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.
Planar graphIn graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
BenzeneBenzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, benzene is classed as a hydrocarbon. Benzene is a natural constituent of petroleum and is one of the elementary petrochemicals. Due to the cyclic continuous pi bonds between the carbon atoms, benzene is classed as an aromatic hydrocarbon.
Taxicab geometryA taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 distance, L1 distance or norm (see Lp space), snake distance, city block distance, Manhattan distance or Manhattan length.
Benzyl groupIn organic chemistry, benzyl is the substituent or molecular fragment possessing the structure . Benzyl features a benzene ring () attached to a methylene group () group. In IUPAC nomenclature, the prefix benzyl refers to a substituent, for example benzyl chloride or benzyl benzoate. Benzyl is not to be confused with phenyl with the formula . The term benzylic is used to describe the position of the first carbon bonded to a benzene or other aromatic ring. For example, is referred to as a "benzylic" carbocation.
Structure theorem for finitely generated modules over a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
Trigonal planar molecular geometryIn chemistry, trigonal planar is a molecular geometry model with one atom at the center and three atoms at the corners of an equilateral triangle, called peripheral atoms, all in one plane. In an ideal trigonal planar species, all three ligands are identical and all bond angles are 120°. Such species belong to the point group D3h. Molecules where the three ligands are not identical, such as , deviate from this idealized geometry. Examples of molecules with trigonal planar geometry include boron trifluoride (), formaldehyde (), phosgene (), and sulfur trioxide ().
Ideal class groupIn number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group.
Principal ideal ringIn mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring.
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.
