Space groupIn mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions.
Molecular orbitalIn chemistry, a molecular orbital (ɒrbədl) is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The terms atomic orbital and molecular orbital were introduced by Robert S. Mulliken in 1932 to mean one-electron orbital wave functions. At an elementary level, they are used to describe the region of space in which a function has a significant amplitude.
Point groupIn geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx.
Dimethyl sulfoxideDimethyl sulfoxide (DMSO) is an organosulfur compound with the formula (CH3)2SO. This colorless liquid is the sulfoxide most widely used commercially. It is an important polar aprotic solvent that dissolves both polar and nonpolar compounds and is miscible in a wide range of organic solvents as well as water. It has a relatively high boiling point. DMSO has the unusual property that many individuals perceive a garlic-like taste in the mouth after DMSO makes contact with their skin.
Nitrogen dioxideNitrogen dioxide is a chemical compound with the formula NO2. It is one of several nitrogen oxides. NO2 is an intermediate in the industrial synthesis of nitric acid, millions of tons of which are produced each year for use primarily in the production of fertilizers. At higher temperatures it is a reddish-brown gas. It can be fatal if inhaled in large quantities. Nitrogen dioxide is a paramagnetic, bent molecule with C2v point group symmetry. It is included in the NOx family of atmospheric pollutants.
Schoenflies notationThe Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead.
DipoleIn physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system is a pair of charges of equal magnitude but opposite sign separated by some typically small distance. (A permanent electric dipole is called an electret.) A magnetic dipole is the closed circulation of an electric current system. A simple example is a single loop of wire with constant current through it.
Symmetry groupIn group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.
NitriteThe nitrite ion has the chemical formula NO2-. Nitrite (mostly sodium nitrite) is widely used throughout chemical and pharmaceutical industries. The nitrite anion is a pervasive intermediate in the nitrogen cycle in nature. The name nitrite also refers to organic compounds having the –ONO group, which are esters of nitrous acid. Sodium nitrite is made industrially by passing a mixture of nitrogen oxides into aqueous sodium hydroxide or sodium carbonate solution: The product is purified by recrystallization.
PyridinePyridine is a basic heterocyclic organic compound with the chemical formula . It is structurally related to benzene, with one methine group replaced by a nitrogen atom. It is a highly flammable, weakly alkaline, water-miscible liquid with a distinctive, unpleasant fish-like smell. Pyridine is colorless, but older or impure samples can appear yellow, due to the formation of extended, unsaturated polymeric chains, which show significant electrical conductivity.
Improper rotationIn geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.
SymmetrySymmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
