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v. t. e. Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land', and μέτρον (métron) 'a measure') [1] is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. [2] Geometry is, along with arithmetic, one of ...
- Overview
- Euclidean geometry
- Analytic geometry
- Projective geometry
- Differential geometry
- Non-Euclidean geometries
- Topology
- History of geometry
- Ancient geometry: practical and empirical
geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning “Earth measurement.” Eventually it was realized that geometry need not be limited to the study of flat surfaces (plane geometry) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms.
This article begins with a brief guidepost to the major branches of geometry and then proceeds to an extensive historical treatment. For information on specific branches of geometry, see Euclidean geometry, analytic geometry, projective geometry, differential geometry, non-Euclidean geometries, and topology.
In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. The Elements epitomized the ax...
Analytic geometry was initiated by the French mathematician René Descartes (1596–1650), who introduced rectangular coordinates to locate points and to enable lines and curves to be represented with algebraic equations. Algebraic geometry is a modern extension of the subject to multidimensional and non-Euclidean spaces.
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Projective geometry originated with the French mathematician Girard Desargues (1591–1661) to deal with those properties of geometric figures that are not altered by projecting their image, or “shadow,” onto another surface.
The German mathematician Carl Friedrich Gauss (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry. Using differential calculus, he characterized the intrinsic properties of curves and surfaces. For instance, he showed that the intrinsic curvature of a cylinder is the same as that of a plane, as can be seen by cutting a cylinder along its axis and flattening, but not the same as that of a sphere, which cannot be flattened without distortion.
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Beginning in the 19th century, various mathematicians substituted alternatives to Euclid’s parallel postulate, which, in its modern form, reads, “given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.” They hoped to show that the alternatives were logically impossible. Instead...
Topology, the youngest and most sophisticated branch of geometry, focuses on the properties of geometric objects that remain unchanged upon continuous deformation—shrinking, stretching, and folding, but not tearing. The continuous development of topology dates from 1911, when the Dutch mathematician L.E.J. Brouwer (1881–1966) introduced methods gen...
The earliest known unambiguous examples of written records—dating from Egypt and Mesopotamia about 3100 bce—demonstrate that ancient peoples had already begun to devise mathematical rules and techniques useful for surveying land areas, constructing buildings, and measuring storage containers. Beginning about the 6th century bce, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, from the combination of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth.
In addition to describing some of the achievements of the ancient Greeks, notably Euclid’s logical development of geometry in the Elements, this article examines some applications of geometry to astronomy, cartography, and painting from classical Greece through medieval Islam and Renaissance Europe. It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in the modern age.
The origin of geometry lies in the concerns of everyday life. The traditional account, preserved in Herodotus’s History (5th century bce), credits the Egyptians with inventing surveying in order to reestablish property values after the annual flood of the Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to evaluate tribute, store oil and grain, and build dams and pyramids. Even the three abstruse geometrical problems of ancient times—to double a cube, trisect an angle, and square a circle, all of which will be discussed later—probably arose from practical matters, from religious ritual, timekeeping, and construction, respectively, in pre-Greek societies of the Mediterranean. And the main subject of later Greek geometry, the theory of conic sections, owed its general importance, and perhaps also its origin, to its application to optics and astronomy.
While many ancient individuals, known and unknown, contributed to the subject, none equaled the impact of Euclid and his Elements of geometry, a book now 2,300 years old and the object of as much painful and painstaking study as the Bible. Much less is known about Euclid, however, than about Moses. In fact, the only thing known with a fair degree of confidence is that Euclid taught at the Library of Alexandria during the reign of Ptolemy I (323–285/283 bce). Euclid wrote not only on geometry but also on astronomy and optics and perhaps also on mechanics and music. Only the Elements, which was extensively copied and translated, has survived intact.
- Performing transformations. Intro to Euclidean geometry: Performing transformations Introduction to rigid transformations: Performing transformations Translations: Performing transformations.
- Transformation properties and proofs. Rigid transformations overview: Transformation properties and proofs Dilation preserved properties: Transformation properties and proofs Properties & definitions of transformations: Transformation properties and proofs.
- Congruence. Transformations & congruence: Congruence Triangle congruence from transformations: Congruence Congruent triangles: Congruence Theorems concerning triangle properties: Congruence.
- Similarity. Definitions of similarity: Similarity Introduction to triangle similarity: Similarity Solving similar triangles: Similarity. Angle bisector theorem: Similarity Solving problems with similar & congruent triangles: Similarity Proving relationships using similarity: Similarity Solving modeling problems with similar & congruent triangles: Similarity.
Interactive, free online geometry tool from GeoGebra: create triangles, circles, angles, transformations and much more!
Aprende geometría con este curso gratuito y online que abarca desde líneas rectas hasta geometría analítica. Explora conceptos, teoremas, demostraciones, ejemplos y ejercicios con videos y actividades interactivas.
Learn about geometry, the branch of math that studies shapes and their properties. Explore plane and solid geometry, angles, polygons, circles, conic sections, transformations, symmetry, tessellations and more with interactive activities and examples.
Aprende los conceptos básicos de geometría con este curso interactivo y gratuito. Explora las unidades sobre líneas, ángulos, figuras, plano coordenado, área, volumen, teorema de Pitágoras y transformaciones.
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