Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid has a regular polygon base and is usually implied to be a right pyramid.

When unspecified, a pyramid is usually assumed to be a regular square pyramid, like the physical pyramid structures. A triangle-based is more often called a tetrahedron.

Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called acute if its apex above the interior of the base, and obtuse if its apex above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base. In a tetrahedron these qualifiers will change based on which face is considered the base.

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

Geometry (disambiguation)

Geometry is a branch of mathematics dealing with spatial relationships.

Geometry or geometric may also refer to:

Geometry (Robert Rich album)

Geometry (1991) is an album by the American ambient and electronic musician Robert Rich. Although completed in 1988, this album was not released until three years later.

This album is more active and structured than any of his previous works. The music was inspired in part by the complex patterns of Islamic designs like those found at the Alhambra, and features complex just intonation. The first three tracks consist of complex layered sequences of electronic notes in rich, organic-sounding chime tones. Rich revisited this style in Gaudí (1991) and Electric Ladder (2006). Tracks 4 through 7 are slow textures more common to Robert Rich’s work. The album ends with another active piece similar to the first three tracks.

Tracks 1, 2, 4 and 8 were mixed by Robert Rich and future collaborator Steve Roach at Roach’s studio in Venice, California. This album was released in a two CD set with Numena in 1997.

Track listing

Pyramid (song)

"Pyramid" is the second single of Filipina pop and R&B singer Charice and the lead single from her self-titled debut album. This song features vocals from Virgin singer Iyaz, was written by Niclas Molinder, Joacim Persson, Johan Alkenäs and R&B singer-songwriter Lyrica Anderson, and produced by Twin and Alke. A remix was said to be released on iTunes on February 16, 2010 which was later changed to February 23, the same day as the release of the album version. But the album version was delayed and was released on March 2. It is Charice's first number one single. Pyramid is a midtempo pop ballad with lyrics that metaphorically describes the strength of a relationship. "Pyramid" peaked at the Billboard Hot 100 chart at number 56, making Charice the second Filipino singer to enter the chart since Jaya's debut single "If You Leave Me Now" peaked on the Billboard Hot 100 at number 44 in 1990.

Pyramid had already received airplay before it was released to iTunes. It aired on 104.3 in Las Vegas before its release as a single. The album version was released to U.S. mainstream radio on March 15, 2010. It is her first single to achieve mainstream airplay. The song was sung in Simlish for The Sims 3: Ambitions, an expansion pack for the strategic life simulation computer game, The Sims 3.

Pyramid (Providence band)

Pyramid is a punk rock band from Providence, Rhode Island, composed of Connie Strychasz (vocals, drums), Florence Wallis (bass), Mackensie Elisabeth (guitar) and Thea Shaheen (ukulele). Members have previously or concurrently played with other Providence bands including The Low Anthem. Founded in 2014, Pyramid has built a strong following in the town thanks to its memorable live shows. A series of demos were released on Soundcloud, and a debut album is forthcoming.

External links

References

Pyramid (The Alan Parsons Project album)

Pyramid is the third album by progressive rock band The Alan Parsons Project, released in 1978. It is a concept album centred on the pyramids of Giza. At the time the album was conceived, interest in pyramid power and Tutankhamun was widespread in the US and the UK. Pyramid was nominated for the 1978 Grammy Award for Best Engineered Album, Non-Classical. Liner notes read "...this album seeks to amplify the haunting echoes of the past and explore the unsolved mysteries of the present. Pyramid...the last remaining wonder of the ancient world."

The album also dabbles with new wave, a genre that was emerging in Britain around the time of the album's recording. Many progressive and soft rock artists incorporated this style into their albums during the time spanning from late 1977 to 1979. In this album, the genre is mildly evident, through rhythm, in songs such as "Can't Take it with You" and immensely so in others such as "Pyramania".

Track listing

All songs written and composed by Alan Parsons and Eric Woolfson.