1.
Crystal
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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
2.
Quasicrystal
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A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all space, but it lacks translational symmetry. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, the discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier, but, until the 1980s, in 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals. Roughly, an ordering is non-periodic if it lacks translational symmetry, symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally, the aperiodicity is revealed in the symmetry of the diffraction pattern. In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the communitys reaction, it took him two years to publish the results for which he was awarded the Nobel Prize in Chemistry in 2011. In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically. Nevertheless, two later, his student Robert Berger constructed a set of some 20,000 square tiles that can tile the plane. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found, in 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry, around the same time Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry. Mathematically, quasicrystals have been shown to be derivable from a method that treats them as projections of a higher-dimensional lattice. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984, the tiling is formed by two tiles with rhombohedral shape. Shechtman first observed ten-fold electron diffraction patterns in 1982, as described in his notebook, the observation was made during a routine investigation, by electron microscopy, of a rapidly cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards. In the summer of the same year Shechtman visited Ilan Blech, Blech responded that such diffractions had been seen before. Around that time, Shechtman also related his finding to John Cahn of NIST who did not offer any explanation, Shechtman quoted Cahn as saying, Danny, this material is telling us something and I challenge you to find out what it is
3.
Birefringence
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Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent, the birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress and this effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it in calcite, a crystal having one of the strongest birefringences. A mathematical description of wave propagation in a birefringent medium is presented below, following is a qualitative explanation of the phenomenon. Thus rotating the material around this axis does not change its optical behavior and this special direction is known as the optic axis of the material. Light whose polarization is perpendicular to the axis is governed by a refractive index no. Light whose polarization is in the direction of the optic axis sees an optical index ne, for any ray direction there is a linear polarization direction perpendicular to the optic axis, and this is called an ordinary ray. The magnitude of the difference is quantified by the birefringence, Δ n = n e − n o, the propagation of the ordinary ray is simply described by no as if there were no birefringence involved. However the extraordinary ray, as its name suggests, propagates unlike any wave in an optical material. Its refraction at a surface can be using the effective refractive index. However it is in fact an inhomogeneous wave whose power flow is not exactly in the direction of the wave vector and this causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the ray, to rotate slightly around that of the ordinary ray. When the light propagates either along or orthogonal to the optic axis, in the first case, both polarizations see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity but is not an inhomogeneous wave, for instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source. The case of so-called biaxial crystals is substantially more complex and these are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices, being extraordinary waves, however, the direction of power flow is not identical to the direction of the wave vector in either case. The two refractive indices can be determined using the index ellipsoids for given directions of the polarization, note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution but is described by three unequal principle refractive indices nα, nβ and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant, for this reason, birefringent materials with three distinct refractive indices are called biaxial
4.
Mirror
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This is different from other light-reflecting objects that do not preserve much of the original wave signal other than color and diffuse reflected light. The most familiar type of mirror is the mirror, which has a flat screen surface. Curved mirrors are used, to produce magnified or diminished images or focus light or simply distort the reflected image. Mirrors are commonly used for personal grooming or admiring oneself, decoration, Mirrors are also used in scientific apparatus such as telescopes and lasers, cameras, and industrial machinery. Most mirrors are designed for light, however, mirrors designed for other wavelengths of electromagnetic radiation are also used. The first mirrors used by people were most likely pools of dark, still water, the requirements for making a good mirror are a surface with a very high degree of flatness, and a surface roughness smaller than the wavelength of the light. The earliest manufactured mirrors were pieces of polished stone such as obsidian, examples of obsidian mirrors found in Anatolia have been dated to around 6000 BC. Mirrors of polished copper were crafted in Mesopotamia from 4000 BC, polished stone mirrors from Central and South America date from around 2000 BC onwards. In China, bronze mirrors were manufactured from around 2000 BC, some of the earliest bronze, Mirrors made of other metal mixtures such as copper and tin speculum metal may have also been produced in China and India. Mirrors of speculum metal or any precious metal were hard to produce and were owned by the wealthy. Stone mirrors often had poor reflectivity compared to metals, yet metals scratch or tarnish easily, depending upon the color, both often yielded reflections with poor color rendering. The poor image quality of ancient mirrors explains 1 Corinthians 13s reference to seeing as in a mirror, glass was a desirable material for mirrors. Because the surface of glass is smooth, it produces reflections with very little blur. In addition, glass is very hard and scratch resistant, however, glass by itself has little reflectivity, so people began coating it with metals to increase the reflectivity. According to Pliny, the people of Sidon developed a technique for creating crude mirrors by coating glass with molten lead. Glass mirrors backed with gold leaf are mentioned by Pliny in his Natural History and these circular mirrors were typically small, from only a fraction of an inch to as much as eight inches in diameter. These small mirrors produced distorted images, yet were prized objects of high value and these ancient glass mirrors were very thin, thus very fragile, because the glass needed to be extremely thin to prevent cracking when coated with a hot, molten metal. Due to the quality, high cost, and small size of these ancient glass mirrors
5.
Hexagonal crystal family
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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
6.
Mineralogy
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Mineralogy is a subject of geology specializing in the scientific study of chemistry, crystal structure, and physical properties of minerals and mineralized artifacts. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, the German Renaissance specialist Georgius Agricola wrote works such as De re metallica and De Natura Fossilium which began the scientific approach to the subject. Systematic scientific studies of minerals and rocks developed in post-Renaissance Europe, the modern study of mineralogy was founded on the principles of crystallography and to the microscopic study of rock sections with the invention of the microscope in the 17th century. Nicholas Steno first observed the law of constancy of interfacial angles in quartz crystals in 1669 and this was later generalized and established experimentally by Jean-Baptiste L. Romé de lIslee in 1783. In 1814, Jöns Jacob Berzelius introduced a classification of minerals based on their chemistry rather than their crystal structure, james D. Dana published his first edition of A System of Mineralogy in 1837, and in a later edition introduced a chemical classification that is still the standard. It, however, retains a focus on the structures commonly encountered in rock-forming minerals. An initial step in identifying a mineral is to examine its physical properties and these can be classified into density, measures of mechanical cohesion, macroscopic visual properties, magnetic and electric properties, radioactivity and solubility in hydrogen chloride. If the mineral is crystallized, it will also have a distinctive crystal habit that reflects the crystal structure or internal arrangement of atoms. It is also affected by crystal defects and twinning. Many crystals are polymorphic, having more than one crystal structure depending on factors such as pressure and temperature. ”Examples of polymorphs are calcite and aragonite - two minerals with identical chemical composition, distinguished by their crystallography, calcite is rhombohedral and aragonite is orthorhombic. The crystal structure is the arrangement of atoms in a crystal and it is represented by a lattice of points which repeats a basic pattern, called a unit cell, in three dimensions. The lattice can be characterized by its symmetries and by the dimensions of the unit cell and these dimensions are represented by three Miller indices. The lattice remains unchanged by certain symmetry operations about any point in the lattice, reflection, rotation, inversion, and rotary inversion. Together, they make up an object called a crystallographic point group or crystal class. There are 32 possible crystal classes, in addition, there are operations that displace all the points, translation, screw axis, and glide plane. In combination with the point symmetries, they form 230 possible space groups, most geology departments have X-ray powder diffraction equipment to analyze the crystal structures of minerals. X-rays have wavelengths that are the order of magnitude as the distances between atoms. In a sample that is ground to a powder, the X-rays sample a random distribution of all crystal orientations, powder diffraction can distinguish between minerals that may appear the same in a hand sample, for example quartz and its polymorphs tridymite and cristobalite
7.
Physicist
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A physicist is a scientist who has specialized knowledge in the field of physics, the exploration of the interactions of matter and energy across the physical universe. A physicist is a scientist who specializes or works in the field of physics, physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists can also apply their knowledge towards solving real-world problems or developing new technologies, some physicists specialize in sectors outside the science of physics itself, such as engineering. The study and practice of physics is based on a ladder of discoveries. Many mathematical and physical ideas used today found their earliest expression in ancient Greek culture and Asian culture, the bulk of physics education can be said to flow from the scientific revolution in Europe, starting with the work of Galileo and Kepler in the early 1600s. New knowledge in the early 21st century includes an increase in understanding physical cosmology. The term physicist was coined by William Whewell in his 1840 book The Philosophy of the Inductive Sciences, many physicist positions require an undergraduate degree in applied physics or a related science or a Masters degree like MSc, MPhil, MPhys or MSci. In a research oriented level, students tend to specialize in a particular field, Physics students also need training in mathematics, and also in computer science and programming. For being employed as a physicist a doctoral background may be required for certain positions, undergraduate students like BSc Mechanical Engineering, BSc Electrical and Computer Engineering, BSc Applied Physics. etc. With physics orientation are chosen as research assistants with faculty members, the highest honor awarded to physicists is the Nobel Prize in Physics, awarded since 1901 by the Royal Swedish Academy of Sciences. The three major employers of career physicists are academic institutions, laboratories, and private industries, with the largest employer being the last, physicists in academia or government labs tend to have titles such as Assistants, Professors, Sr. /Jr. As per the American Institute for Physics, some 20% of new physics Ph. D. s holds jobs in engineering development programs, while 14% turn to computer software, a majority of physicists employed apply their skills and training to interdisciplinary sectors. For industry or self-employment. and also in science and programming. Hence a majority of Physics bachelors degree holders are employed in the private sector, other fields are academia, government and military service, nonprofit entities, labs and teaching
8.
Johann F. C. Hessel
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Johann Friedrich Christian Hessel was a German physician and professor of mineralogy at the University of Marburg. The origins of geometric crystallography, for which Hessels work was noteworthy, Hessel also made contributions to classical mineralogy, as well. A crystal form here denotes a set of symmetrically equivalent planes with Miller indices enclosed in braces, for example, a cube-shaped crystal of fluorite has six equivalent faces. The entire set is denoted as, the indices for each of the individual six faces are enclosed by parentheses and these are designated, and. The cube belongs to the isometric or tessular class, as do an octahedron and tetrahedron, the essential symmetry elements of the isometric class is the existence of a set of three 4-fold, four 3-fold, and six 2-fold rotation axes. In the earlier classification schemes by the German mineralogists Christian Samuel Weiss and Friedrich Mohs the isometric class had been designated sphäroedrisch and tessularisch, as of Hessels time, not all of the 32 possible symmetries had actually been observed in real crystals. Hessels work originally appeared in 1830 as an article in Gehler’s Physikalische Wörterbuch and it went unnoticed until it was republished in 1897 as part of a collection of papers on crystallography in Oswald’s Klassiker der Exakten Wissenschaften. Prior to this posthumous re-publication of Hessels investigations, similar findings had been reported by the French scientist Auguste Bravais in Extrait J. Math, pures et Applique ́es and by the Russian crystallographer Alex V. Gadolin in 1867. However, the 32 classes of symmetry are one-and-the-same as the 32 crystallographic point groups. Briefly, a crystal is similar to three-dimensional wallpaper, in that it is a repetition of some motif. The motif is created by point group operations, while the wallpaper, the symmetry of the motif is the true point group symmetry of the crystal and it causes the symmetry of the external forms. Specifically, the crystals external morphological symmetry must conform to the components of the space group symmetry operations. Under favorable circumstances, point groups can be determined solely by examination of the crystal morphology and this is not always possible because, of the many forms normally apparent or expected in a typical crystal specimen, some forms may be absent or show unequal development. The word habit is used to describe the external shape of a crystal specimen. In general, a substance may crystallize in different habits because the rates of the various faces need not be the same. Following the work of the Swiss mathematician Simon Antoine Jean LHuilier, in this case, the sum of the valence and the number of faces does not equal two plus the number of edges. Such exceptions can occur when a polyhedron possesses internal cavities, which, in turn, Hessel found this to be true with lead sulfide crystals inside calcium fluoride crystals. Hessel also found Eulers formula disobeyed with interconnected polyhedra, for example, in the field of classical mineralogy, Hessel showed that the plagioclase feldspars could be considered solid solutions of albite and anorthite
9.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
10.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
11.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
12.
Cubic crystal system
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In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common