The continuous development of topology dates from 1911, when the Dutch mathematician L.E.J. 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. Instead, they discovered that consistent non-Euclidean geometries exist. Non-Euclidean geometriesīeginning 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. 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. Using differential calculus, he characterized the intrinsic properties of curves and surfaces. The German mathematician Carl Friedrich Gauss (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry. 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. Get a Britannica Premium subscription and gain access to exclusive content. For information on specific branches of geometry, see Euclidean geometry, analytic geometry, projective geometry, differential geometry, non-Euclidean geometries, and topology. This article begins with a brief guidepost to the major branches of geometry and then proceeds to an extensive historical treatment. 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. Geometry, the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space. © Massachusetts Institute of Technology ( A Britannica Publishing Partner) See all videos for this article Britannica Explains In these videos, Britannica explains a variety of topics and answers frequently asked questions.Įxploring how civil and environmental engineers use geometry to study processes of deformation in projects of various scales.This Time in History In these videos, find out what happened this month (or any month!) in history.#WTFact Videos In #WTFact Britannica shares some of the most bizarre facts we can find.Demystified Videos In Demystified, Britannica has all the answers to your burning questions.Britannica Classics Check out these retro videos from Encyclopedia Britannica’s archives.