geometry

geometry

[jee-om-i-tree]
geometry [Gr.,=earth measuring], branch of mathematics 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, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus 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 by J. V. Poncelet (1822) and of non-Euclidean geometry 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. 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 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, 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 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. 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).

Any theory of the nature of geometric space differing from the traditional view held since Euclid's time. These geometries arose in the 19th century when several mathematicians working independently explored the possibility of rejecting Euclid's parallel postulate. Different assumptions about how many lines through a point not on a given line could be parallel to that line resulted in hyperbolic geometry and elliptic geometry. Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity. Seealso Nikolay Lobachevsky, Bernhard Riemann.

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Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. Its importance lies less in its results than in the systematic method Euclid used to develop and present them. This axiomatic method has been the model for many systems of rational thought, even outside mathematics, for over 2,000 years. From 10 axioms and postulates, Euclid deduced 465 theorems, or propositions, concerning aspects of plane and solid geometric figures. This work was long held to constitute an accurate description of the physical world and to provide a sufficient basis for understanding it. During the 19th century, rejection of some of Euclid's postulates resulted in two non-Euclidean geometries that proved just as valid and consistent.

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Branch of mathematics that deals with the relationships between geometric figures and the images (mappings) of them that result from projection. Examples of projections include motion pictures, maps of the Earth's surface, and shadows cast by objects. One stimulus for the subject's development was the need to understand perspective in drawing and painting. Every point of the projected object and the corresponding point of its image must lie on the projection ray, a line that passes through the centre of projection. Modern projective geometry emphasizes the mathematical properties (such as straightness of lines and points of intersection) preserved in projections despite the distortion of lengths, angles, and shapes.

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Non-Euclidean geometry, useful in modeling interstellar space, that rejects the parallel postulate, proposing instead that at least two lines through any point not on a given line are parallel to that line. Though many of its theorems are identical to those of Euclidean geometry, others differ. For example, two parallel lines converge in one direction and diverge in the other, and the angles of a triangle add up to less than 180°.

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In mathematics, the study of complex shapes with the property of self-similarity, known as fractals. Rather like holograms that store the entire image in each part of the image, any part of a fractal can be repeatedly magnified, with each magnification resembling all or part of the original fractal. This phenomenon can be seen in objects like snowflakes and tree bark. The term fractal was coined by Benoit B. Mandelbrot in 1975. This new system of geometry has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics; fractals can describe irregularly shaped objects or spatially nonuniform phenomena that cannot be described by Euclidean geometry. Fractal simulations have been used to plot the distributions of galactic clusters and to generate lifelike images of complicated, irregular natural objects, including rugged terrains and foliage used in films. Seealso chaos theory.

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Non-Euclidean geometry that rejects Euclid's fifth postulate (the parallel postulate) and modifies his second postulate. It is also known as Riemannian geometry, after Bernhard Riemann. It asserts that no line passing through a point not on a given line is parallel to that line. It also states that while any straight line of finite length can be extended indefinitely, all straight lines are the same length. Though many of elliptic geometry's theorems are identical to those of Euclidean geometry, others differ (e.g., the angles in a triangle add up to more than 180°). It can most easily be pictured as geometry done on the surface of a sphere where all lines are great circles.

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Field of mathematics in which methods of calculus are applied to the local geometry of curves and surfaces (i.e., to a small portion of a surface or curve around a point). A simple example is finding the tangent line on a two-dimensional curve at a given point. Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.

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Investigation of geometric objects using coordinate systems. Because René Descartes was the first to apply algebra to geometry, it is also known as Cartesian geometry. It springs from the idea that any point in two-dimensional space can be represented by two numbers and any point in three-dimensional space by three. Because lines, circles, spheres, and other figures can be thought of as collections of points in space that satisfy certain equations, they can be explored via equations and formulas rather than graphs. Most of analytic geometry deals with the conic sections. Because these are defined using the notion of fixed distance, each section can be represented by a general equation derived from the distance formula.

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Study of geometric objects expressed as equations and represented by graphs in a given coordinate system. In contrast to Euclidean geometry, algebraic geometry represents geometric objects using algebraic equations (e.g., a circle of radius math.r is defined by math.x2 + math.y2 = math.r2). Objects so defined can then be analyzed for symmetries, intercepts, and other properties without having to refer to a graph.

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In geometry, two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).

Definition of congruence in analytic geometry

In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.

A more formal definition: two subsets A and B of Euclidean space Rn are called congruent if there exists an isometry f : RnRn (an element of the Euclidean group E(n)) with f(A) = B. Congruence is an equivalence relation.

Congruence of triangles

Two triangles are congruent if their corresponding sides and angles are equal. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles.

If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:

triangle mathrm{ABC} cong triangle mathrm{DEF}

Determining congruence

Congruence between two triangles can be shown through the following comparisons:

  • SAS (Side-Angle-Side): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.
  • SSS (Side-Side-Side): Two triangles are congruent if their corresponding sides are equal.
  • ASA (Angle-Side-Angle): Two triangles are congruent if a pair of corresponding angles and the included side are equal. The ASA Postulate was contributed by Thales of Miletus (Greek). In most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates.
  • AAS (Angle-Angle-Side): Two triangles are congruent if a pair of corresponding angles and a not-included side are equal, since the 3rd angle would have to be equal, and therefore the side would be included. This one is valid only in Euclidean geometry.

Side-Side-Angle

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS (Angle-Side-Side)) does not always prove congruence.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter or equal to the adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases, SSA proves congruence. Notice that the opposite side cannot be smaller than the adjacent side times the sine of the angle as this could not describe a triangle.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition), we can calculate the third side and fall back on SSS.

The SSA condition proves congruence if the angle is acute and the opposite side either equals the adjacent side times the sine of the angle (right triangle) or is longer than the adjacent side.

Angle-Angle-Angle

AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only similarity and not congruence. However, in spherical geometry and hyperbolic geometry this is sufficient for congruence.

See also

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