Important concepts in
geometry
The following are some of the most important concepts in
geometry.
Axioms
Euclid took an abstract approach to geometry in his
Elements, one of the most influential books ever
written. Euclid introduced certain axioms, or
postulates, expressing primary or self-evident
properties of points, lines, and planes. He proceeded to
rigorously deduce other properties by mathematical
reasoning. The characteristic feature of Euclid's
approach to geometry was its rigor, and it has come to
be known as axiomatic or synthetic geometry. At the
start of the 19th century, the discovery of
non-Euclidean geometries by Nikolai Ivanovich
Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl
Friedrich Gauss (1777–1855) and others led to a revival
of interest in this discipline, and in the 20th century,
David Hilbert (1862–1943) employed axiomatic reasoning
in an attempt to provide a modern foundation of
geometry.
Points
Points are considered fundamental objects in Euclidean
geometry. They have been defined in a variety of ways,
including Euclid's definition as 'that which has no
part' and through the use of algebra or nested sets. In
many areas of geometry, such as analytic geometry,
differential geometry, and topology, all objects are
considered to be built up from points. However, there
has been some study of geometry without reference to
points.
Lines
Euclid described a line as "breadthless length" which
"lies equally with respect to the points on itself". In
modern mathematics, given the multitude of geometries,
the concept of a line is closely tied to the way the
geometry is described. For instance, in analytic
geometry, a line in the plane is often defined as the
set of points whose coordinates satisfy a given linear
equation, but in a more abstract setting, such as
incidence geometry, a line may be an independent object,
distinct from the set of points which lie on it. In
differential geometry, a geodesic is a generalization of
the notion of a line to curved spaces.
Planes
A plane is a flat, two-dimensional surface that extends
infinitely far. Planes are used in every area of
geometry. For instance, planes can be studied as a
topological surface without reference to distances or
angles; it can be studied as an affine space, where
collinearity and ratios can be studied but not
distances; it can be studied as the complex plane using
techniques of complex analysis; and so on. |
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Angles
Euclid defines a plane angle as the inclination to each
other, in a plane, of two lines which meet each other,
and do not lie straight with respect to each other. In
modern terms, an angle is the figure formed by two rays,
called the sides of the angle, sharing a common
endpoint, called the vertex of the angle.
In Euclidean geometry, angles are used to study polygons
and triangles, as well as forming an object of study in
their own right. The study of the angles of a triangle
or of angles in a unit circle forms the basis of
trigonometry.
In differential geometry and calculus, the angles
between plane curves or space curves or surfaces can be
calculated using the derivative.
Curves
A curve is a 1-dimensional object that may be straight
(like a line) or not; curves in 2-dimensional space are
called plane curves and those in 3-dimensional space are
called space curves.
In topology, a curve is defined by a function from an
interval of the real numbers to another space. In
differential geometry, the same definition is used, but
the defining function is required to be differentiable
Algebraic geometry studies algebraic curves, which are
defined as algebraic varieties of dimension one.
Surfaces
A surface is a two-dimensional object, such as a sphere
or paraboloid. In differential geometry and topology,
surfaces are described by two-dimensional 'patches' (or
neighborhoods) that are assembled by diffeomorphisms or
homeomorphisms, respectively. In algebraic geometry,
surfaces are described by polynomial equations.
Manifolds
A manifold is a generalization of the concepts of curve
and surface. In topology, a manifold is a topological
space where every point has a neighborhood that is
homeomorphic to Euclidean space. In differential
geometry, a differentiable manifold is a space where
each neighborhood is diffeomorphic to Euclidean space.
Manifolds are used extensively in physics, including in
general relativity and string theory
Topologies and metrics
A topology is a mathematical structure on a set that
tells how elements of the set relate spatially to each
other. The best-known examples of topologies come from
metrics, which are ways of measuring distances between
points. For instance, the Euclidean metric measures the
distance between points in the Euclidean plane, while
the hyperbolic metric measures the distance in the
hyperbolic plane. Other important examples of metrics
include the Lorentz metric of special relativity and the
semi-Riemannian metrics of general relativity.
Compass and straightedge
constructions
Classical geometers paid special attention to
constructing geometric objects that had been described
in some other way. Classically, the only instruments
allowed in geometric constructions are the compass and
straightedge. Also, every construction had to be
complete in a finite number of steps. However, some
problems turned out to be difficult or impossible to
solve by these means alone, and ingenious constructions
using parabolas and other curves, as well as mechanical
devices, were found.
Dimension
Where the traditional geometry allowed dimensions 1 (a
line), 2 (a plane) and 3 (our ambient world conceived of
as three-dimensional space), mathematicians have used
higher dimensions for nearly two centuries. Dimension
has gone through stages of being any natural number n,
possibly infinite with the introduction of Hilbert
space, and any positive real number in fractal geometry.
Dimension theory is a technical area, initially within
general topology, that discusses definitions; in common
with most mathematical ideas, dimension is now defined
rather than an intuition. Connected topological
manifolds have a well-defined dimension; this is a
theorem (invariance of domain) rather than anything a
priori.
The issue of dimension still matters to geometry, in the
absence of complete answers to classic questions.
Dimensions 3 of space and 4 of space-time are special
cases in geometric topology. Dimension 10 or 11 is a key
number in string theory. Research may bring a
satisfactory geometric reason for the significance of 10
and 11 dimensions.
Symmetry
The theme of symmetry in geometry is nearly as old as
the science of geometry itself. Symmetric shapes such as
the circle, regular polygons and platonic solids held
deep significance for many ancient philosophers and were
investigated in detail before the time of Euclid.
Symmetric patterns occur in nature and were artistically
rendered in a multitude of forms, including the graphics
of M. C. Escher. Nonetheless, it was not until the
second half of 19th century that the unifying role of
symmetry in foundations of geometry was recognized.
Felix Klein's Erlangen program proclaimed that, in a
very precise sense, symmetry, expressed via the notion
of a transformation group, determines what geometry is.
Symmetry in classical Euclidean geometry is represented
by congruences and rigid motions, whereas in projective
geometry an analogous role is played by collineations,
geometric transformations that take straight lines into
straight lines. However it was in the new geometries of
Bolyai and Lobachevsky, Riemann, Clifford and Klein, and
Sophus Lie that Klein's idea to 'define a geometry via
its symmetry group' proved most influential. Both
discrete and continuous symmetries play prominent roles
in geometry, the former in topology and geometric group
theory, the latter in Lie theory and Riemannian
geometry.
A different type of symmetry is the principle of duality
in projective geometry (see Duality (projective
geometry)) among other fields. This meta-phenomenon can
roughly be described as follows: in any theorem,
exchange point with plane, join with meet, lies in with
contains, and you will get an equally true theorem. A
similar and closely related form of duality exists
between a vector space and its dual space.
Non-Euclidean geometry
In the nearly two thousand years since Euclid, while the
range of geometrical questions asked and answered
inevitably expanded, the basic understanding of space
remained essentially the same. Immanuel Kant argued that
there is only one, absolute, geometry, which is known to
be true a priori by an inner faculty of mind: Euclidean
geometry was synthetic a priori. This dominant view was
overturned by the revolutionary discovery of
non-Euclidean geometry in the works of Bolyai,
Lobachevsky, and Gauss (who never published his theory).
They demonstrated that ordinary Euclidean space is only
one possibility for development of geometry. A broad
vision of the subject of geometry was then expressed by
Riemann in his 1867 inauguration lecture Über die
Hypothesen, welche der Geometrie zu Grunde liegen (On
the hypotheses on which geometry is based), published
only after his death. Riemann's new idea of space proved
crucial in Einstein's general relativity theory, and
Riemannian geometry, that considers very general spaces
in which the notion of length is defined, is a mainstay
of modern geometry. |
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 Kiddle: Geometry
Wikipedia: Geometry |
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