GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Differential geometryDifferential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.
Non-Euclidean geometryIn mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.
Euclidean geometryEuclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
Elliptic geometryElliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
Hyperbolic geometryIn mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.
Projective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.
Synthetic geometrySynthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms. The term "synthetic geometry" has been coined only after the 17th century, and the introduction by René Descartes of the coordinate method, which was called analytic geometry.
Alcohol (chemistry)In chemistry, an alcohol is a type of organic compound that carries at least one hydroxyl () functional group bound to a saturated carbon atom. Alcohols range from the simple, like methanol and ethanol, to complex, like sucrose and cholesterol. The presence of an OH group strongly modifies the properties of hydrocarbons, conferring hydrophilic (water-loving) properties. The OH group provides a site at which many reactions can occur.
Organic chemistryOrganic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms. Study of structure determines their structural formula. Study of properties includes physical and chemical properties, and evaluation of chemical reactivity to understand their behavior.
EtherIn organic chemistry, ethers are a class of compounds that contain an ether group—an oxygen atom connected to two alkyl or aryl groups. They have the general formula , where R and R′ represent the alkyl or aryl groups. Ethers can again be classified into two varieties: if the alkyl or aryl groups are the same on both sides of the oxygen atom, then it is a simple or symmetrical ether, whereas if they are different, the ethers are called mixed or unsymmetrical ethers.
IUPAC nomenclature of organic chemistryIn chemical nomenclature, the IUPAC nomenclature of organic chemistry is a method of naming organic chemical compounds as recommended by the International Union of Pure and Applied Chemistry (IUPAC). It is published in the Nomenclature of Organic Chemistry (informally called the Blue Book). Ideally, every possible organic compound should have a name from which an unambiguous structural formula can be created. There is also an IUPAC nomenclature of inorganic chemistry.
Organic compoundIn chemistry, many authors consider an organic compound to be any chemical compound that contains carbon-hydrogen or carbon-carbon bonds, however, some authors consider an organic compound to be any chemical compound that contains carbon. The definition of "organic" versus "inorganic" varies from author to author, and is a topic of debate. For example, methane () is considered organic, but whether some other carbon-containing compounds are organic or inorganic varies from author to author, for example halides of carbon without carbon-hydrogen and carbon-carbon bonds (e.
Fatty alcoholFatty alcohols (or long-chain alcohols) are usually high-molecular-weight, straight-chain primary alcohols, but can also range from as few as 4–6 carbons to as many as 22–26, derived from natural fats and oils. The precise chain length varies with the source. Some commercially important fatty alcohols are lauryl, stearyl, and oleyl alcohols. They are colourless oily liquids (for smaller carbon numbers) or waxy solids, although impure samples may appear yellow.
Enol etherIn organic chemistry an enol ether is an alkene with an alkoxy substituent. The general structure is R2C=CR-OR where R = H, alkyl or aryl. A common subfamily of enol ethers are vinyl ethers, with the formula ROCH=CH2. Important enol ethers include the reagent 3,4-dihydropyran and the monomers methyl vinyl ether and ethyl vinyl ether. Akin to enamines, enol ethers are electron-rich alkenes by virtue of the electron-donation from the heteroatom via pi-bonding. Enol ethers have oxonium ion character.
Organic reactionOrganic reactions are chemical reactions involving organic compounds. The basic organic chemistry reaction types are addition reactions, elimination reactions, substitution reactions, pericyclic reactions, rearrangement reactions, photochemical reactions and redox reactions. In organic synthesis, organic reactions are used in the construction of new organic molecules. The production of many man-made chemicals such as drugs, plastics, food additives, fabrics depend on organic reactions.
Absolute geometryAbsolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems (such as Hilbert's axioms without the parallel axiom) are used instead.
Diethyl etherDiethyl ether, or simply ether, is an organic compound in the ether class with the formula or , sometimes abbreviated as . It is a colourless, highly volatile, sweet-smelling ("ethereal odour"), extremely flammable liquid. It is commonly used as a solvent in laboratories and as a starting fluid for some engines. It was formerly used as a general anesthetic, until non-flammable drugs were developed, such as halothane. It has been used as a recreational drug to cause intoxication.
Ether lipidIn an organic chemistry general sense, an ether lipid implies an ether bridge between an alkyl group (a lipid) and an unspecified alkyl or aryl group, not necessarily glycerol. If glycerol is involved, the compound is called a glyceryl ether, which may take the form of an alkylglycerol, an alkyl acyl glycerol, or in combination with a phosphatide group, a phospholipid. In a biochemical sense, an ether lipid usually implies glycerophospholipids of various type, also called phospholipids, in which the sn-1 position of the glycerol backbone has a lipid attached by an ether bond and a lipid attached to the sn-2 position via an acyl group.
ThermoacidophileA thermoacidophile is an extremophilic microorganism that is both thermophilic and acidophilic; i.e., it can grow under conditions of high temperature and low pH. The large majority of thermoacidophiles are archaea (particularly the Thermoproteota and "Euryarchaeota") or bacteria, though occasional eukaryotic examples have been reported. Thermoacidophiles can be found in hot springs and solfataric environments, within deep sea vents, or in other environments of geothermal activity.
