Direction (geometry)

Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference (the violins in a full orchestra are typically seated to the left of the conductor), or absolute according to some previously agreed upon frame of reference (New York City lies due west of Madrid). Direction is often indicated manually by an extended index finger or written as an arrow. On a vertically oriented sign representing a horizontal plane, such as a road sign, "forward" is usually indicated by an upward arrow. Mathematically, direction may be uniquely specified by a unit vector, or equivalently by the angles made by the most direct path with respect to a specified set of axes.

See also

Geometry

Geometry (from the Ancient Greek: γεωμετρία; 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.

Geometry (Ivo Perelman album)

Geometry is an album by Brazilian jazz saxophonist Ivo Perelman featuring American pianist Borah Bergman, which was recorded in 1996 and released on the English Leo label.

Reception

In his review for AllMusic, Alex Henderson says that "this CD doesn't quite fall into the 'essential' category... Nonetheless, Geometry is an enjoyable release that Perelman's more-devoted followers will want."

The Penguin Guide to Jazz notes that "Bergman is wily enough to find ways of both supporting and undercutting the mighty sound of the tenor."

Track listing

Personnel

References

Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References

Direction

Direction may refer to:

Directions (Miles Davis album)

Directions is a compilation album by American jazz musician Miles Davis, released in 1981 by Columbia Records. It collects previously unreleased outtakes that Davis recorded between 1960 and 1970. Directions was the last of a series of compilation albums of Davis' music that Columbia released to bridge the hiatus that ended with The Man with the Horn in July 1981.

Music

Directions is a double album that features previously unreleased outtakes recorded over a 10-year period by Davis. Apart from "Song of Our Country" from the recording sessions for Sketches of Spain (1960), a 1961 recording of "'Round Midnight", and "So Near, So Far" from 1963, the album's songs are from Davis' transitional period during 1967 to 1970, when he was experimenting with a fusion of jazz and rock. They feature sidemen such as saxophonist Wayne Shorter, guitarist John McLaughlin, and keyboardists Herbie Hancock, Joe Zawinul, and Chick Corea.

Critical reception

In his review of the CD reissue for Allmusic, critic Scott Yanow gave Directions two-and-a-half out of five stars and said that, although some of the tracks "ramble on a bit too long", the music is "mostly quite fascinating" and "highly recommend[able] to collectors with an open ear toward fusion."J. D. Considine gave the album four out of five stars in The Rolling Stone Album Guide (2004).

Direction Records (British label)

The Direction record label was a subsidiary label established by CBS Records in the UK in spring 1967. It primarily released American soul and R&B recordings in the UK, but also issued records by British and Jamaican artists. Among its most successful releases were those by Sly & the Family Stone, The Bandwagon, and The Chambers Brothers.

References