Question: 5 fold diffraction pattern symmetries were discovered in 1982 by dan shechtman when he discovered this phase which opened new field of study in quasiperiodic crystals

Context paragraphs:
Quasicrystal: A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic.  A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry.  While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

Chaoite: Chaoite, or white carbon, is a mineral described as an allotrope of carbon whose existence is disputed.  It was discovered in shock-fused graphite gneiss from the Ries crater in Bavaria.  It has been described as slightly harder than graphite, with a reflection colour of grey to white.  From its electron diffraction pattern, the mineral has been considered to have a carbyne structure, the linear acetylenic carbon allotrope of carbon.  A later report has called this identification, and the very existence of carbyne phases, into question, arguing that the new reflections in the diffraction pattern are due to clay impurities.

Fraunhofer diffraction: In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.  In contrast, the diffraction pattern created near the object, in the near field region, is given by the Fresnel diffraction equation.

Low-energy electron microscopy: Low-energy electron microscopy, or LEEM, is an analytical surface science technique invented by Ernst Bauer in 1962, however, not fully developed (by Ernst Bauer and Wolfgang Telieps) until 1985.  LEEM is a technique used by surface scientists to image atomically clean surfaces, atom-surface interactions, and thin (crystalline) films.  In LEEM, high-energy electrons (15-20 keV) are emitted from an electron gun, focused using a set of condenser optics, and sent through a magnetic beam deflector (usually 60˚ or 90˚).  The “fast” electrons travel through an objective lens and begin decelerating to low energies (1-100 eV) near the sample surface because the sample is held at a potential near that of the gun.  The low-energy electrons are now termed “surface-sensitive” and the near-surface sampling depth can be varied by tuning the energy of the incident electrons (difference between the sample and gun potentials minus the work functions of the sample and system).  The low-energy elastically backscattered electrons travel back through the objective lens, reaccelerate to the gun voltage (because the objective lens is grounded), and pass through the beam separator again.  However, now the electrons travel away from the condenser optics and into the projector lenses.  Imaging of the back focal plane of the objective lens into the object plane of the projector lens (using an intermediate lens) produces a diffraction pattern (low-energy electron diffraction, LEED) at the imaging plane and recorded in a number of different ways.  The intensity distribution of the diffraction pattern will depend on the periodicity at the sample surface and is a direct result of the wave nature of the electrons.  One can produce individual images of the diffraction pattern spot intensities by turning off the intermediate lens and inserting a contrast aperture in the back focal plane of the objective lens (or, in state-of-the-art instruments, in the center of the separator, as chosen by the excitation of the objective lens), thus allowing for real-time observations of dynamic processes at surfaces.  Such phenomena include (but are not limited to): tomography, phase transitions, adsorption, reaction, segregation, thin film growth, etching, strain relief, sublimation, and magnetic microstructure.  These investigations are only possible because of the accessibility of the sample; allowing for a wide variety of in situ studies over a wide temperature range.

Crystallographic restriction theorem: The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold.  However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

George-ericksenite: George-ericksenite is a mineral with the chemical formula NaCaMg(IO)(CrO)(HO).  It is vitreous, pale yellow to bright lemon yellow, brittle, and features a prismatic to acicular crystal habit along [001] and somewhat flattened crystal habit on {110}.  It was first encountered in 1984 at the Pinch Mineralogical Museum.  One specimen of dietzeite from Oficina Chacabuco, Chile had bright lemon-yellow micronodules on it.  These crystals produced an X-ray powder diffraction pattern that did not match any XRD data listed for inorganic compounds.  The X-ray diffraction pattern and powder mount were set aside until 1994.  By then, the entire mineral collection from the Pinch Mineralogical Museum had been purchased by the Canadian Museum of Nature.  The specimen was then retrieved and studied further.  This study was successful and the new mineral george-ericksenite was discovered.  The mineral was named for George E. Ericksen who was a research economic geologist with the U.S. Geological Survey for fifty years.  The mineral and name have been approved by Commission on New Minerals and Mineral Names (IMA).  The specimen, polished thin section, and the actual crystal used for the structure determination are kept in the Display Series of the National Mineral Collection of Canada at the Canadian Museum of Nature, Ottawa, Ontario.

Alan Lindsay Mackay: Alan Lindsay Mackay FRS (born 6 September 1926) is a British crystallographer, born in Wolverhampton.  He spent his scientific career at Birkbeck College, founded by George Birkbeck, one of the Colleges of the University of London, where he was immersed in a liberal scientific atmosphere under the leadership of John Desmond Bernal.  He became a Fellow of the Royal Society (FRS) on 17 March 1988 and a Fellow of Birkbeck College on 2 March 2002.  He is also a Fellow of the Mexican Academy of Sciences.  Mackay has made important scientific contributions related to the structure of materials: In 1962 he published a manuscript that showed how to pack atoms in an icosahedral fashion; a first step towards five-fold symmetry in materials science.  These arrangements are now called Mackay icosahedra.  He is a pioneer in the introduction of five-fold symmetry in materials and in 1981 predicted quasicrystals in a paper (in Russian) entitled "De Nive Quinquangula" in which he used a Penrose tiling in two and three dimensions to predict a new kind of ordered structures not allowed by traditional crystallography.  In a later manuscript, in 1982, he took the optical Fourier transform of a 2-D Penrose tiling decorated with atoms, obtaining a pattern with sharp spots and five-fold symmetry.  This brought the possibility of identifying quasiperiodic order in a material through diffraction.  Quasicrystals with icosahedral symmetry were found by Dan Shechtman and co-workers in 1984.  For his contributions to quasicrystals in 2010 Mackay was awarded the Buckley Prize,

Fresnel diffraction: In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field.  It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object.  In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.

Dan Shechtman: Dan Shechtman (Hebrew: דן שכטמן; born January 24, 1941) is the Philip Tobias Professor of Materials Science at the Technion – Israel Institute of Technology, an Associate of the US Department of Energy's Ames Laboratory, and Professor of Materials Science at Iowa State University.  On April 8, 1982, while on sabbatical at the U.S. National Bureau of Standards in Washington, D.C., Shechtman discovered the icosahedral phase, which opened the new field of quasiperiodic crystals.  Shechtman was awarded the 2011 Nobel Prize in Chemistry for the discovery of quasicrystals, making him one of six Israelis who have won the Nobel Prize in Chemistry.

Fiber diffraction: Fiber diffraction is a subarea of scattering, an area in which molecular structure is determined from scattering data (usually of X-rays, electrons or neutrons).  In fiber diffraction the scattering pattern does not change, as the sample is rotated about a unique axis (the fiber axis).  Such uniaxial symmetry is frequent with filaments or fibers consisting of biological or man-made macromolecules.  In crystallography fiber symmetry is an aggravation regarding the determination of crystal structure, because reflexions are smeared and may overlap in the fiber diffraction pattern.  Materials science considers fiber symmetry a simplification, because almost the complete obtainable structure information is in a single two-dimensional (2D) diffraction pattern exposed on photographic film or on a 2D detector.  2 instead of 3 co-ordinate directions suffice to describe fiber diffraction.

Supporting paragraphs: Crystallographic restriction theorem Dan Shechtman
Answer: icosahedral phase