url=https://web.archive.org/web/20210122012725/https://www.repository.cam.ac.uk/handle/1810/295873|url-status=live}}</ref>

<ref name="natq">{{Cite journal | doi = 10.1038/srep09111| pmid = 25765857| pmc = 4357871| title = Natural quasicrystal with decagonal symmetry| journal = Scientific Reports| volume = 5| pages = 9111| year = 2015| last1 = Bindi | first1 = L. | last2 = Yao | first2 = N. | last3 = Lin | first3 = C. | last4 = Hollister | first4 = L.S. | last5 = Andronicos | first5 = C.L. | last6 = Distler | first6 = V.V. | last7 = Eddy | first7 = M.P. | last8 = Kostin | first8 = A. | last9 = Kryachko | first9 = V. | last10 = MacPherson | first10 = G.J. | last11 = Steinhardt | first11 = W.M. | last12 = Yudovskaya | first12 = M. | last13 = Steinhardt | first13 = P.J. | bibcode = 2015NatSR...5.9111B}}</ref>

<ref name="bloomberg">{{cite news|url=https://www.bloomberg.com/news/2011-10-05/technion-s-shechtman-wins-chemistry-nobel-for-discovery-of-quasicrystals.html|title=Tecnion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery|last=Gerlin|first=Andrea|date=October 5, 2011|website=[[Bloomberg News|Bloomberg]]|archive-url=https://web.archive.org/web/20141205014945/http://www.bloomberg.com/news/2011-10-05/technion-s-shechtman-wins-chemistry-nobel-for-discovery-of-quasicrystals.html|archive-date=December 5, 2014|access-date=January 4, 2019}}</ref>

<ref name="mrs">{{cite journal|doi=10.1557/mrs.2011.190|title=Sputtering technique forms versatile quasicrystalline coatings|journal=MRS Bulletin|volume=36|issue=8|pages=581|year=2011|doi-access=free}}</ref>

<ref name="nature.com">{{cite journal|last2=Kuczera|first2=Pawel|last3=Sologubenko|first3=Alla|last4=Sumigawa|first4=Takashi|last5=Kitamura|first5=Takayuki|last6=Steurer|first6=Walter|last7=Spolenak|first7=Ralph|year=2016|title=Superior room-temperature ductility of typically brittle quasicrystals at small sizes|journal=Nature Communications|volume=7|pages=12261|doi=10.1038/ncomms12261|pmc=4990631|pmid=27515779|last1=Zou|first1=Yu|bibcode=2016NatCo...712261Z}}</ref>

<ref name="mitrev">{{cite web |url=https://www.technologyreview.com/s/425729/the-quasicrystal-laureate/ |title=The Quasicrystal Laureate |newspaper=MIT Technology Review |date=12 October 2011 |author=Kalman, Matthew |access-date=12 February 2016 |archive-date=28 July 2020 |archive-url=https://web.archive.org/web/20200728104035/https://www.technologyreview.com/2011/10/12/190784/the-quasicrystal-laureate/ |url-status=live }}</ref>
}}

==Further reading==
* {{cite arXiv   |last1=Kramer |first1=Peter |date=2010 |title=Gateways towards quasicrystals |eprint=1101.0061 |class=cond-mat.mtrl-sci }}
* {{cite journal |last=Senechal |first=Marjorie |year=2006 |url=https://www.ams.org/notices/200608/whatis-senechal.pdf |title=What is... a Quasicrystal? |journal=[[Notices of the AMS]] |volume=53 |issue=8 |pages=886–887}}
* {{Cite journal |last1=Shaginyan |first1=V. R. |last2=Msezane |first2=A. Z. |last3=Popov |first3=K. G. |last4=Japaridze |first4=G. S. |last5=Khodel |first5=V. A. |year=2013 |title=Common quantum phase transition in quasicrystals and heavy-fermion metals |journal=Physical Review B |volume=87 |issue=24 |article-number=245122 |doi=10.1103/PhysRevB.87.245122 |arxiv=1302.1806 |bibcode=2013PhRvB..87x5122S |s2cid=119239115}}
* {{cite journal |last=Thiel |first=P. A. |year=2008 |title=Quasicrystal Surfaces |journal=[[Annual Review of Physical Chemistry]] |volume=59 |pages=129–152 |doi=10.1146/annurev.physchem.59.032607.093736 |bibcode = 2008ARPC...59..129T |pmid=17988201 }}

==External links==
* [https://web.archive.org/web/20150429115101/http://www.quasi.ameslab.gov/Bib.html A Partial Bibliography of Literature on Quasicrystals] (1996–2008).
* [http://www.tau.ac.il/~ronlif/quasicrystals.html Quasicrystals: an introduction by R. Lifshitz]
* [http://www.jcrystal.com/steffenweber/qc.html Quasicrystals: an introduction by S. Weber]
* [http://www.physics.princeton.edu/~steinh/quasi/ Steinhardt's proposal] {{Webarchive |url=https://web.archive.org/web/20161018121732/http://www.physics.princeton.edu/~steinh/quasi/ |date=2016-10-18 }}
* [http://vimeo.com/29590068 Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart]
* [http://www.ideasroadshow.com/issues/indiana-steinhardt-and-the-quest-for-quasicrystals-part-i "Indiana Steinhardt and the Quest for Quasicrystals – A Conversation with Paul Steinhardt"] {{Webarchive |url=https://web.archive.org/web/20161104003344/http://www.ideasroadshow.com/issues/indiana-steinhardt-and-the-quest-for-quasicrystals-part-i |date=2016-11-04 }}, ''Ideas Roadshow'', 2016
* [http://news.bbc.co.uk/1/shared/spl/hi/pop_ups/08/sci_nat_enl_1260381987/html/1.stm BBC webpage] showing pictures of Quasicrystals
* [https://thinkzone.wlonk.com/Quasicrystals/QCBlocks-CutFoldInstructions.pdf Quasicrystal Blocks: Description and Cut & Fold Instructions] Space-filling models

{{Patterns in nature}}
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[[Category:Crystallography]]
[[Category:Condensed matter physics]]
[[Category:Tessellation]]
[[Category:Quasicrystals | ]]