Difference between revisions of "Geometry"

Latest revision as of 06:02, 3 April 2016

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, volume, relative position of figures, and the properties of space.

History

Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC).

Euclid

By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment -- Euclidean geometry -- set a standard for many centuries to follow.

Archimedes

Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus.

Astronomy

The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.

Quadrivium

In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

Coordinates, algebra

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century.

Projective geometry

The theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.

Topology, Differential geometry

The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

Distinction between physical and geometrical space

In Euclid's time, there was no clear distinction between physical and geometrical space.

Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space.

Formal mathematics

With the rise of formal mathematics in the 20th century, 'space' (whether 'point', 'line', or 'plane') lost its intuitive contents, so today one has to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meanings) and abstract spaces.

Contemporary geometry

Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales.

These spaces may be endowed with additional structure which allow one to speak about length.

Modern geometry

Modern geometry has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry and general relativity.

Exotic applications

While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance.

Examples include:

• Fractal geometry
• Algebraic geometry

See also

• Analytic geometry
• Circle
• Close-packing of equal spheres
• Computer graphics
• Coordinate system
• Degrees of freedom
• Descriptive geometry
• Differential geometry
• Differential geometry of surfaces
• Link (geometry)
• Mathematical notation
• Mathematical object
• Mathematics
• Measurement
• Pattern
• Perspective (graphical)
• Pi
• Scaling (geometry)
• Sphere
• Simplicial complex
• Tensor
• Topology
• Two-dimensional
• Vertex (geometry)
• Visual arts
• Weaire–Phelan structure

External links

• Geometry @ Wikipedia