Euclid's The Elements of Geometry (c.300 BCE), was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry.
In the early 17th century, there were two important developments in geometry. The first and mostm iportant was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596–1650) and Pierre de Fermat (1601–1665). The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other.
Geometry is still feeling the effects of two developments from the nineteenth century. Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.
As a consequence of these major changes in the conception of geometry, the concept of 'space' became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics. The traditional type of geometry was recognised as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same
Contemporary geometrySome of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov, and William Thurston. Geometry now is, in large part, the study ofstructures on manifolds, that have a geometric meaning in the sense of the principle of covariance that lies at the root of general relativity theory, in theoretical physics.
Contemporary Euclidean geometry
The study of traditional Euclidean geometry is by no means dead. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean group of rigid motions. The fundamental formulae of geometry, such as the Pythagorean theorem, can be presented in this way for a general inner product space.
Euclidean geometry has become closely connected with computational geometry, computer graphics, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
Algebraic geometryThe field of algebraic geometry is the modern incarnation of the
Cartesian geometry of co-ordinates.
The geometric style which was traditionally called the Italian school is now known as birational geometry. Objects from algebraic geometry are now commonly applied in string theory, as well as diophantine geometry.
Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra. For practical applications, Gröbner basis theory and real algebraic geometry are major subfields.
Differential geometryDifferential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Contemporary differential geometry isintrinsic, meaning that space is
a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point. Topology and geometry
The field of topology, which saw massive development in the twentieth century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. Contemporary geometric topology and differential topology, and particular
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