Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m²), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

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

Area (LDS Church)

An area is an administrative unit of The Church of Jesus Christ of Latter-day Saints (LDS Church), which typically is composed of multiple stakes and missions. Areas are the primary church administrative unit between individual stakes and the church as a whole.

History

The areas as they now exist were formed in January 1984. Prior to that time, general authorities served as "area supervisors" and at times resided outside of Salt Lake City. In 1984, 13 initial areas were created; by 1992 there were 22, and by early 2007 there were 31. As of August 2012 there are 25 areas.

Administration

Until 2003, each area had a president and two counselors, all of whom were typically general authorities (area seventies were sometimes asked to be counselors). This three-man body was known as the area presidency. In that year, the church eliminated area presidencies for all areas located in the United States and Canada. Each of these areas were placed under the direct supervision of one of the seven members of the Presidency of the Seventy, thus freeing more general authorities from specific area assignments. Since these areas were previously administered by area presidencies located at church headquarters in Salt Lake City, the administrative change was not as drastic as it might seem.

Area (graph drawing)

In graph drawing, the area used by a drawing is a commonly used way of measuring its quality.

Definition

For a drawing style in which the vertices are placed on the integer lattice, the area of the drawing may be defined as the area of the smallest axis-aligned bounding box of the drawing: that is, it the product of the largest difference in x-coordinates of two vertices with the largest difference in y-coordinates. For other drawing styles, in which vertices are placed more freely, the drawing may be scaled so that the closest pair of vertices have distance one from each other, after which the area can again be defined as the area of a smallest bounding box of a drawing. Alternatively, the area can be defined as the area of the convex hull of the drawing, again after appropriate scaling.

Polynomial bounds

For straight-line drawings of planar graphs with n vertices, the optimal worst-case bound on the area of a drawing is Θ(n²). The nested triangles graph requires this much area no matter how it is embedded, and several methods are known that can draw planar graphs with at most quadratic area.Binary trees, and trees of bounded degree more generally, have drawings with linear or near-linear area, depending on the drawing style. Every outerplanar graph has a straight-line outerplanar drawing with area subquadratic in its number of vertices, If bends or crossings are allowed, then outerplanar graphs have drawings with near-linear area. However, drawing series-parallel graphs requires an area larger than n by a superpolylogarithmic factor, even if edges can be drawn as polylines.

Area (architecture)

In architecture, an area (areaways in North America) is an excavated, subterranean space around the walls of a building, designed to admit light into a basement, often providing access to the house for tradesmen and deliveries and access to vaults beneath the pavement for storage of coal and ash.

Function

The term is most commonly applied to urban houses of the Georgian period in the UK, where it was normal for the service rooms, such as the kitchen, scullery and laundry, to be in the basement. Areas were commonly enclosed for safety reasons by wrought iron or cast iron railings, which became one of the principal decorative features of the astylar terraced houses of this period. Areas are also found in the English and French country house, where basements were popular in the 18th century as a way of accommodating service functions while allowing all four faces of a symmetrical Classical building to relate directly to its landscape setting, as at Mereworth Castle in Kent or The Abbey, Skirwith in Cumberland. Basements, and consequently areas, decreased in popularity in the 19th century, as attitudes to servants changed, although they continued to be constructed as service accommodation in urban settings where land was at a premium until the early 20th century. A suburban residential application can be seen at the Gamble House in Pasadena, California, to light the laundry and service areas of the expansive basement.