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Geometry 𓆩⚪𓆪｜Definition｜1st｜20251119205401-00-⌔
Geometry
Geometry12 is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.3 Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry,4 which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.5
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics.6 Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles’s proof of Fermat’s Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
During the 19th century, several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss’s Theorema Egregium (“remarkable theorem”) that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word “space”, which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined.
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Link to original Footnotes
(from Ancient Greek γεωμετρία * (geōmetría)* ‘land measurement’; from γῆ * (gê)* ‘earth, land’ and μέτρον * (métron)*‘a measure’) ↩
“Geometry - Formulas, Examples | Plane and Solid Geometry”. Cuemath. Retrieved 31 August 2023. ↩
Vincenzo De Risi (2015). Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age. Birkhäuser. pp. 1–. ISBN 978-3-319-12102-4. Archived from the original on 20 February 2021. Retrieved 14 September 2019. ↩
Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development of hyperbolic geometry by Lobachevsky and other non-Euclidean geometries by Gauss and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of Desargues in the 17th century, all the way back to the implicit use of spherical geometry to understand the Earth’s geodesy and to navigate the oceans since antiquity. ↩
Tabak, John (2014). Geometry: the language of space and form. Infobase Publishing. p. xiv. ISBN 978-0-8160-4953-0. ↩
Walter A. Meyer (2006). Geometry and Its Applications. Elsevier. ISBN 978-0-08-047803-6. Archived from the original on 1 September 2021. Retrieved 14 September 2019. ↩
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