Geometry is the branch of mathematics that studies shape, size, the positions of figures relative to one another, and the properties of space. The word comes from the Greek geō ("earth") and metron ("measure"), which reflects its early use in land surveying and building. Geometry is one of the oldest mathematical disciplines. It began as a set of practical rules for measuring fields, volumes and angles. In ancient Greece it became a deductive science, and it is now a large group of fields that includes Euclidean, non-Euclidean, analytic, projective and differential geometry, as well as topology. Its ideas are central to physics, engineering, architecture, cartography and computer science.
Origins
The earliest recorded geometry was practical. In ancient Egypt, scribes calculated the areas of fields and the volumes of granaries and pyramids. The Rhind Mathematical Papyrus, from about the 16th century BCE, includes a method for approximating the area of a circle. In Mesopotamia, Babylonian scribes knew the relationship now called the Pythagorean theorem more than a thousand years before Pythagoras, and they recorded sets of numbers that fit it on clay tablets. In China, the Nine Chapters on the Mathematical Art, compiled by the time of the Han dynasty, dealt with areas, volumes and right triangles. The 3rd-century commentator Liu Hui later improved the estimate of π by inscribing polygons in a circle.
Greek geometry and Euclid
In ancient Greece, geometry became a subject in which results had to be justified by argument. Tradition credits Thales and the Pythagoreans with the first proofs of geometric statements. Around 300 BCE, Euclid wrote the Elements. Its thirteen books derive hundreds of propositions about plane and solid figures, ratios and numbers from a small set of definitions, common notions and five postulates. The Elements was the standard model of deductive reasoning for more than two thousand years, and it is often described as one of the most influential textbooks ever written.
Euclid's fifth postulate, the parallel postulate, was clearly different from the other four. In the form later popularised by Playfair, it says that through a point not on a given line there is exactly one line parallel to it. Euclid avoided using it for as long as he could: the first 28 propositions of the Elements are proved without it. Later Greek mathematicians extended geometry further. Archimedes calculated areas and volumes bounded by curves and gave upper and lower limits for π. Apollonius of Perga wrote a systematic treatise on conic sections, meaning ellipses, parabolas and hyperbolas.
Medieval and early modern developments
Scholars working under the Abbasid Caliphate and its successor states translated, preserved and extended Greek geometry. They developed trigonometry for use in astronomy. Ibn al-Haytham, Omar Khayyam and Nasir al-Din al-Tusi each attempted to prove the parallel postulate, and in doing so they studied quadrilaterals whose properties later turned out to belong to non-Euclidean geometry. During the Renaissance, painters and architects worked out the mathematics of linear perspective. In the 17th century, Girard Desargues built on that work to create the beginnings of projective geometry.
In 1637 René Descartes published La Géométrie. Together with independent work by Pierre de Fermat, it founded analytic geometry, in which points are described by coordinates and curves by equations. This joined geometry with algebra and made the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz possible. Calculus in turn gave new ways to study tangents, curvature and arc length.
Non-Euclidean geometry
In the 18th century, Giovanni Saccheri and Johann Lambert tried to prove the parallel postulate by contradiction. Without realising it, they derived many theorems of a different geometry. In the early 19th century, Carl Friedrich Gauss, János Bolyai and Nikolai Lobachevsky independently concluded that a consistent geometry could be built by replacing the parallel postulate. In this geometry, now called hyperbolic, more than one parallel passes through a given point, and the angles of a triangle add up to less than 180°. In 1854 Bernhard Riemann gave a lecture that generalised geometry to curved spaces of any dimension, which he called manifolds, and introduced a geometry in which there are no parallel lines at all. In 1868 Eugenio Beltrami built a model in which Euclid's first four postulates hold and the fifth fails. This settled the old question: the parallel postulate cannot be proved from the other four. Together these results came to be known as non-Euclidean geometry. They also raised philosophical questions about whether the geometry of physical space is known a priori, which mattered for epistemology and the philosophy of science.
Unification and foundations
By the 1870s there were many different geometries, and mathematicians looked for a way to relate them. In his Erlangen Program of 1872, Felix Klein proposed defining each geometry by a group of transformations and the properties that stay unchanged under those transformations. On this view, Euclidean, hyperbolic and elliptic geometries can all be treated within projective geometry. In 1899 David Hilbert published Grundlagen der Geometrie (Foundations of Geometry), which gave Euclidean geometry a complete and rigorous set of axioms. It shaped the modern axiomatic approach in mathematics and in logic.
Major branches
- Euclidean geometry: points, lines, circles, polygons and polyhedra in flat space, including congruence, similarity and constructions with compass and straightedge.
- Analytic and algebraic geometry: geometric objects defined by equations. Algebraic geometry studies the solution sets of polynomial equations and is closely connected to number theory.
- Differential geometry: curves, surfaces and manifolds studied with calculus. It grew out of Gauss's 1827 work on the curvature of surfaces and Riemann's metric geometry.
- Topology: properties that stay the same under continuous deformation, such as connectedness and the number of holes. Its origins are often traced to Leonhard Euler's 1736 solution of the Königsberg bridges problem.
- Projective, affine and convex geometry, as well as discrete and computational geometry, which have applications in computer graphics and robotics.
Applications
Geometry is used throughout science and technology. Surveying, navigation and astronomy depend on trigonometry and spherical geometry. Optics models light rays geometrically. In Albert Einstein's general theory of relativity, gravity is described as the curvature of four-dimensional spacetime, using Riemannian geometry. Modern physics, including gauge theories and string theory, relies on differential geometry and topology. Practical uses include engineering design, architecture, computer-aided design, computer vision, satellite positioning and medical imaging.
References
- Non-Euclidean geometry - MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
- Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy/Fall 2016 Edition)plato.stanford.edu
- Felix Klein: A Legacy of Innovation in Mathematics and Educationdcn.nat.fau.eu
- non-euclidean - geometryfiles.eric.ed.gov
- Illustrations of non-Euclidean geometry in virtual realityarxiv.org
- History Of Non Euclidean Geometryslideshare.net