Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries) can be developed without introducing any contradiction. 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. One of the oldest such discoveries is Carl Friedrich Gauss' Theorema Egregium code: lat promoted to code: la ('remarkable theorem') that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space.

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.ĭuring the 19th century several discoveries enlarged dramatically the scope of geometry. Geometry also has applications in areas of mathematics that are apparently unrelated. 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. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

A mathematician who works in the field of geometry is called a geometer. Geometry is, along with arithmetic, one of the oldest branches of mathematics.

Geometry (from Ancient Greek γεωμετρία ( geōmetría) 'land measurement' from γῆ ( gê) 'earth, land', and μέτρον ( métron) 'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

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