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Quadrivium Geometry

Objective   3/1/2016

Documentation on the Quadrivium Geometry.

Geometry   Math Open Reference   6/14/2016

Wikipedia   Math Open Reference

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

A mathematician who works in the field of geometry is called a geometer.

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 treatment—Euclidean geometry—set a standard for many centuries to follow.

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

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 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.

Furthermore, 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.

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).

Foucault pendulum   3/15/2016


The Foucault pendulum or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth.

While it had long been known that the Earth rotates, the introduction of the Foucault pendulum in 1851 was the first simple proof of the rotation in an easy-to-see experiment.

Today, Foucault pendulums are popular displays in science museums and universities.

Geometric modeling   3/16/2016


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)   3/20/2016

Wikipedia   Archimedes Geo3D

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.

Pendulum   3/8/2016

Wikipedia   Leon Foucault

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 seismometers.

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 friction.

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.

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