geometries definition of geometries in the Free Online Encyclopedia.

geometry [Gr.,=earth measuring], branch of mathematics mathematics, deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or
..... Click the link for more information. concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.

Types of Geometry

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic.
..... Click the link for more information. , of which algebraic geometry algebraic geometry, branch of geometry , based on analytic geometry , that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates ).
..... Click the link for more information. is a further development (see Cartesian coordinates Cartesian coordinates (kärtē`zhən)
..... Click the link for more information. ). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry descriptive geometry, branch of geometry concerned with the two-dimensional representation of three-dimensional objects; it was introduced in 1795 by Gaspard Monge. By means of such representations, geometrical problems in three dimensions may be solved in the plane.
..... Click the link for more information. for this purpose in the late 18th cent. differential geometry r moves along a curve at arc length s from some fixed point, then t = dr/ds is a unit tangent vector to the curve at r. The normal vector n
..... Click the link for more information. , in which the concepts of the calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit —the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry projective geometry, branch of geometry concerned with those properties of geometric figures that remain invariant under projection. The basic elements are points, lines, and planes, and the following statements are usually taken as assumptions: (1) two points lie in
..... Click the link for more information. by J. V. Poncelet (1822) and of non-Euclidean geometry non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
..... Click the link for more information. by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

Their Relationship to Each Other

The different geometries are classified and related to one another in various ways. The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly. Both Euclidean and non-Euclidean geometry are types of metric geometry, in which the lengths of line segments and the sizes of angles may be measured and compared. Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.

The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry. Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity relativity, physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.
..... Click the link for more information. . Ordered geometry consists of all propositions common to both absolute geometry and affine geometry; this geometry includes the notion on intermediacy ("betweenness") but not that of measurement.

An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given group group, in mathematics, system consisting of a set of elements and a binary operation a+b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e.
..... Click the link for more information. of transformations. For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations). Topology topology, branch of mathematics , formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size.
..... Click the link for more information. , perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.

The Axiomatic Approach to Geometry

Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts. Euclid first defined his basic terms, such as point and line, then stated without proof certain axioms axiom, in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
..... Click the link for more information. and postulates about them that seemed to be self-evident or obvious truths, and finally derived a number of statements (theorems) from the postulates by means of deductive logic logic, the systematic study of valid inference. A distinction is drawn between logical validity and truth. Validity merely refers to formal properties of the process of inference.
..... Click the link for more information. . This axiomatic method has since been adopted not only throughout mathematics but in many other fields as well. The close examination of the axioms and postulates of Euclidean geometry during the 19th cent. resulted in the realization that the logical basis of geometry was not as firm as had previously been supposed. New axiom and postulate systems were developed by various mathematicians, notably David Hilbert (1899).

Bibliography

See H. G. Forder, The Foundations of Euclidean Geometry (1927); H. S. M. Coxeter, Introduction to Geometry (2d ed. 1969).

geometry

1. the branch of mathematics concerned with the properties, relationships, and measurement of points, lines, curves, and surfaces

a. any branch of geometry using a particular notation or set of assumptions

b. any branch of geometry referring to a particular set of objects

3. Arts the shape of a solid or a surface

geometry [jē′äm·ə·trē]

(mathematics)

The qualitative study of shape and size.