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).
By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatmentâ€”Euclidean geometryâ€”set a
standard for many centuries to follow.
Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern
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.
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.
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.
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.
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 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 has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry
and general relativity.
One of the youngest physical theories, string theory, is also very geometric in flavour.
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 (for
example, in fractal geometry and algebraic geometry).
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and
algorithms for the mathematical description of shapes.
The shapes studied in geometric modeling are mostly two- or three-dimensional, although many of its tools and
principles can be applied to sets of any finite dimension.
Today most geometric modeling is done with computers and for computer-based applications.
Two-dimensional models are important in computer typography and technical drawing.
Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM), and widely used in many
applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing.
Geometric models are usually distinguished from procedural and object-oriented models, which define the shape
implicitly by an opaque algorithm that generates its appearance.
They are also contrasted with digital images and volumetric models which represent the shape as a subset of a fine
regular partition of space; and with fractal models that give an infinitely recursive definition of the shape.
However, these distinctions are often blurred: for instance, a digital image can be interpreted as a collection of
colored squares; and geometric shapes such as circles are defined by implicit mathematical equations. Also, a fractal
model yields a parametric or implicit model when its recursive definition is truncated to a finite depth.
Notable awards of the area are the John A. Gregory Memorial Award and the Bezier award.
Interactive geometry software (IGS, or dynamic geometry environments, DGEs)
Interactive geometry software (IGS, or dynamic geometry environments, DGEs) are computer programs which
allow one to create and then manipulate geometric constructions, primarily in plane geometry.
In most IGS, one starts construction by putting a few points and using them to define new objects such as lines,
circles or other points.
After some Yconstruction is done, one can move the points one started with and see how the construction changes.
A pendulum is a weight suspended from a pivot so that it can swing freely.
When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due
to gravity that will accelerate it back toward the equilibrium position.
When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium
position, swinging back and forth.
The time for one complete cycle, a left swing and a right swing, is called the period.
The period depends on the length of the pendulum, and also to a slight degree on the amplitude, the width of
the pendulum's swing.
From its examination in around 1602 by Galileo Galilei, the regular motion of pendulums was used for
timekeeping, and was the world's most accurate timekeeping technology until the 1930s.
Pendulums are used to regulate pendulum clocks, and are used in scientific instruments such as accelerometers and
Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even
as a standard of length.
The word "pendulum" is new Latin, from the Latin pendulus, meaning 'hanging'.
The simple gravity pendulum is an idealized mathematical model of a pendulum.
This is a weight (or bob) on the end of a massless cord suspended from a pivot, without
When given an initial push, it will swing back and forth at a constant amplitude.
Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.