1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

2.
Chemistry
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Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the science because it bridges other natural sciences, including physics. For the differences between chemistry and physics see comparison of chemistry and physics, the history of chemistry can be traced to alchemy, which had been practiced for several millennia in various parts of the world. The word chemistry comes from alchemy, which referred to a set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism. An alchemist was called a chemist in popular speech, and later the suffix -ry was added to this to describe the art of the chemist as chemistry, the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία and this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Chemi or Kimi, which is the ancient name of Egypt in Egyptian. Alternately, al-kīmīā may derive from χημεία, meaning cast together, in retrospect, the definition of chemistry has changed over time, as new discoveries and theories add to the functionality of the science. The term chymistry, in the view of noted scientist Robert Boyle in 1661, in 1837, Jean-Baptiste Dumas considered the word chemistry to refer to the science concerned with the laws and effects of molecular forces. More recently, in 1998, Professor Raymond Chang broadened the definition of chemistry to mean the study of matter, early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didnt develop a systematic theory. Greek atomism dates back to 440 BC, arising in works by such as Democritus and Epicurus. In 50 BC, the Roman philosopher Lucretius expanded upon the theory in his book De rerum natura, unlike modern concepts of science, Greek atomism was purely philosophical in nature, with little concern for empirical observations and no concern for chemical experiments. Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis. He formulated Boyles law, rejected the four elements and proposed a mechanistic alternative of atoms. Before his work, though, many important discoveries had been made, the Scottish chemist Joseph Black and the Dutchman J. B. English scientist John Dalton proposed the theory of atoms, that all substances are composed of indivisible atoms of matter. Davy discovered nine new elements including the alkali metals by extracting them from their oxides with electric current, british William Prout first proposed ordering all the elements by their atomic weight as all atoms had a weight that was an exact multiple of the atomic weight of hydrogen. The inert gases, later called the noble gases were discovered by William Ramsay in collaboration with Lord Rayleigh at the end of the century, thereby filling in the basic structure of the table. Organic chemistry was developed by Justus von Liebig and others, following Friedrich Wöhlers synthesis of urea which proved that organisms were, in theory

3.
Polyhedral skeletal electron pair theory
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In chemistry the polyhedral skeletal electron pair theory provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were formulated by Kenneth Wade and were further developed by Michael Mingos and others. The rules are based on a molecular orbital treatment of the bonding and these rules have been extended and unified in the form of the Jemmis mno rules. Different rules are invoked depending on the number of electrons per vertex, the 4n rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many boranes and carboranes. For such clusters, the structures are based on deltahedra, which are polyhedra in which every face is triangular. The 4n clusters are classified as closo-, nido-, arachno- or hypho-, based on whether they represent a complete deltahedron, or a deltahedron that is missing one, two or three vertices. However, hypho clusters are uncommon due to the fact that the electron count is high enough to start to fill antibonding orbitals. If the electron count is close to 5 electrons per vertex, the structure changes to one governed by the 5n rules. As the electron count increases further, the structures of clusters with 5n electron counts become unstable, the 6n clusters have structures that are based on rings. A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4n, 5n, the following polyhedra are closo polyhedra, and are the basis for the 4n rules, each of these have triangular faces. The number of vertices in the cluster determines what polyhedron the structure is based on, using the electron count, the predicted structure can be found. N is the number of vertices in the cluster, the 4n rules are enumerated in the following table. When counting electrons for each cluster, the number of electrons is enumerated. For each transition metal present,10 electrons are subtracted from the electron count. For example, in Rh616 the total number of electrons would be 6 ×9 +16 ×2 −6 ×10 =86 –6 ×10 =26, therefore, the cluster is a closo polyhedron because n =6, with 4n +2 =26. Larger and more electropositive atoms tend to occupy vertices of high connectivity, in the special case of boron hydride clusters, each boron connected to 3 or more vertices has one terminal hydride, while a boron connected to 2 other vertices has 2 terminal hydrogens. If more hydrogens are present, they are placed in open face positions to even out the number of the vertices. In general, closo structures with n vertices are n-vertex polyhedra, Example, Pb2−10 Electron count,10 × Pb +2 =10 ×4 +2 =42 electrons

4.
Boron
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Boron is a chemical element with symbol B and atomic number 5. Produced entirely by cosmic ray spallation and supernovae and not by stellar nucleosynthesis, it is an element in the Solar system. Boron is concentrated on Earth by the water-solubility of its more common naturally occurring compounds and these are mined industrially as evaporites, such as borax and kernite. The largest known deposits are in Turkey, the largest producer of boron minerals. Elemental boron is a metalloid that is found in small amounts in meteoroids, industrially, very pure boron is produced with difficulty because of refractory contamination by carbon or other elements. Several allotropes of boron exist, amorphous boron is a powder, crystalline boron is silvery to black, extremely hard. The primary use of boron is as boron filaments with applications similar to carbon fibers in some high-strength materials. Boron is primarily used in chemical compounds, about half of all consumption globally, boron is used as an additive in glass fibers of boron-containing fiberglass for insulation and structural materials. The next leading use is in polymers and ceramics in high-strength, lightweight structural, borosilicate glass is desired for its greater strength and thermal shock resistance than ordinary soda lime glass. Boron compounds are used as fertilizers in agriculture and in sodium perborate bleaches, a small amount of boron is used as a dopant in semiconductors, and reagent intermediates in the synthesis of organic fine chemicals. A few boron-containing organic pharmaceuticals are used or are in study, natural boron is composed of two stable isotopes, one of which has a number of uses as a neutron-capturing agent. In biology, borates have low toxicity in mammals, but are toxic to arthropods and are used as insecticides. Boric acid is mildly antimicrobial, and several natural boron-containing organic antibiotics are known, small amounts of boron compounds play a strengthening role in the cell walls of all plants, making boron a necessary plant nutrient. Boron is involved in the metabolism of calcium in both plants and animals and it is considered an essential nutrient for humans, and boron deficiency is implicated in osteoporosis. The word boron was coined from borax, the mineral from which it was isolated, by analogy with carbon, marco Polo brought some glazes back to Italy in the 13th century. Agricola, around 1600, reports the use of borax as a flux in metallurgy, in 1777, boric acid was recognized in the hot springs near Florence, Italy, and became known as sal sedativum, with primarily medical uses. The rare mineral is called sassolite, which is found at Sasso, Sasso was the main source of European borax from 1827 to 1872, when American sources replaced it. Boron compounds were relatively rarely used until the late 1800s when Francis Marion Smiths Pacific Coast Borax Company first popularized and produced them in volume at low cost

5.
Hydrogen
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Hydrogen is a chemical element with chemical symbol H and atomic number 1. With a standard weight of circa 1.008, hydrogen is the lightest element on the periodic table. Its monatomic form is the most abundant chemical substance in the Universe, non-remnant stars are mainly composed of hydrogen in the plasma state. The most common isotope of hydrogen, termed protium, has one proton, the universal emergence of atomic hydrogen first occurred during the recombination epoch. At standard temperature and pressure, hydrogen is a colorless, odorless, tasteless, non-toxic, nonmetallic, since hydrogen readily forms covalent compounds with most nonmetallic elements, most of the hydrogen on Earth exists in molecular forms such as water or organic compounds. Hydrogen plays an important role in acid–base reactions because most acid-base reactions involve the exchange of protons between soluble molecules. In ionic compounds, hydrogen can take the form of a charge when it is known as a hydride. The hydrogen cation is written as though composed of a bare proton, Hydrogen gas was first artificially produced in the early 16th century by the reaction of acids on metals. Industrial production is mainly from steam reforming natural gas, and less often from more energy-intensive methods such as the electrolysis of water. Most hydrogen is used near the site of its production, the two largest uses being fossil fuel processing and ammonia production, mostly for the fertilizer market, Hydrogen is a concern in metallurgy as it can embrittle many metals, complicating the design of pipelines and storage tanks. Hydrogen gas is flammable and will burn in air at a very wide range of concentrations between 4% and 75% by volume. The enthalpy of combustion is −286 kJ/mol,2 H2 + O2 →2 H2O +572 kJ Hydrogen gas forms explosive mixtures with air in concentrations from 4–74%, the explosive reactions may be triggered by spark, heat, or sunlight. The hydrogen autoignition temperature, the temperature of spontaneous ignition in air, is 500 °C, the detection of a burning hydrogen leak may require a flame detector, such leaks can be very dangerous. Hydrogen flames in other conditions are blue, resembling blue natural gas flames, the destruction of the Hindenburg airship was a notorious example of hydrogen combustion and the cause is still debated. The visible orange flames in that incident were the result of a mixture of hydrogen to oxygen combined with carbon compounds from the airship skin. H2 reacts with every oxidizing element, the ground state energy level of the electron in a hydrogen atom is −13.6 eV, which is equivalent to an ultraviolet photon of roughly 91 nm wavelength. The energy levels of hydrogen can be calculated fairly accurately using the Bohr model of the atom, however, the atomic electron and proton are held together by electromagnetic force, while planets and celestial objects are held by gravity. The most complicated treatments allow for the effects of special relativity

6.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does

7.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two

8.
Hexadecagon
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In mathematics, a hexadecagon or 16-gon is a sixteen-sided polygon. A regular hexadecagon is a hexadecagon in which all angles are equal and its Schläfli symbol is and can be constructed as a truncated octagon, t, and a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon, as 16 =24, a regular hexadecagon is constructible using compass and straightedge, this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, the area of a regular hexadecagon with edge length t is A =4 t 2 cot π16 =4 t 2. Since the area of the circumcircle is π R2, the regular hexadecagon fills approximately 97. 45% of its circumcircle, the regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups, Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups, Z16, Z8, Z4, Z2, and Z1, on the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1, the dihedral symmetries are divided depending on whether they pass through vertices or edges Cyclic symmetries in the middle column are labeled as g for their central gyration orders. These two forms are duals of each other and have half the order of the regular hexadecagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g16 subgroup has no degrees of freedom but can seen as directed edges. A skew hexadecagon is a polygon with 24 vertices and edges. The interior of such an hexadecagon is not generally defined, a skew zig-zag hexadecagon has vertices alternating between two parallel planes. A regular skew hexadecagon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, symmetry, order 32. The octagrammic antiprism, s and octagrammic crossed-antiprism, s also have regular skew octagons, there are three regular star polygons, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds, is reduced to 2 as two octagons, is reduced to 4 as four squares and reduces to 2 as two octagrams, and finally is reduced to 8 as eight digons. Deeper truncations of the octagon and octagram can produce isogonal intermediate hexadecagram forms with equally spaced vertices. A truncated octagon is a hexadecagon, t=, a quasitruncated octagon, inverted as, is a hexadecagram, t=. A truncated octagram is a hexadecagram, t= and a quasitruncated octagram, inverted as, is a hexadecagram, hexadecagrams are included in the Girih patterns in the Alhambra. An octagonal star can be seen as a concave hexadecagon, Weisstein, Eric W. Hexadecagon

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