geometry
American Heritage Dictionary:
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ge·om·e·try
(jē-ŏm'ĭ-trē) 
n., pl., -tries.
n., pl., -tries.
- The mathematics of the properties, measurement, and relationships of points, lines, angles, surfaces, and solids.
- A system of geometry: Euclidean geometry.
- A geometry restricted to a class of problems or objects: solid geometry.
- A book on geometry.
- Configuration; arrangement.
- A surface shape.
- A physical arrangement suggesting geometric forms or lines.
[Middle English geometrie, from Old French, from Latin geōmetria, from Greek geōmetriā, from geōmetrein, to measure land : geō-, geo- + metron, measure.]
geometrician ge·om'e·tri'cian (jē-ŏm'ĭ-trĭsh'ən, jē'ə-mĭ-) or ge·om'e·ter n.
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A branch of mathematics concerned with the properties of space, including points, lines, curves, planes and surfaces in space, and figures bounded by them. For discussion of various branches of geometry See also Algebraic geometry; Differential geometry; Euclidean geometry; Projective geometry; Riemannian geometry.
Oxford Dictionary of Philosophy:
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geometry
Although various laws concerning lines and angles were known to the Egyptians and the Pythagoreans, the systematic treatment of geometry by the axiomatic method began with the Elements of Euclid. From a small number of explicit axioms, postulates, and definitions Euclid deduces theorems concerning the various figures of geometrical interest. Until the 19th century this work stood as a supreme example of the exercise of reason, which all other intellectual achievements ought to take as a model. With increasing standards of formal rigour it was recognized that Euclid does contain gaps, but fully formalized versions of his geometry have been provided. For example, in the axiomatization of David Hilbert, there are six primitive terms, in that of E. V. Huntington only two: ‘sphere’ and ‘includes’.
In the work of Kant, Euclidean geometry stands as the supreme example of a synthetic a priori construction, representing the way the mind has to think about space, because of the mind's own intrinsic structure. However, only shortly after Kant was writing non-Euclidean geometries were contemplated. They were foreshadowed by the mathematician K. F. Gauss (1777-1855), but the first serious non-Euclidean geometry is usually attributed to the Russian mathematician N. I. Lobachevsky, writing in the 1820s. Euclid's fifth axiom, the axiom of parallels, states that through any point not falling on a straight line, one straight line can be drawn that does not intersect the first. In Lobachevsky's geometry several such lines can exist. Later G. F. B. Riemann (1822-66) realized that the two-dimensional geometry that would be hit upon by persons confined to the surface of a sphere would be different from that of persons living on a plane: for example, π would be smaller, since the diameter of a circle, as drawn on a sphere, is relatively large compared to the circumference. In the figure, BCB, the circumference of the circle, is less than 2π AB, where AB is the radius. Generalizing, Riemann reached the idea of a geometry in which there are no straight lines that do not intersect a given straight line, just as on a sphere all great circles (the shortest distance between two points) intersect.
The way then lay open to separating the question of the mathematical nature of a purely formal geometry from the question of its physical application. In 1854 Riemann showed that space of any curvature could be described by a set of numbers known as its metric tensor. For example, ten numbers suffice to describe the point of any four-dimensional manifold. To apply a geometry means finding coordinative definitions correlating the notions of the geometry, notably those of a straight line and an equal distance, with physical phenomena such as the path of a light ray, or the size of a rod at different times and places. The status of these definitions has been controversial, with some such as Poincaré seeing them simply as conventions, and others seeing them as important empirical truths. With the general rise of holism in the philosophy of science the question of status has abated a little, it being recognized simply that the co-ordination plays a fundamental role in physical science. See also relativity theory, space-time.
Columbia Encyclopedia:
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geometry
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).
Oxford Companion to the Mind:
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geometry
Even the earliest history of geometry as a theoretical study of spatial relationships and spatial structures reveals that it had, in addition, a powerful symbolic role in both imaginative and theoretically speculative thought. Thus, although the history of geometry can be told in purely mathematical terms, it is also, from another point of view, a history of forms of representation, of ways of thinking about the world and of views on the nature of thought itself.
We can never recapture the origins of geometry. Certainly there were in ancient cultures, such as those of the Babylonians and Egyptians, systematically codified empirical procedures relating to land measurement, the construction of temples, and astronomical observation which could be regarded as antecedents. But it is only with the Greeks that a sophisticated theoretical and demonstrative geometry of the kind we find in Euclid's Elements emerges. It would be difficult to overestimate the impact of the emergence of geometry as a deductively organized discipline, one in which it is shown that, for example, the internal angles of a triangle must equal two right angles, given the apparently undeniable truth of the kind of propositions which Euclid adopts as axioms and postulates and the definition of a triangle. Such a demonstration provides knowledge not merely that something happens to be the case, but also an understanding of why it must be so, given the nature of the things concerned. Geometry thus provided, and to some extent continues to provide, the paradigm of what it is to have scientific knowledge or understanding, moulding, for example, Aristotle's discussion in the Posterior Analytics of the demonstration of causes.
But the very feature of geometry that singles it out as providing a conception of the ideal at which all other putative sciences should aim was also, from the outset, the source of philosophical problems concerning the status of geometrical knowledge and its relation to the ever-changing physical world. Many sketches of the history of geometry suggest that the Greeks regarded geometry as a theory of physical space and that they thus thought they had found a way of discovering truths about the physical world by mere contemplation. But this is to project onto the Greeks a way of thinking about geometry, physical space, and their relation, which is a product of the Renaissance and of the new scientific outlook that emerged from it. For while geometry did provide the Greeks and subsequent generations of Western thinkers with their ideal of what it is to have scientific understanding (theoretical knowledge involving demonstrations from accounts of essence), it is clear that geometry itself was not regarded as yielding a scientific understanding of the ever-changing physical world, even though it finds application in that world, especially in the applied mathematical disciplines of astronomy, optics, and harmony; geometry deals with a timeless, unchanging world of pure shapes and sizes (forms).
From this point of view, the problems raised by geometry concern the relation between forms and items in the physical world, how knowledge of forms is possible, and how this knowledge can be so useful in everyday dealings with the physical world. To the Pythagoreans and to Plato the example of mathematics suggested that the knowable, intelligible reality, must be an unchanging realm behind that of changing appearances. Aristotle, on the other hand, insisted that shapes and sizes have no existence except as aspects of physical, changeable things which are the things that have primary reality. The mathematician then deals with the possible shapes and sizes of physical things but qua measurable, not qua changeable, whereas the physicist, being primarily concerned with change and its explanation, seeks an understanding of natural objects qua changeable. Geometry is the science not just of spatial magnitudes but of all continuous magnitudes which, being represented by lines, can then be handled geometrically. Arithmetic is the science of discrete magnitudes.
Thus, far from taking geometry to be the science of physical space, the Greeks tended to see it as the science of continuous magnitudes, essentially concerned with ratios and proportions and with methods of construction enabling these to be determined. The physical space of the Aristotelian universe is not the infinite, homogeneous space of Euclidean geometry, but the highly structured bounded set of nested spheres centred on the earth. It was the cumulative effect of a number of diverse factors operating from the beginning of the Renaissance that led, by the 18th century, to that identification of physical with geometric space which is most clearly apparent in the works of Isaac Newton when he talks of absolute space. Perhaps the two most important of these factors were the coordinate development of geometrical theories of perspective and of geometrical optics. The work of Italian artists such as Leonardo da Vinci gave geometry a role in accounts of the mechanisms of visual perception. The geometrical treatment of problems of perspective, involving two-dimensional representations of three-dimensional spatial relationships, brings with it (a) a tendency to treat geometry as descriptive of both perceptual and physical space, and (b) the development of the methods of projective geometry (needed to handle the various possible projections of three-dimensional solids into two dimensions). In projective geometry there is a shift of emphasis away from figures and their fixed shapes and sizes towards descriptions of the possible types of projections, the behaviour of shapes under such transformations, and a move towards a consideration of the more global, structural properties of spaces. The adoption of the Copernican, sun-centred view of the universe shattered once and for all the crystalline spheres, leaving the earth spinning through an infinite, homogeneous three-dimensional Euclidean space.
But if geometry is to be a theory of physical space then its axioms must be true of this space, and there is a problem of just how it is that we can come to recognize truths about physical space, the space of the world of experience, as necessary truths. To account for this necessity, it would seem that they must be knowable a priori, but without proof, for they are first principles. But then how can any truths about the world of experience be knowable prior to, or independently of, experience? There are two problems to be separated here. First there is a psychological problem of how we come by our grasp of spatial relations and spatial concepts, and of what exactly we do acquire simply by experience. Secondly there is a philosophical problem which concerns not how we in the first instance come by our beliefs about space, but how, if at all, these beliefs can be proved or justified.
So long as geometry was synonymous with Euclidean geometry it was possible to think that we have some innate knowledge, or a faculty of geometric intuition which makes it possible for us to recognize the Euclidean axioms as necessarily and self-evidently true. But Euclid's fifth postulate (which says, in effect, that parallel lines never meet however far they are extended) had never seemed entirely self-evident, and there is a long history of unsuccessful attempts to prove this postulate from the other four. In a work published in 1733, the Italian mathematician Girolamo Saccheri adopted a new strategy which was that of combining the first four of Euclid's postulates with the negation of the fifth, with the aim of deriving a contradiction and thus showing indirectly that the fifth postulate is a logical consequence of the remaining four. Although he thought he had succeeded in doing this, his proofs were faulty, as was shown by Bolyai, Lobachevski, Riemann, and others who almost simultaneously demonstrated the existence of non-Euclidean geometries, ones in which the fifth postulate is false. These were shown to be consistent if Euclidean geometry is consistent, by interpreting them as the geometries of curved surfaces of various kinds (e.g. the surface of a sphere). When Einstein made use of non-Euclidean geometries in his theories of Special and General Relativity, treating space–time as a non-Euclidean four-dimensional space, it was no longer possible to regard the Euclidean axioms as self-evident, necessary truths concerning physical reality. This opened up once again questions concerning the nature and status of geometry and its role in physical theories. Some have argued that the choice of geometry is merely a matter of convention; it is just a question of the form of representation we want to use. Others argue that it is an empirical matter, a question of seeking empirically to determine a correct account of physical space.
It is possible, however, to articulate more sophisticated intermediate positions as a result of the work carried out in pursuit of Felix Klein's Erlangen programme (1872). Klein proposed that every geometry can be defined by specifying its group of transformations, and thereby its invariants. This leads to a hierarchy of geometrical theories of increasing generality: metrical geometry, affine geometry, projective geometry, topology (roughly speaking). Given these distinctions it is possible to ask more precise questions about which parts of the full metrical geometry we use could be regarded as factually constrained and which are a matter of convention.
(Published 1987)
We can never recapture the origins of geometry. Certainly there were in ancient cultures, such as those of the Babylonians and Egyptians, systematically codified empirical procedures relating to land measurement, the construction of temples, and astronomical observation which could be regarded as antecedents. But it is only with the Greeks that a sophisticated theoretical and demonstrative geometry of the kind we find in Euclid's Elements emerges. It would be difficult to overestimate the impact of the emergence of geometry as a deductively organized discipline, one in which it is shown that, for example, the internal angles of a triangle must equal two right angles, given the apparently undeniable truth of the kind of propositions which Euclid adopts as axioms and postulates and the definition of a triangle. Such a demonstration provides knowledge not merely that something happens to be the case, but also an understanding of why it must be so, given the nature of the things concerned. Geometry thus provided, and to some extent continues to provide, the paradigm of what it is to have scientific knowledge or understanding, moulding, for example, Aristotle's discussion in the Posterior Analytics of the demonstration of causes.
But the very feature of geometry that singles it out as providing a conception of the ideal at which all other putative sciences should aim was also, from the outset, the source of philosophical problems concerning the status of geometrical knowledge and its relation to the ever-changing physical world. Many sketches of the history of geometry suggest that the Greeks regarded geometry as a theory of physical space and that they thus thought they had found a way of discovering truths about the physical world by mere contemplation. But this is to project onto the Greeks a way of thinking about geometry, physical space, and their relation, which is a product of the Renaissance and of the new scientific outlook that emerged from it. For while geometry did provide the Greeks and subsequent generations of Western thinkers with their ideal of what it is to have scientific understanding (theoretical knowledge involving demonstrations from accounts of essence), it is clear that geometry itself was not regarded as yielding a scientific understanding of the ever-changing physical world, even though it finds application in that world, especially in the applied mathematical disciplines of astronomy, optics, and harmony; geometry deals with a timeless, unchanging world of pure shapes and sizes (forms).
From this point of view, the problems raised by geometry concern the relation between forms and items in the physical world, how knowledge of forms is possible, and how this knowledge can be so useful in everyday dealings with the physical world. To the Pythagoreans and to Plato the example of mathematics suggested that the knowable, intelligible reality, must be an unchanging realm behind that of changing appearances. Aristotle, on the other hand, insisted that shapes and sizes have no existence except as aspects of physical, changeable things which are the things that have primary reality. The mathematician then deals with the possible shapes and sizes of physical things but qua measurable, not qua changeable, whereas the physicist, being primarily concerned with change and its explanation, seeks an understanding of natural objects qua changeable. Geometry is the science not just of spatial magnitudes but of all continuous magnitudes which, being represented by lines, can then be handled geometrically. Arithmetic is the science of discrete magnitudes.
Thus, far from taking geometry to be the science of physical space, the Greeks tended to see it as the science of continuous magnitudes, essentially concerned with ratios and proportions and with methods of construction enabling these to be determined. The physical space of the Aristotelian universe is not the infinite, homogeneous space of Euclidean geometry, but the highly structured bounded set of nested spheres centred on the earth. It was the cumulative effect of a number of diverse factors operating from the beginning of the Renaissance that led, by the 18th century, to that identification of physical with geometric space which is most clearly apparent in the works of Isaac Newton when he talks of absolute space. Perhaps the two most important of these factors were the coordinate development of geometrical theories of perspective and of geometrical optics. The work of Italian artists such as Leonardo da Vinci gave geometry a role in accounts of the mechanisms of visual perception. The geometrical treatment of problems of perspective, involving two-dimensional representations of three-dimensional spatial relationships, brings with it (a) a tendency to treat geometry as descriptive of both perceptual and physical space, and (b) the development of the methods of projective geometry (needed to handle the various possible projections of three-dimensional solids into two dimensions). In projective geometry there is a shift of emphasis away from figures and their fixed shapes and sizes towards descriptions of the possible types of projections, the behaviour of shapes under such transformations, and a move towards a consideration of the more global, structural properties of spaces. The adoption of the Copernican, sun-centred view of the universe shattered once and for all the crystalline spheres, leaving the earth spinning through an infinite, homogeneous three-dimensional Euclidean space.
But if geometry is to be a theory of physical space then its axioms must be true of this space, and there is a problem of just how it is that we can come to recognize truths about physical space, the space of the world of experience, as necessary truths. To account for this necessity, it would seem that they must be knowable a priori, but without proof, for they are first principles. But then how can any truths about the world of experience be knowable prior to, or independently of, experience? There are two problems to be separated here. First there is a psychological problem of how we come by our grasp of spatial relations and spatial concepts, and of what exactly we do acquire simply by experience. Secondly there is a philosophical problem which concerns not how we in the first instance come by our beliefs about space, but how, if at all, these beliefs can be proved or justified.
So long as geometry was synonymous with Euclidean geometry it was possible to think that we have some innate knowledge, or a faculty of geometric intuition which makes it possible for us to recognize the Euclidean axioms as necessarily and self-evidently true. But Euclid's fifth postulate (which says, in effect, that parallel lines never meet however far they are extended) had never seemed entirely self-evident, and there is a long history of unsuccessful attempts to prove this postulate from the other four. In a work published in 1733, the Italian mathematician Girolamo Saccheri adopted a new strategy which was that of combining the first four of Euclid's postulates with the negation of the fifth, with the aim of deriving a contradiction and thus showing indirectly that the fifth postulate is a logical consequence of the remaining four. Although he thought he had succeeded in doing this, his proofs were faulty, as was shown by Bolyai, Lobachevski, Riemann, and others who almost simultaneously demonstrated the existence of non-Euclidean geometries, ones in which the fifth postulate is false. These were shown to be consistent if Euclidean geometry is consistent, by interpreting them as the geometries of curved surfaces of various kinds (e.g. the surface of a sphere). When Einstein made use of non-Euclidean geometries in his theories of Special and General Relativity, treating space–time as a non-Euclidean four-dimensional space, it was no longer possible to regard the Euclidean axioms as self-evident, necessary truths concerning physical reality. This opened up once again questions concerning the nature and status of geometry and its role in physical theories. Some have argued that the choice of geometry is merely a matter of convention; it is just a question of the form of representation we want to use. Others argue that it is an empirical matter, a question of seeking empirically to determine a correct account of physical space.
It is possible, however, to articulate more sophisticated intermediate positions as a result of the work carried out in pursuit of Felix Klein's Erlangen programme (1872). Klein proposed that every geometry can be defined by specifying its group of transformations, and thereby its invariants. This leads to a hierarchy of geometrical theories of increasing generality: metrical geometry, affine geometry, projective geometry, topology (roughly speaking). Given these distinctions it is possible to ask more precise questions about which parts of the full metrical geometry we use could be regarded as factually constrained and which are a matter of convention.
(Published 1987)
— Mary Elizabeth Tiles
- Bibliography
- Boyer, C. B. (1968). A History of Mathematics.
- Koyré, A. (1957). From the Closed World to the Infinite Universe.
- Lanczos, C. (1970). Space through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein.
- Nerlich, G. (1976). The Shape of Space.
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geometer
IN BRIEF: n. - A mathematician specializing in a branch of mathematics that deals with measurement, properties, and relationships of points, lines, angles, surfaces, and solids.
Euclid was a renown geometer.
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The branch of mathematics that treats the properties, measurement, and relations of points, lines, angles, surfaces, and solids. (See Euclid and plane geometry.)
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categories related to 'geometry'
- Branches and Computational Systems - geometry: branch of mathematics that deals with deduction of properties, measurement, and relationships of points, lines, angles, surfaces, and solid figures in space
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Geometry
For other uses, see Geometry (disambiguation).
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metria "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.
While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).[2]
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Overview
The recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.
Practical geometry
Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.
Axiomatic geometry
See also: Euclidean geometry
Euclid took a more abstract approach in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of the 19th century the discovery of non-Euclidean geometries by Gauss, Lobachevsky, Bolyai, and others led to a revival of interest, and in the 20th century David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
Geometric constructions
Main article: Compass and straightedge constructions
Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
Numbers in geometry
In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon abstract numbers in favor of concrete geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation, and whom the Cartesian plane is named after. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.
Geometry of position
Main articles: Projective geometry and Topology
Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.
Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.
Geometry beyond Euclid
For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),[4] published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
Dimension
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory geometric reason for the significance of 10 and 11 dimensions.
Symmetry
The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in projective geometry (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.
Modern geometry
Modern geometry is the title of a popular textbook by Dubrovin, Novikov and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.
History of geometry
Main article: History of geometry
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.[5][6] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000-1800 BC) and Moscow Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.[7] South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.[8][9]
In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.[10] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,[11] though the statement of the theorem has a long history[12][13] Eudoxus (408–c.355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[14] as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,[15] introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[16] The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.[17] Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.[18] He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.
In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry[19][page needed] and geometric algebra.[20] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[21] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[22] Omar Khayyám (1048–1131) found geometric solutions to cubic equations.[23] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c.1230–c.1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[24]
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is a geometry without measurement or parallel lines, just the study of how points are related to each other.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860) and Carl Friedrich Gauss (1777–1855) and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
Contemporary geometry
Euclidean geometry
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
Differential geometry
Differential geometry has been of increasing importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved. Contemporary differential geometry is intrinsic, meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point, and not a priori parts of some ambient flat Euclidean space.
Topology and geometry
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.
Algebraic geometry
The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.
The study of low dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.
See also
Lists
- List of geometers
- List of geometry topics
- List of important publications in geometry
- List of mathematics articles
Related topics
- Flatland, a book written by Edwin Abbott Abbott about two- and three-dimensional space, to understand the concept of four dimensions
- Interactive geometry software
- Why 10 dimensions?
- Shulba Sutras
- Trigonometry
References
Notes
- ^ Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews (1998). "Fractal geometry in digital imaging". Academic Press. p.1. ISBN 0-12-703970-8
- ^ It is quite common in algebraic geometry to speak about geometry of algebraic varieties over finite fields, possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary spheres or cones.
- ^ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
- ^ http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/
- ^ J. Friberg, "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.
- ^ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-0-486-22332-2. http://books.google.com/?id=JVhTtVA2zr8C. Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.
- ^ (Boyer 1991, "Egypt" p. 19)
- ^ The Journal of Egyptian Archaeology. Vol. 84, 1998 Gnomons at Meroë and Early Trigonometry. pg. 171
- ^ Neolithic Skywatchers. May 27, 1998 by Andrew L. Slayman Archaeology.org
- ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
- ^ Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0.
- ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
- ^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal.
- ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 92)
- ^ (Boyer 1991, "Euclid of Alexandria" p. 119)
- ^ (Boyer 1991, "Euclid of Alexandria" p. 104)
- ^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible, has been more widely used...."
- ^ O'Connor, J.J. and Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html. Retrieved 2007-08-07.
- ^ R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, London
- ^ Boyer (1991). "The Arabic Hegemony". pp. 241–242. "Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""
- ^ O'Connor, John J.; Robertson, Edmund F., "Al-Mahani", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Mahani.html.
- ^ O'Connor, John J.; Robertson, Edmund F., "Al-Sabi Thabit ibn Qurra al-Harrani", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Thabit.html.
- ^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html.
- ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
Sources
- Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
- Nikolai I. Lobachevsky, Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society, 2010.
Bibliography
- Mlodinow, M.; Euclid's window (the story of geometry from parallel lines to hyperspace), UK edn. Allen Lane, 1992.
External links
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- "4000 Years of Geometry", lecture by Robin Wilson given at Gresham College, 3 October 2007 (available for MP3 and MP4 download as well as a text file)
- Finitism in Geometry at the Stanford Encyclopedia of Philosophy
- The Geometry Junkyard
- Interactive Geometry Applications (Java and Cabri 3D)
- Interactive geometry reference with hundreds of applets
- Dynamic Geometry Sketches (with some Student Explorations)
- Geometry classes at Khan Academy
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Translations:
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Geometry
Nederlands (Dutch)
meetkunde, relatieve rangschikking van delen/ voorwerpen
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n. - géométrie
Deutsch (German)
n. - Geometrie
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n. - γεωμετρία, γεωμετρική διάταξη αντικειμένων
Português (Portuguese)
n. - geometria (f)
Español (Spanish)
n. - geometría
Svenska (Swedish)
n. - geometri
中文（简体）(Chinese (Simplified))
几何学
中文（繁體）(Chinese (Traditional))
n. - 幾何學
العربيه (Arabic)
(الاسم) هندسه
עברית (Hebrew)
n. - הנדסת-המישור, גיאומטריה
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