1.
Quasi-crystals (supramolecular)
–
Quasi-crystals are supramolecular aggregates exhibiting both crystalline properties as well as amorphous, liquid-like properties. In his 1978 paper Krongauz coined the term “Quasi-Crystals” for the new self-organized colloidal particles, the supramolecular Quasi-crystals are produced in photochemical reaction by exposing solutions of photochromic spiropyran molecules to UV radiation. The ultraviolet light induces the conversion of the spiropyrans to merocyanine molecules that manifest electric dipole moments, the quasi-crystals have external shape of submicron globules and their internal structure consists of crystals enveloped by an amorphous matter. In an applied field, quasi-crystals form macroscopic threads that show linear optical dichroism. Later Krongauz described unusual phase transitions of molecules composedof mesogenic and spiropyran moieties, a micrograph of their mesophase appeared on the cover of Nature in a 1984 paper, “Quasi-Liquid Crystals. ”The investigation of spiropyran-merocyanineself-organized systems, including macromolecules, has continued over the years. These studies have resulted in discoveries of unusual and practically significant phenomena, thus, in the electrostatic field, quasi-crystals and quasi-liquid crystals have exhibited 2nd order non-linear optical properties. Potential applications of these materials have been described and patented. Work on spiropyran-merocyanineself-assemblies currently continues in several laboratories –see, for example, reference and the references cited in that study
2.
Aluminium
–
Aluminium or aluminum is a chemical element in the boron group with symbol Al and atomic number 13. It is a silvery-white, soft, nonmagnetic, ductile metal, Aluminium metal is so chemically reactive that native specimens are rare and limited to extreme reducing environments. Instead, it is combined in over 270 different minerals. The chief ore of aluminium is bauxite, Aluminium is remarkable for the metals low density and its ability to resist corrosion through the phenomenon of passivation. Aluminium and its alloys are vital to the industry and important in transportation and structures, such as building facades. The oxides and sulfates are the most useful compounds of aluminium, despite its prevalence in the environment, no known form of life uses aluminium salts metabolically, but aluminium is well tolerated by plants and animals. Because of these salts abundance, the potential for a role for them is of continuing interest. Aluminium is a soft, durable, lightweight, ductile. It is nonmagnetic and does not easily ignite, a fresh film of aluminium serves as a good reflector of visible light and an excellent reflector of medium and far infrared radiation. The yield strength of aluminium is 7–11 MPa, while aluminium alloys have yield strengths ranging from 200 MPa to 600 MPa. Aluminium has about one-third the density and stiffness of steel and it is easily machined, cast, drawn and extruded. Aluminium atoms are arranged in a cubic structure. Aluminium has an energy of approximately 200 mJ/m2. Aluminium is a thermal and electrical conductor, having 59% the conductivity of copper. Aluminium is capable of superconductivity, with a critical temperature of 1.2 kelvin. Aluminium is the most common material for the fabrication of superconducting qubits, the strongest aluminium alloys are less corrosion resistant due to galvanic reactions with alloyed copper. This corrosion resistance is reduced by aqueous salts, particularly in the presence of dissimilar metals. In highly acidic solutions, aluminium reacts with water to form hydrogen, primarily because it is corroded by dissolved chlorides, such as common sodium chloride, household plumbing is never made from aluminium
3.
Palladium
–
Palladium is a chemical element with symbol Pd and atomic number 46. It is a rare and lustrous silvery-white metal discovered in 1803 by William Hyde Wollaston and he named it after the asteroid Pallas, which was itself named after the epithet of the Greek goddess Athena, acquired by her when she slew Pallas. Palladium, platinum, rhodium, ruthenium, iridium and osmium form a group of elements referred to as the platinum group metals and these have similar chemical properties, but palladium has the lowest melting point and is the least dense of them. More than half the supply of palladium and its congener platinum is used in catalytic converters, Palladium is also used in electronics, dentistry, medicine, hydrogen purification, chemical applications, groundwater treatment, and jewelry. Palladium is a key component of cells, which react hydrogen with oxygen to produce electricity, heat. Ore deposits of palladium and other PGMs are rare, recycling is also a source, mostly from scrapped catalytic converters. The numerous applications and limited supply sources result in considerable investment interest, Palladium belongs to group 10 in the periodic table, but the configuration in the outermost electron shells is atypical for group 10. Fewer electron shells are filled than the directly preceding it. The valence shell has eighteen electrons – ten more than the eight found in the shells of the noble gases from neon onward. Palladium is a soft metal that resembles platinum. It is the least dense and has the lowest melting point of the platinum group metals and it is soft and ductile when annealed and is greatly increased in strength and hardness when cold-worked. Palladium dissolves slowly in concentrated acid, in hot, concentrated sulfuric acid. It dissolves readily at room temperature in aqua regia, common oxidation states of palladium are 0, +1, +2 and +4. Palladium was first observed in 2002, Palladium films with defects produced by alpha particle bombardment at low temperature exhibit superconductivity having Tc=3.2 K. Naturally occurring palladium is composed of seven isotopes, six of which are stable, the most stable radioisotopes are 107Pd with a half-life of 6.5 million years, 103Pd with 17 days, and 100Pd with 3.63 days. Eighteen other radioisotopes have been characterized with atomic weights ranging from 90.94948 u to 122.93426 u and these have half-lives of less than thirty minutes, except 101Pd, 109Pd, and 112Pd. For isotopes with atomic mass unit values less than that of the most abundant stable isotope, 106Pd, the primary mode of decay for those isotopes of Pd with atomic mass greater than 106 is beta decay with the primary product of this decay being silver. Radiogenic 107Ag is a product of 107Pd and was first discovered in 1978 in the Santa Clara meteorite of 1976
4.
Manganese
–
Manganese is a chemical element with symbol Mn and atomic number 25. It is not found as an element in nature, it is often found in minerals in combination with iron. Manganese is a metal with important industrial metal alloy uses, particularly in stainless steels, by the mid-18th century, Swedish chemist Carl Wilhelm Scheele had used pyrolusite to produce chlorine. Scheele and others were aware that pyrolusite contained a new element, johan Gottlieb Gahn was the first to isolate an impure sample of manganese metal in 1774, which he did by reducing the dioxide with carbon. Manganese phosphating is used for rust and corrosion prevention on steel, ionized manganese is used industrially as pigments of various colors, which depend on the oxidation state of the ions. The permanganates of alkali and alkaline earth metals are powerful oxidizers, Manganese dioxide is used as the cathode material in zinc-carbon and alkaline batteries. In biology, manganese ions function as cofactors for a variety of enzymes with many functions. Manganese enzymes are essential in detoxification of superoxide free radicals in organisms that must deal with elemental oxygen. Manganese also functions in the complex of photosynthetic plants. The element is a trace mineral for all known living organisms but is a neurotoxin. In larger amounts, and apparently with far greater effectiveness through inhalation, Manganese is a silvery-gray metal that resembles iron. It is hard and very brittle, difficult to fuse, Manganese metal and its common ions are paramagnetic. Manganese tarnishes slowly in air and oxidizes like iron in water containing dissolved oxygen, naturally occurring manganese is composed of one stable isotope, 55Mn. Eighteen radioisotopes have been isolated and described, the most stable being 53Mn with a half-life of 3.7 million years, 54Mn with a half-life of 312.3 days, and 52Mn with a half-life of 5.591 days. All of the radioactive isotopes have half-lives of less than three hours, and the majority of less than one minute. Manganese also has three meta states, Manganese is part of the iron group of elements, which are thought to be synthesized in large stars shortly before the supernova explosion. 53Mn decays to 53Cr with a half-life of 3.7 million years, because of its relatively short half-life, 53Mn is relatively rare, produced by cosmic rays impact on iron. Manganese isotopic contents are combined with chromium isotopic contents and have found application in isotope geology
5.
Crystal
–
A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification, the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both ice and rock crystal, from κρύος, icy cold, frost. Examples of large crystals include snowflakes, diamonds, and table salt, most inorganic solids are not crystals but polycrystals, i. e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
6.
Structure
–
Structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as biological organisms, abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy, a network featuring many-to-many links, buildings, aircraft, skeletons, anthills, beaver dams and salt domes are all examples of load-bearing structures. The results of construction are divided into buildings and non-building structures, the effects of loads on physical structures are determined through structural analysis, which is one of the tasks of structural engineering. The structural elements can be classified as one-dimensional, two-dimensional, or three-dimensional, the latter was the main option available to early structures such as Chichen Itza. Two-dimensional elements with a third dimension have little of either. The structure elements are combined in structural systems, the majority of everyday load-bearing structures are section-active structures like frames, which are primarily composed of one-dimensional structures. In biology, structures exist at all levels of organization, ranging hierarchically from the atomic and molecular to the cellular, tissue, organ, organismic, population, usually, a higher-level structure is composed of multiple copies of a lower-level structure. Structural biology is concerned with the structure of macromolecules, particularly proteins. The function of molecules is determined by their shape as well as their composition. Protein structure has a four-level hierarchy, the primary structure is the sequence of amino acids that make it up. It has a backbone made up of a repeated sequence of a nitrogen. The secondary structure consists of repeated patterns determined by hydrogen bonding, the two basic types are the α-helix and the β-pleated sheet. The tertiary structure is a back and forth bending of the chain. Chemical structure refers to both molecular geometry and electronic structure, the structure can be represented by a variety of diagrams called structural formulas. Lewis structures use a dot notation to represent the valence electrons for an atom, bonds between atoms can be represented by lines with one line for each pair of electrons that is shared. In a simplified version of such a diagram, called a skeletal formula, only carbon-carbon bonds, atoms in a crystal have a structure that involves repetition of a basic unit called a unit cell. The atoms can be modeled as points on a lattice, and one can explore the effect of symmetry operations that include rotations about a point, reflections about a symmetry planes, and translations
7.
Order and disorder
–
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures, upon heating, the degree of freedom that is ordered or disordered can be translational, rotational, or a spin state. The order can consist either in a crystalline space group symmetry. Depending on how the correlations decay with distance, one speaks of long range order or Short range order, if a disordered state is not in thermodynamic equilibrium, one speaks of quenched disorder. For instance, a glass is obtained by quenching a liquid, by extension, other quenched states are called spin glass, orientational glass. In some contexts, the opposite of quenched disorder is annealed disorder, the strictest form of order in a solid is lattice periodicity, a certain pattern is repeated again and again to form a translationally invariant tiling of space. This is the property of a crystal. Possible symmetries have been classified in 14 Bravais lattices and 230 space groups, besides structural order, one may consider charge ordering, spin ordering, magnetic ordering, and compositional ordering. Magnetic ordering is observable in neutron diffraction and it is a thermodynamic entropy concept often displayed by a second-order phase transition. Generally speaking, high energy is associated with disorder and low thermal energy with ordering. Ordering peaks become apparent in experiments at low energy. Long-range order characterizes physical systems in remote portions of the same sample exhibit correlated behavior. This can be expressed as a function, namely the spin-spin correlation function. Where s is the quantum number and x is the distance function within the particular system. This function is equal to unity when x = x ′, typically, it decays exponentially to zero at large distances, and the system is considered to be disordered. If, however, the correlation decays to a constant value at large | x − x ′ | then the system is said to possess long-range order. If it decays to zero as a power of the distance then it is called quasi-long-range order, note that what constitutes a large value of | x − x ′ | is understood in the sense of asymptotics. In statistical physics, a system is said to present quenched disorder when some parameters defining its behaviour are random variables which do not evolve with time, spin glasses are a typical example
8.
Bravais lattice
–
This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox
9.
Translational symmetry
–
In geometry, a translation slides a thing by a, Ta = p + a. In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation, discrete translational symmetry is invariant under discrete translation. More precisely it must hold that ∀ δ A f = A, laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noethers theorem, space translational symmetry of a system is equivalent to the momentum conservation law. Translational symmetry of a means that a particular translation does not change the object. Fundamental domains are e. g. H + a for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. In spaces with higher than 1, there may be multiple translational symmetry. For each set of k independent translation vectors the symmetry group is isomorphic with Zk, in particular the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions, in this case the set of all translations forms a lattice. The absolute value of the determinant of the matrix formed by a set of vectors is the hypervolume of the n-dimensional parallelepiped the set subtends. This parallelepiped is a region of the symmetry, any pattern on or in it is possible. E. g. in 2D, instead of a and b we can take a. In general in 2D, we can take pa + qb and ra + sb for integers p, q, r and this ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair, each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object, without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers, for two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice. With rotational symmetry of order two of the pattern on the tile we have p2, the rectangle is a more convenient unit to consider as fundamental domain than a parallelogram consisting of part of a tile and part of another one
10.
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, crystals are modeled as discrete lattices, generated by a list of independent finite translations. Because discreteness requires that the spacings between lattice points have a bound, the group of rotational symmetries of the lattice at any point must be a finite group. The strength of the theorem is not all finite groups are compatible with a discrete lattice, in any dimension. The special cases of 2D and 3D are most heavily used in applications, a rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts, now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon. If a displacement exists between any two points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point, but this is impossible, because the new octagon is about 80% as large as the original. The significance of the shrinking is that it is unlimited, the same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like, thus no discrete lattice can have 8-fold symmetry. The same argument applies to any rotation, for k greater than 6. A shrinking argument also eliminates 5-fold symmetry, Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and assemble a 5-point star, the vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original. The existence of quasicrystals and Penrose tilings shows that the assumption of a translation is necessary. And without the discrete lattice assumption, the construction not only fails to reach a contradiction. Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of those assumptions, Consider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation of angle α about any point is a symmetry of the lattice. Rotating about point B by α maps point A to a new point A. Similarly, since both rotations mentioned are symmetry operations, A and B must both be lattice points
11.
Rotational symmetry
–
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
12.
Bragg's law
–
In physics, Braggs law, or Wulff–Braggs condition in post-Soviet countries, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave, the movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects, this phenomenon is known as Rayleigh scattering. The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible, a similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively, the resulting wave interference pattern is the basis of diffraction analysis. This analysis is called Bragg diffraction and they found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes, both neutron and X-ray wavelengths are comparable with inter-atomic distances and thus are an excellent probe for this length scale. Sir William Lawrence Bragg explained this result by modeling the crystal as a set of parallel planes separated by a constant parameter d. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively. Although simple, Braggs law confirmed the existence of particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS and they are the only father-son team to jointly win. William Lawrence Bragg was 25 years old, making him then, for a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance d. When the scattered waves interfere constructively, they remain in phase since the difference between the lengths of the two waves is equal to an integer multiple of the wavelength. The path difference between two waves undergoing interference is given by 2dsinθ, where θ is the scattering angle, the effect of the constructive or destructive interference intensifies because of the cumulative effect of reflection in successive crystallographic planes of the crystalline lattice. Note that moving particles, including electrons, protons and neutrons, have an associated wavelength called de Broglie wavelength, a diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the pattern at the points where the scattering angles satisfy Bragg condition. Suppose that a single wave is incident on aligned planes of lattice points, with separation d. Points A and C are on one plane, and B is on the plane below, there will be a path difference between the ray that gets reflected along AC and the ray that gets transmitted, then reflected, along AB and BC respectively. Putting everything together, n λ =2 d sin θ =2 d sin θ sin 2 θ, which simplifies to n λ =2 d sin θ, which is Braggs law
13.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
14.
Aperiodic tiling
–
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types is aperiodic if copies of these tiles can form only non-periodic tilings, the Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known, consider a periodic tiling by unit squares. Now cut one square into two rectangles, the tiling obtained in this way is non-periodic, there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling, in order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrary large periodic parts. A tiling is called if its hull contains only non-periodic tilings. The hull of a tiling T ∈ R d contains all translates T+x of T, formally this is the closure of the set in the local topology. In the local topology two tiles are ε -close if they agree in a ball of radius 1 / ε around the origin, to give an even simpler example than above, consider a one-dimensional tiling T of the line that looks like. aaaaaabaaaaa. Where a represents an interval of one, b represents an interval of length two. Thus the tiling T consists of many copies of a. Now all translates of T are the tilings with one b somewhere, the sequence of tilings where b is centred at 1,2,4, …,2 n, … converges - in the local topology - to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling. aaaaaa, for many well-behaved tilings holds, if a tiling is non-periodic and repetitive then it is aperiodic. In 1964 Robert Berger found a set of prototiles from which he demonstrated that the tiling problem is in fact not decidable. This first such set, used by Berger in his proof of undecidability, Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles. An even smaller set of six aperiodic tiles was discovered by Raphael M. Robinson in 1971, Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977. The aperiodic Penrose tilings can be generated not only by a set of prototiles
15.
Paradigm shift
–
A paradigm shift, a concept identified by the American physicist and philosopher Thomas Kuhn, is a fundamental change in the basic concepts and experimental practices of a scientific discipline. Kuhn contrasted these shifts, which characterize a scientific revolution, to the activity of normal science, in this context, the word paradigm is used in its original meaning, as example. The nature of scientific revolutions has been studied by modern philosophy since Immanuel Kant used the phrase in the preface to his Critique of Pure Reason and he referred to Greek mathematics and Newtonian physics. In the 20th century, new crises in the concepts of mathematics, physics. It was against this background that Kuhn published his work. Kuhn presented his notion of a shift in his influential book The Structure of Scientific Revolutions. As one commentator summarizes, Kuhn acknowledges having used the term paradigm in two different meanings, in the second sense, the paradigm is a single element of a whole, say for instance Newton’s Principia, which, acting as a common model or an example. Stands for the rules and thus defines a coherent tradition of investigation. Thus the question is for Kuhn to investigate by means of the paradigm what makes possible the constitution of what he calls normal science and that is to say, the science which can decide if a certain problem will be considered scientific or not. This is precisely the meaning of the term paradigm, which Kuhn considered the most new and profound. An epistemological paradigm shift was called a revolution by epistemologist. The paradigm, in Kuhns view, is not simply the current theory, but the entire worldview in which it exists and this is based on features of landscape of knowledge that scientists can identify around them. There are anomalies for all paradigms, Kuhn maintained, that are brushed away as acceptable levels of error, or simply ignored, rather, according to Kuhn, anomalies have various levels of significance to the practitioners of science at the time. When enough significant anomalies have accrued against a current paradigm, the discipline is thrown into a state of crisis. During this crisis, new ideas, perhaps ones previously discarded, are tried, eventually a new paradigm is formed, which gains its own new followers, and an intellectual battle takes place between the followers of the new paradigm and the hold-outs of the old paradigm. Some found Arthur Eddingtons photographs of light bending around the sun to be compelling, while some questioned their accuracy, after a given discipline has changed from one paradigm to another, this is called, in Kuhns terminology, a scientific revolution or a paradigm shift. If this is correct, Kuhns claims must be taken in a weaker sense than they often are, at that time, physics seemed to be a discipline filling in the last few details of a largely worked-out system. In 1900, Lord Kelvin famously told an assemblage of physicists at the British Association for the Advancement of Science, all that remains is more and more precise measurement
16.
Crystallography
–
Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein to write. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography, X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to face is plotted on the net. Each point is labelled with its Miller index, the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
17.
Icosahedrite
–
Icosahedrite is the first known naturally occurring quasicrystal phase. It has the composition Al63Cu24Fe13 and is an approved by the International Mineralogical Association in 2010. Its discovery followed a 10-year-long systematic search by a team of scientists led by Luca Bindi. The rock sample also contains spinel, diopside, forsterite, nepheline, sodalite, corundum, stishovite, khatyrkite, cupalite, evidence shows that the sample is actually extraterrestrial in origin, delivered to the Earth by a CV3 carbonaceous chondrite asteroid that dates back 4.5 Gya. A geological expedition has identified the place of the original discovery. The same Al-Cu-Fe quasicrystal phase had previously created in the laboratory by Japanese experimental metallurgists in the late 1980s
18.
Linear independence
–
These concepts are central to the definition of dimension. A vector space can be of finite-dimension or infinite-dimension depending on the number of independent basis vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a space is linearly dependent are central to determining a basis for a vector space. Thus, v1 is shown to be a combination of the remaining vectors. This implies that no vector in the set can be represented as a combination of the remaining vectors in the set. In other words, a set of vectors is independent if the only representations of 0 → as a linear combination of its vectors is the trivial representation in which all the scalars ai are zero. In order to allow the number of linearly independent vectors in a space to be countably infinite. More generally, let V be a space over a field K. A set X of elements of V is linearly independent if the corresponding family x∈X is linearly independent. Equivalently, a family is dependent if a member is in the span of the rest of the family. The trivial case of the empty family must be regarded as independent for theorems to apply. A set of vectors which is independent and spans some vector space. For example, the space of all polynomials in x over the reals has the subset as a basis. A geographic example may help to clarify the concept of linear independence, a person describing the location of a certain place might say, It is 3 miles north and 4 miles east of here. This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space, the person might add, The place is 5 miles northeast of here. Although this last statement is true, it is not necessary, in this example the 3 miles north vector and the 4 miles east vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third 5 miles northeast vector is a combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary
19.
Materials science
–
The interdisciplinary field of materials science, also commonly termed materials science and engineering, involves the discovery and design of new materials, with an emphasis on solids. Materials science still incorporates elements of physics, chemistry, and engineering, as such, the field was long considered by academic institutions as a sub-field of these related fields. Materials science is a syncretic discipline hybridizing metallurgy, ceramics, solid-state physics and it is the first example of a new academic discipline emerging by fusion rather than fission. Many of the most pressing scientific problems humans currently face are due to the limits of the materials that are available, thus, breakthroughs in materials science are likely to affect the future of technology significantly. Materials scientists emphasize understanding how the history of a material influences its structure, the understanding of processing-structure-properties relationships is called the § materials paradigm. This paradigm is used to advance understanding in a variety of areas, including nanotechnology, biomaterials. Such investigations are key to understanding, for example, the causes of various accidents and incidents. The material of choice of a given era is often a defining point, phrases such as Stone Age, Bronze Age, Iron Age, and Steel Age are great examples. Originally deriving from the manufacture of ceramics and its putative derivative metallurgy, materials science is one of the oldest forms of engineering, modern materials science evolved directly from metallurgy, which itself evolved from mining and ceramics and the use of fire. Materials science has driven, and been driven by, the development of technologies such as rubbers, plastics, semiconductors. Before the 1960s, many materials science departments were named metallurgy departments, reflecting the 19th, a material is defined as a substance that is intended to be used for certain applications. There are a myriad of materials around us—they can be found in anything from buildings to spacecraft, Materials can generally be divided into two classes, crystalline and non-crystalline. The traditional examples of materials are metals, semiconductors, ceramics, New and advanced materials that are being developed include nanomaterials and biomaterials, etc. The basis of science involves studying the structure of materials. Once a materials scientist knows about this structure-property correlation, they can go on to study the relative performance of a material in a given application. The major determinants of the structure of a material and thus of its properties are its constituent chemical elements and these characteristics, taken together and related through the laws of thermodynamics and kinetics, govern a materials microstructure, and thus its properties. As mentioned above, structure is one of the most important components of the field of materials science, Materials science examines the structure of materials from the atomic scale, all the way up to the macro scale. Characterization is the way materials scientists examine the structure of a material, structure is studied at various levels, as detailed below
20.
Dan Shechtman
–
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. Dan Shechtman was born in 1941 in Tel Aviv, in what was then Mandatory Palestine and he is married to Prof. Tzipora Shechtman, Head of the Department of Counseling and Human Development at Haifa University, and author of two books on psychotherapy. They have a son Yoav Shechtman and three daughters, Tamar Finkelstein, Ella Shechtman-Cory, and Ruth Dougoud-Nevo, in 1975, he joined the department of materials engineering at Technion. In 1981–1983 he was on sabbatical at Johns Hopkins University, where he studied rapidly solidified aluminum transition metal alloys, during this study he discovered the Icosahedral Phase which opened the new field of quasiperiodic crystals. In 1992–1994 he was on sabbatical at National Institute of Standards and Technology, Shechtmans Technion research is conducted in the Louis Edelstein Center, and in the Wolfson Centre which is headed by him. He served on several Technion Senate Committees and headed one of them, Shechtman joined the Iowa State faculty in 2004. He currently spends about five months a year in Ames on a part-time appointment, since 2014 he has been the head of the International Scientific Council of Tomsk Polytechnic University. From the day Shechtman published his findings on quasicrystals in 1984 to the day Linus Pauling died, for a long time it was me against the world, he said. I was a subject of ridicule and lectures about the basics of crystallography, the leader of the opposition to my findings was the two-time Nobel Laureate Linus Pauling, the idol of the American Chemical Society and one of the most famous scientists in the world. For years, til his last day, he fought against quasi-periodicity in crystals and he was wrong, and after a while, I enjoyed every moment of this scientific battle, knowing that he was wrong. Linus Pauling is noted saying There is no such thing as quasicrystals, Pauling was apparently unaware of a paper in 1981 by H. Kleinert and K. Maki which had pointed out the possibility of a non-periodic Icosahedral Phase in quasicrystals. The head of Shechtmans research group told him to go back and read the textbook, on publication of his paper, other scientists began to confirm and accept empirical findings of the existence of quasicrystals. Quasicrystals have also found naturally. A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered, a quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. Underlying it is a worked out by Fibonacci in the 13th century. But presently have no technological applications, the Nobel prize was 10 million Swedish krona. On January 17,2014, in an interview with Israels Channel One, Shechtman received the endorsement of the ten Members of Knesset required to run
21.
Penrose tiling
–
A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s, the aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right. A Penrose tiling has many properties, most notably, It is non-periodic. It is self-similar, so the same patterns occur at larger and larger scales, thus, the tiling can be obtained through inflation and every finite patch from the tiling occurs infinitely many times. It is a quasicrystal, implemented as a structure a Penrose tiling will produce Bragg diffraction. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes, Penrose tilings are simple examples of aperiodic tilings of the plane. The most familiar tilings are periodic, a copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling, more informally, if a tiling has no periods it is said to be non-periodic. A set of prototiles is said to be if it tiles the plane but every such tiling is non-periodic. The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings and he observed that if this problem were undecidable, then there would have to exist an aperiodic set of Wang dominoes. At the time, this implausible, so Wang conjectured no such set could exist. Wangs student Robert Berger proved that the Domino Problem was undecidable in his 1964 thesis, raphael Robinson, in a 1971 paper which simplified Bergers techniques and undecidability proof, used this technique to obtain an aperiodic set of just six prototiles. The first Penrose tiling is a set of six prototiles, introduced by Roger Penrose in a 1974 paper. Traces of these ideas can also be found in the work of Albrecht Dürer, acknowledging inspiration from Kepler, Penrose found matching rules for these shapes, obtaining an aperiodic set. His tiling can be viewed as a completion of Keplers finite Aa pattern, Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling and the rhombus tiling. The rhombus tiling was discovered by Robert Ammann in 1976. In this approach, the Penrose tiling is viewed as a set of points, its vertices, the three types of Penrose tiling, P1–P3, are described individually below
22.
Wang tile
–
Wang tiles, first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems. They are modelled visually by square tiles with a color on each side, a set of such tiles is selected. Then copies of the tiles are arranged side by side with matching colors, the basic question about a set of Wang tiles is whether it can tile the plane or not, i. e. whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern and he also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane. The idea of constraining adjacent tiles to each other occurs in the game of dominoes. The algorithmic problem of determining whether a set can tile the plane became known as the domino problem. According to Wangs student, Robert Berger, The Domino Problem deals with the class of all domino sets and it consists of deciding, for each domino set, whether or not it is solvable. In other words, the problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. In 1966, Wangs student Robert Berger solved the problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if, the undecidability of the halting problem then implies the undecidability of Wangs tiling problem. Combining Bergers undecidability result with Wangs observation shows that there must exist a set of Wang tiles that tiles the plane. This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal, in later years, increasingly smaller sets were found. For example, the set of 13 tiles given in the image above is an aperiodic set published by Karel Culik II in 1996 and it can tile the plane, but not periodically. Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense, for example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes, winfree et al. have demonstrated the feasibility of creating molecular tiles made from DNA that can act as Wang tiles. Mittal et al. have shown that these tiles can also be composed of nucleic acid. Wang tiles have also used in cellular automata theory decidability proofs. Edge-matching puzzle Eternity II puzzle Percy Alexander MacMahon TetraVex Grünbaum, Branko, Shephard, tilings and Patterns, New York, W. H. Freeman, ISBN 0-7167-1193-1
23.
Roger Penrose
–
Sir Roger Penrose OM FRS is an English mathematical physicist, mathematician and philosopher of science. He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, Penrose is known for his work in mathematical physics, in particular for his contributions to general relativity and cosmology. He has received prizes and awards, including the 1988 Wolf Prize for physics. Penrose told a Russian audience that his grandmother had left St. Petersburg in the late 1880s and his uncle was artist Roland Penrose, whose son with photographer Lee Miller is Antony Penrose. Penrose is the brother of physicist Oliver Penrose and of chess Grandmaster Jonathan Penrose, Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951. He devised and popularised the Penrose triangle in the 1950s, describing it as impossibility in its purest form, Escher, whose earlier depictions of impossible objects partly inspired it. Eschers Waterfall, and Ascending and Descending were in inspired by Penrose. As reviewer Manjit Kumar puts it, As a student in 1954, soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isnt. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up, an article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of illusions was inspired to produce his two masterpieces. One approach to issue was by the use of perturbation theory. The importance of Penroses epoch-making paper Gravitational collapse and space-time singularities was not only its result, following up his weak cosmic censorship hypothesis, Penrose went on, in 1979, to formulate a stronger version called the strong censorship hypothesis. Together with the BKL conjecture and issues of stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979 dates Penroses influential Weyl curvature hypothesis on the conditions of the observable part of the universe. Penrose and James Terrell independently realised that objects travelling near the speed of light appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation, in 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature. Penrose developed these ideas based on the article Deux types fondamentaux de distribution statistique by Czech geographer, demographer, in 1984, such patterns were observed in the arrangement of atoms in quasicrystals
24.
Alan Lindsay Mackay
–
Alan Lindsay Mackay FRS is a British crystallographer, born in Wolverhampton. He became a Fellow of the Royal Society on 17 March 1988 and he is also a Fellow of the Mexican Academy of Sciences. These arrangements are now called Mackay icosahedra, 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, of the American Physical Society, with Dov Levine, the Nobel Prize in Chemistry was awarded in 2011 to Dan Shechtman for the discovery of quasicrystals. Mackay has been interested in a generalized crystallography, which can not only crystals. He has also written a book of poetry and has translated from the German, with commentaries and he produces scientifically inspired visual art under his artistic pseudonym Sho Takahashi. Some of his 3D printed minimal surface designs can be found at his shapeways store, published in De Nive Quinquangula,1981. Original signed by Roger Penrose,2005
25.
Fourier transform
–
The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics
26.
Robert Ammann
–
Robert Ammann was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann attended Brandeis University, but generally did not go to classes and he worked as a programmer for Honeywell. After ten years, his position was eliminated as part of a routine cutback, in 1975, Ammann read an announcement by Martin Gardner of new work by Roger Penrose. Penrose had discovered two sets of aperiodic tiles, each consisting of just two quadrilaterals. Since Penrose was taking out a patent, he wasnt ready to publish them, Ammann wrote a letter to Gardner, describing his own work, which duplicated one of Penroses sets, plus a foursome of golden rhombohedra that formed aperiodic tilings in space. More letters followed, and Ammann became a correspondent with many of the professional researchers and he discovered several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles. He also showed how to generate tilings using lines in the plane as guides for lines marked on the tiles, the discovery of quasicrystals in 1982 changed the status of aperiodic tilings and Ammanns work from mere recreational mathematics to respectable academic research. After more than ten years of coaxing, he agreed to meet various professionals in person, afterwards, Ammann dropped out of sight, and died of a heart attack a few years later. News of his death did not reach the community for a few more years. Ammanns discoveries came to only after Penrose had published his own discovery. In 1981 de Bruijn exposed the cut and project method and in 1984 came the news about Shechtman quasicrystals which promoted the Penrose tiling to fame. But in 1982 Beenker published a mathematical explanation for the octagonal case which became known as the Ammann–Beenker tiling. In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry and it is acknowledged however that Robert Ammann first proposed the construction of rhombic prisms which is the three-dimensional model of Shechtmans quasicrystals. Senechal, Marjorie, The Mysterious Mr. Ammann, The Mathematical Intelligencer,26, 10–21, doi,10. 1007/BF02985414, amman tilings and references at the Tilings encyclopedia
27.
Conic section
–
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
28.
Hexagonal crystal family
–
In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral, the hexagonal crystal family consists of two lattice systems, hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice, hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes, the unit cell is a rhombohedron. This is a cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used, the hexagonal crystal family consists of two crystal systems, trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves, the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation
29.
Electron diffraction
–
Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a used to study matter by firing electrons at a sample. This phenomenon is known as wave–particle duality, which states that a particle of matter can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves and this technique is similar to X-ray and neutron diffraction. Electron diffraction is most frequently used in solid state physics and chemistry to study the structure of solids. Experiments are usually performed in an electron microscope, or a scanning electron microscope as electron backscatter diffraction. In these instruments, electrons are accelerated by a potential in order to gain the desired energy. The periodic structure of a crystalline solid acts as a grating, scattering the electrons in a predictable manner. Working back from the diffraction pattern, it may be possible to deduce the structure of the crystal producing the diffraction pattern. However, the technique is limited by the phase problem, the de Broglie hypothesis, formulated in 1924, predicts that particles should also behave as waves. De Broglies formula was confirmed three years later for electrons with the observation of diffraction in two independent experiments. At the University of Aberdeen, George Paget Thomson passed a beam of electrons through a metal film. Around the same time at Bell Labs, Clinton Joseph Davisson, in 1937, Thomson and Davisson shared the Nobel Prize for Physics for their discovery. Unlike other types of radiation used in studies of materials, such as X-rays and neutrons, electrons are charged particles. This means that the incident electrons feel the influence of both the positively charged nuclei and the surrounding electrons. In comparison, X-rays interact with the distribution of the valence electrons. In addition, the moment of neutrons is non-zero. Because of these different forms of interaction, the three types of radiation are suitable for different studies, in the kinematical approximation for electron diffraction, the intensity of a diffracted beam is given by, I g = | ψ g |2 ∝ | F g |2
30.
National Institute of Standards and Technology
–
The National Institute of Standards and Technology is a measurement standards laboratory, and a non-regulatory agency of the United States Department of Commerce. Its mission is to promote innovation and industrial competitiveness, in 1821, John Quincy Adams had declared Weights and measures may be ranked among the necessities of life to every individual of human society. From 1830 until 1901, the role of overseeing weights and measures was carried out by the Office of Standard Weights and Measures, president Theodore Roosevelt appointed Samuel W. Stratton as the first director. The budget for the first year of operation was $40,000, a laboratory site was constructed in Washington, DC, and instruments were acquired from the national physical laboratories of Europe. In addition to weights and measures, the Bureau developed instruments for electrical units, in 1905 a meeting was called that would be the first National Conference on Weights and Measures. Quality standards were developed for products including some types of clothing, automobile brake systems and headlamps, antifreeze, during World War I, the Bureau worked on multiple problems related to war production, even operating its own facility to produce optical glass when European supplies were cut off. Between the wars, Harry Diamond of the Bureau developed a blind approach radio aircraft landing system, in 1948, financed by the Air Force, the Bureau began design and construction of SEAC, the Standards Eastern Automatic Computer. The computer went into operation in May 1950 using a combination of vacuum tubes, about the same time the Standards Western Automatic Computer, was built at the Los Angeles office of the NBS and used for research there. A mobile version, DYSEAC, was built for the Signal Corps in 1954, due to a changing mission, the National Bureau of Standards became the National Institute of Standards and Technology in 1988. Following 9/11, NIST conducted the investigation into the collapse of the World Trade Center buildings. NIST had a budget for fiscal year 2007 of about $843.3 million. NISTs 2009 budget was $992 million, and it also received $610 million as part of the American Recovery, NIST employs about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 1,800 NIST associates complement the staff, in addition, NIST partners with 1,400 manufacturing specialists and staff at nearly 350 affiliated centers around the country. NIST publishes the Handbook 44 that provides the Specifications, tolerances, the Congress of 1866 made use of the metric system in commerce a legally protected activity through the passage of Metric Act of 1866. NIST is headquartered in Gaithersburg, Maryland, and operates a facility in Boulder, nISTs activities are organized into laboratory programs and extramural programs. Effective October 1,2010, NIST was realigned by reducing the number of NIST laboratory units from ten to six, nISTs Boulder laboratories are best known for NIST‑F1, which houses an atomic clock. NIST‑F1 serves as the source of the official time. NIST also operates a neutron science user facility, the NIST Center for Neutron Research, the NCNR provides scientists access to a variety of neutron scattering instruments, which they use in many research fields
31.
Polyhedral symmetry
–
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three groups, The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. The reflection symmetries have 6,9, and 15 mirrors respectively, the octahedral symmetry, can be seen as the union of 6 tetrahedral symmetry mirrors, and 3 mirrors of dihedral symmetry Dih2. Pyritohedral symmetry is another doubling of tetrahedral symmetry, S. M. Regular Polytopes, 3rd ed
32.
Journal of Applied Physics
–
The Journal of Applied Physics is a peer-reviewed scientific journal published since 1931 by the American Institute of Physics. Its emphasis is on the understanding of the physics underpinning modern technology, according to the 2015 Journal Citation Reports, the journal has a 2015 impact factor of 2.101. The journal was owned by the American Physical Society and was, for its first seven volumes. In January 1937 ownership was transferred to the American Institute of Physics in line with the efforts of the American Physical Society to enhance the standing of physics as a profession
33.
Khatyrka
–
Khatyrka is a rural locality in Anadyrsky District of Chukotka Autonomous Okrug, Russia, located on the shore of the Bering Sea southwest of Beringovsky. Population, 377 , with a population as of 1 January 2015 of 349. Municipally, it is incorporated as Khatyrka Rural Settlement, Khatyrka is one of the oldest settlements in Anadyrsky district, having been founded around 1756. The name of the village and the river was taken from the cape first mapped by the naval officer Sarychyevym. The Chuckchi call the place Vatyrkan, meaning dry or exhausted as the surrounding the village is used for reindeer grazing. There are currently two operating as the Khatyrskoye municipal agricultural unitary enterprise, tending for around 2000 reindeer on the tundra. Within the village, the cultural centre supports the national ensemble Chukotka Dawns, the main occupations revolve around traditional indigenous economic activities of fishing and hunting, both on land and sea. There is also a reindeer herding enterprise on the pastures surrounding Khatyrka with around four hundred heads. Although this is a traditional enterprise among many Arctic peoples, these formerly nomadic enterprises were grouped together in the Soviet Union to form a collective farm. Following the dissolution of the Soviet Union, almost all government support was removed from the farms as the market economy took over. As with many Chukotkan villages, Khatyrka has a traditional Chukchi ensemble, additionally, the village has a new secondary boarding school, kindergarten, local hospital, recreation center, library, post office, communications center, shop and a bakery. The main occupation of the inhabitants is livestock, brigades herd reindeer on pastures in the vicinity of the village above the river. In the village the municipal agricultural enterprise Khatyrskoye is based to run the reindeer herding operations, the village is also involved in traditional fishing and hunting of sea mammals, and a fish processing plant operates to service this. In 2011 the company launched a new slaughterhouse complex, electricity is provided to the village by a local generator. Khatyrka is not connected to any inhabited place by any permanent road being situated on the coast at the mouth of the River Khatyrka. Until July 2008, Khatyrka was a part of both Beringovsky Municipal and Beringnovsky Administrative District, in May 2008, Beringovsky Municipal District was merged into Anadyrsky Municipal District, however, this change did not affect the borders of Beringovsky Administrative District. Khatyrka continued to administratively be a part of the latter until June 2011, Khatyrka has a tundra climate because the warmest month has an average temperature between 0 °C and 10 °C. List of inhabited localities in Anadyrsky District Дума Чукотского автономного округа, Закон №148-ОЗ от24 ноября2008 г
34.
Russia
–
Russia, also officially the Russian Federation, is a country in Eurasia. The European western part of the country is more populated and urbanised than the eastern. Russias capital Moscow is one of the largest cities in the world, other urban centers include Saint Petersburg, Novosibirsk, Yekaterinburg, Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a range of environments. It shares maritime borders with Japan by the Sea of Okhotsk, the East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, in 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus ultimately disintegrated into a number of states, most of the Rus lands were overrun by the Mongol invasion. The Soviet Union played a role in the Allied victory in World War II. The Soviet era saw some of the most significant technological achievements of the 20th century, including the worlds first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the second largest economy, largest standing military in the world. It is governed as a federal semi-presidential republic, the Russian economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2015. Russias extensive mineral and energy resources are the largest such reserves in the world, making it one of the producers of oil. The country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction, Russia is a great power as well as a regional power and has been characterised as a potential superpower. The name Russia is derived from Rus, a state populated mostly by the East Slavs. However, this name became more prominent in the later history, and the country typically was called by its inhabitants Русская Земля. In order to distinguish this state from other states derived from it, it is denoted as Kievan Rus by modern historiography, an old Latin version of the name Rus was Ruthenia, mostly applied to the western and southern regions of Rus that were adjacent to Catholic Europe. The current name of the country, Россия, comes from the Byzantine Greek designation of the Kievan Rus, the standard way to refer to citizens of Russia is Russians in English and rossiyane in Russian. There are two Russian words which are translated into English as Russians
35.
Electron crystallography
–
Electron crystallography is a method to determine the arrangement of atoms in solids using a transmission electron microscope. Protein structures are determined from either 2-dimensional crystals, polyhedrons such as viral capsids. Electrons can be used in situations, whereas X-rays cannot. Thus, X-rays will travel through a thin 2-dimensional crystal without diffracting significantly, conversely, the strong interaction between electrons and protons makes thick crystals impervious to electrons, which only penetrate short distances. One of the difficulties in X-ray crystallography is determining phases in the diffraction pattern. Because of the complexity of X-ray lenses, it is difficult to form an image of the crystal being diffracted, the Fourier transform of an atomic resolution image is similar, but different, to a diffraction pattern—with reciprocal lattice spots reflecting the symmetry and spacing of a crystal. For this, and his studies on virus structures and transfer-RNA and this is especially troublesome in the setting of electron crystallography, where that radiation damage is focused on far fewer atoms. One technique used to limit radiation damage is electron cryomicroscopy, in which the samples undergo cryofixation, because of this problem, X-ray crystallography has been much more successful in determining the structure of proteins that are especially vulnerable to radiation damage. Electron crystallographic studies on inorganic crystals using high-resolution electron microscopy images were first performed by Aaron Klug in 1978 and by Sven Hovmöller, HREM images were used because they allow to select only the very thin regions close to the edge of the crystal for structure analysis. This is of importance since in the thicker parts of the crystal the exit-wave function is no longer linearly related to the projected crystal structure. Moreover, not only do the HREM images change their appearance with increasing crystal thickness, to cope with this complexity Michael OKeefe started in the early 1970s to develop image simulation software which allowed to understand an interpret the observed contrast changes in HREM images. The reason for these views is that the word phase has been used with different meanings in the two communities of physicists and crystallographers. The physicists are more concerned about the wave phase - the phase of a wave moving through the sample during exposure by the electrons. This wave has a wavelength of about 0. 02-0.03 Ångström and its phase is related to the phase of the undiffracted direct electron beam. The crystallographers, on the hand, mean the crystallographic structure factor phase when they simply say phase. This phase is the phase of standing waves of potential in the crystal, each of these waves have their specific wavelength, called d-value for distance between so-called Bragg planes of low/high potential. These d-values range from the cell dimensions to the resolution limit of the electron microscope. Their phases are related to a point in the crystal
36.
Sodium carbonate
–
Sodium carbonate, Na2CO3, is the water-soluble sodium salt of carbonic acid. It most commonly occurs as a crystalline decahydrate, which readily effloresces to form a white powder, pure sodium carbonate is a white, odorless powder that is hygroscopic. It has a strongly alkaline taste, and forms a basic solution in water. Sodium carbonate is known domestically for its everyday use as a water softener. Historically it was extracted from the ashes of plants growing in soils, such as vegetation from the Middle East, kelp from Scotland. Because the ashes of these plants were noticeably different from ashes of timber. It is synthetically produced in quantities from salt and limestone by a method known as the Solvay process. The manufacture of glass is one of the most important uses of sodium carbonate, Sodium carbonate acts as a flux for silica, lowering the melting point of the mixture to something achievable without special materials. This soda glass is mildly water-soluble, so some calcium carbonate is added to the mixture to make the glass produced insoluble. This type of glass is known as soda lime glass, soda for the sodium carbonate, soda lime glass has been the most common form of glass for centuries. Sodium carbonate is used as a relatively strong base in various settings. For example, it is used as a pH regulator to maintain stable alkaline conditions necessary for the action of the majority of film developing agents. It acts as an alkali because when dissolved in water, it dissociates into the acid, carbonic acid. This gives sodium carbonate in solution the ability to attack such as aluminium with the release of hydrogen gas. It is an additive in swimming pools used to raise the ph which can be lowered by chlorine tablets. In cooking, it is used in place of sodium hydroxide for lyeing, especially with German pretzels. These dishes are treated with a solution of a substance to change the pH of the surface of the food. In chemistry, it is used as an electrolyte
37.
Reciprocal lattice
–
In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, this first lattice is usually a periodic spatial function in real-space and is known as the direct lattice. The reciprocal lattice plays a role in most analytic studies of periodic structures. In neutron and X-ray diffraction due to the Laue conditions the momentum difference between incoming and diffracted X-rays of a crystal is a lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice, using this process, one can infer the atomic arrangement of a crystal. The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice, assuming an ideal Bravais lattice R n = n 1 ⋅ a 1 + n 2 ⋅ a 2 where n 1, n 2 ∈ Z. Any quantity, e. g. the electronic density in a crystal can be written as a periodic function f = f Due to the periodicity it is useful to write it in Fourier expansions. Mathematically, we can describe the lattice as the set of all vectors G m that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the lattice is the original lattice. Using column vector representation of vectors, the formulae above can be rewritten using matrix inversion. This method appeals to the definition, and allows generalization to arbitrary dimensions, the cross product formula dominates introductory materials on crystallography. The above definition is called the physics definition, as the factor of 2 π comes naturally from the study of periodic structures. The crystallographers definition has the advantage that the definition of b 1 is just the reciprocal magnitude of a 1 in the direction of a 2 × a 3 and this can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice. The direction of the lattice vector corresponds to the normal to the real space planes. The magnitude of the lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. Reciprocal lattices for the crystal system are as follows. The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a primitive cell of side 2 π a
38.
Paul Steinhardt
–
Paul Joseph Steinhardt is an American theoretical physicist and cosmologist who is currently the Albert Einstein Professor in Science at Princeton University. In addition, Steinhardt co-founded and is the current Director of the Princeton Center for Theoretical Science, Steinhardts research has spanned problems in particle physics, astrophysics, cosmology, condensed matter physics, geoscience and photonics. In cosmology, some of his important contributions include, He is one of the architects of inflationary cosmology which has become an part of the big bang theory. He also presented the first example of eternal inflation, which revealed the multiverse. He then became one of the developers of the leading competing alternatives to the big bang, the ekpyrotic, several years later, the conclusion was confirmed by supernova observations. Working with various collaborators, he introduced the concept of dark energy to further explain the accelerating expansion of the universe. In condensed matter physics and geophysics, Steinhardt and his then-student invented the concept of quasicrystals. Their discovery was recognized by the International Mineralogical Association as the first natural quasicrystal, over the next year, disparate clues were pieced together about its origin, finally pointing to a remote mountain range in Chukotka, on Russias Kamchatka peninsula. Two years later, Steinhardt initiated his first-ever field expedition, and they also discovered a second quasicrystal named decagonite, along with a new crystalline mineral that was named steinhardtite, in his honor. In photonics, Steinhardt has been an innovator in using quasicrystals and other patterns to design photonic materials. Steinhardt and his Princeton colleagues later helped form a company named Etaphase, and thats always what hes after. Beginning in the early 1980s, Steinhardt co-authored seminal papers that helped to lay the foundations of inflationary cosmology, in 1982, Steinhardt and Andreas Albrecht constructed the first inflationary models to produce sufficient accelerated expansion to smooth and flatten the universe and then gracefully exit. The inflation is caused by the energy stored in a field that permeates space. This work is significant, because it has been the prototype for most subsequent inflationary models. Hubble friction and slow-roll is an element in most subsequent inflationary models. Contemporaneous calculations by other groups obtained similar conclusions using less rigorous methods. This was significant research, because Steinhardt would eventually argue that if every outcome is allowed, consequently, eternal inflation may be the Achilles heel of the inflationary picture. His argument that the multiverse was a sign of the failure of the model was largely based in his belief in the long-standing scientific method
39.
Girih tiles
–
Girih tiles are a set of five tiles that were used in the creation of Islamic geometric patterns using strapwork for decoration of buildings in Islamic architecture. They have been used since about the year 1200 and their arrangements found significant improvement starting with the Darb-i Imam shrine in Isfahan in Iran built in 1453, all sides of these figures have the same length, and all their angles are multiples of 36°. All of them, except the pentagon, have bilateral symmetry through two perpendicular lines, specifically, the decagon has tenfold rotational symmetry, and the pentagon has fivefold rotational symmetry. Girih are lines which decorate the tiles, the tiles are used to form girih patterns, from the Persian word گره, meaning knot. In most cases, only the girih are visible rather than the boundaries of the tiles themselves, the girih are piece-wise straight lines which cross the boundaries of the tiles at the center of an edge at 54° to the edge. Two intersecting girih cross each edge of a tile, most tiles have a unique pattern of girih inside the tile which are continuous and follow the symmetry of the tile. However, the decagon has two possible girih patterns one of which has only fivefold rather than tenfold rotational symmetry and this finding was supported both by analysis of patterns on surviving structures, and by examination of 15th century Persian scrolls. However, we have no indication of how much more the architects may have known about the mathematics involved and it is generally believed that such designs were constructed by drafting zigzag outlines with only a straightedge and a compass. Templates found on such as the 97 foot long Topkapi Scroll may have been consulted. There is no text, but there is a pattern and color-coding used to highlight symmetries. In this way, craftsmen could make highly complex designs without resorting to mathematics and this use of repeating patterns created from a limited number of geometric shapes available to craftsmen of the day is similar to the practice of contemporary European Gothic artisans. Designers of both styles were concerned with using their inventories of geometrical shapes to create the maximum diversity of forms and this demanded a skill and practice very different from mathematics. Medieval Islamic tiling reveals mathematical savvy
40.
Isfahan
–
Isfahan, historically also rendered in English as Ispahan, Sepahan, Esfahan or Hispahan, is the capital of Isfahan Province in Iran, located about 340 kilometres south of Tehran. The Greater Isfahan Region had a population of 3,793,104 in the 2011 Census, the counties of Isfahan, Borkhar, Najafabad, Khomeynishahr, Shahinshahr, Mobarakeh, Falavarjan, Tiran o Karvan, Lenjan and Jay all constitute the metropolitan city of Isfahan. Isfahan is located on the main north–south and east–west routes crossing Iran and it flourished from 1050 to 1722, particularly in the 16th and 17th centuries under the Safavid dynasty, when it became the capital of Persia for the second time in its history. Even today, the city retains much of its past glory and it is famous for its Persian–Islamic architecture, with many beautiful boulevards, covered bridges, palaces, mosques, and minarets. This led to the Persian proverb Esfahān nesf-e- jahān ast, the Naghsh-e Jahan Square in Isfahan is one of the largest city squares in the world. It has been designated by UNESCO as a World Heritage Site, the city also has a wide variety of historic monuments and is known for the paintings, history and architecture. Isfahan City Center is also the 5th largest shopping mall in the world, see also, Names of Isfahan The name of the region derives from Middle Persian Spahān. Spahān is attested in various Middle Persian seals and inscriptions, including that of Zoroastrian Magi Kartir, the present-day name is the Arabicized form of Ispahan. The region appears with the abbreviation GD on Sasanian numismatics, in Ptolemys Geographia it appears as Aspadana, translating to place of gathering for the army. It is believed that Spahān derives from spādānām the armies, Old Persian plural of spāda, the history of Isfahan can be traced back to the Palaeolithic period. In recent discoveries, archaeologists have found artifacts dating back to the Palaeolithic, Mesolithic, Neolithic, Bronze and Iron ages. It is said that after Cyrus the Great freed the Jews from Babylon some Jews returned to Jerusalem whereas some others decided to live in Persia, but, actually this happened later in the Sasanid period when a Jewish colony was made in the vicinity of the Sasanid. They did not settle anywhere or in any city without examining the water. They did all along until they reached the city of Isfahan, there they rested, examined the water and soil and found that both resembled Jerusalem. Upon they settled there, cultivated the soil, raised children and grandchildren, under the Parthians, Arsacid governors administered a large province from Isfahan, and the citys urban development accelerated to accommodate the needs of a capital city. The next empire to rule Persia, the Sassanids, presided over changes in their realm, instituting sweeping agricultural reform and reviving Iranian culture. The city was called by the name and the region by the name Aspahan or Spahan. The city was governed by Espoohrans or the members of seven noble Iranian families who had important royal positions, extant foundations of some Sassanid-era bridges in Isfahan suggest that the kings were also fond of ambitious urban planning projects
.svg)
. PMID 25765857.
.jpg)
