Dictionary

curvature

Sum of the angles of a triangle. In a plane, the sum of the three angles in a triangle formed by three straight lines is always 180 degrees. In a more general surface, the sum of the angles of a more general triangle formed by three straightest-possible lines (i.e. geodesics) can be larger or smaller than 180 degrees. The difference (the surplus or deficit), divided by the area of the triangle, is a measure for the curvature of that region of the surface.

Second possibility: the circumference of a circle. In the plane, that circumference is equal to 2 times pi times the circle’s radius. On a more general surface, it can be larger or smaller. The difference, divided by the third power of the radius, leads to the same measure for the curvature as the first definition.

Simple examples for curved surfaces are the surface of a sphere (positive curvature, that is to say: sum of the angles in a triangle larger than 180 degrees, circumference of a circle smaller than 2 times pi times radius) and that of a saddle (negative curvature, that is to say: sum of the angles in a triangle smaller than 180 degrees, circumference of a circle larger than 2 times pi times radius).

Curvature cannot only be defined for surfaces, but also for higher-dimensional, more general spaces or spacetimes. However, the generalized definition is substantially more complicated, and curvature is defined not by a single number, but by a set of numbers (that, together, form the “curvature tensor”). It’s basic meaning, however, is the same: it measures the space’s deviation from a flat space of the same dimension.

For physics, an important aspect of curvature is its connection with gravity, as described in Einstein’s general theory of relativity. Basic information about this can be found in the spotlight text Gravity: From weightlessness to curvature.

Definitioner

triangle

In a plane and other flat space: Geometrical object consisting of three points ("vertices") with three connecting straight lines. The definition can be made more general, in a way that applies to curved spaces, as well: A geometric object consisting of three points ("vertices") connected by three segments of geodesics.

surface

Geometric space with two dimensions. Examples include the plane or the surface of a sphere.

sphere

A spherical surface is a simple example for a curved surface. It is easily pictured as a surface embedded in three-dimensional space: a spherical surface is the set of all points at a certain fixed distance from a given point (the point being the centre of the sphere). Mathematically, a spherical surface can be described without recourse to three-dimensional space - when mathematicians talk of the geometry of such a surface, they (almost) always mean the "inner geometry": Those properties of the surface noticeable to two-dimensional beings, living and working in that surface, capable of measuring distances and angles in it. Sphere is regularly used as a synonym for spherical surface (instead of describing a solid, three-dimensional ball). And not only for the two-dimensional spherical surface described above, but also for its analogues in lower and higher dimensions. A one-sphere, for instance, is the same as a circle, a two-sphere is the spherical surface defined above, a three-sphere its three-dimensional analogue.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

plane

A surface within which the axioms of Euclidean geometry (synonym: plane geometry) hold - the rules of geometry as they are taught in high school, with well-known formulae such as Pythagoras's theorem and "the perimeter of a circle is 2 times pi times its radius" hold.

geometry

That part of mathematics concerning itself with surfaces or more general spaces as well as objects defined on such spaces, such as points or lines as well as the objects constructable from points and lines, such as triangles.

dimension

The number of independent directions within a set of points, alternatively: the number of coordinates needed to give each point a unique name. This is rather abstract - time for some examples: A line is one-dimensional. There's only one direction to go on the line (the opposite direction isn't counted extra): Back-forth. A single number is sufficient to define a point of the line. For instance, on a motorway, given the statement "the accident happened 4 kilometres from the beginning of the I95 (or M1, or whatever)" is sufficient information for the rescue workers to know exactly where to go. Surfaces are two-dimensional, as there are two independent directions: back-forth and left-right, say. On the earth's surface, the two coordinate numbers geographical longitude and latitude uniquely define each location. The space that surrounds us is three-dimensional. There are three independent directions, say back-forth, left-right and up-down. In order to define a location in space, one needs to specify three numbers - for instance, two to specify where a house is located on the earth's surface (latitude/longitude, see above) and one floor number (or, more precisely, the height above the earth's surface). Adding time to the three space coordinates (a must for defining an appointment - where and when?), the result is four-dimensional spacetime. In order to define an event in spacetime, one needs to give four numbers: three of them determine where in space the event happens, the fourth gives the time where it happens. According to some of the models that have been studied as candidates for a theory of quantum gravity, our world should have even more space dimensions than the usual three. Some information about these extra dimensions can be found in the spotlight topics "Extra dimensions, and how to hide them", "Hunting for extra dimensions" and Extra dimensions and simplicity.

curvature

For a two-dimensional surface: criterion that allows us to decide whether that surface is a plane, or part of a plane (i.e. a surface on which the usual rules of high school geometry apply), or not. Two possibilities to define the curvature of a plane are the following: Sum of the angles of a triangle. In a plane, the sum of the three angles in a triangle formed by three straight lines is always 180 degrees. In a more general surface, the sum of the angles of a more general triangle formed by three straightest-possible lines (i.e. geodesics) can be larger or smaller than 180 degrees. The difference (the surplus or deficit), divided by the area of the triangle, is a measure for the curvature of that region of the surface. Second possibility: the circumference of a circle. In the plane, that circumference is equal to 2 times pi times the circle's radius. On a more general surface, it can be larger or smaller. The difference, divided by the third power of the radius, leads to the same measure for the curvature as the first definition. Simple examples for curved surfaces are the surface of a sphere (positive curvature, that is to say: sum of the angles in a triangle larger than 180 degrees, circumference of a circle smaller than 2 times pi times radius) and that of a saddle (negative curvature, that is to say: sum of the angles in a triangle smaller than 180 degrees, circumference of a circle larger than 2 times pi times radius). Curvature cannot only be defined for surfaces, but also for higher-dimensional, more general spaces or spacetimes. However, the generalized definition is substantially more complicated, and curvature is defined not by a single number, but by a set of numbers (that, together, form the "curvature tensor"). It's basic meaning, however, is the same: it measures the space's deviation from a flat space of the same dimension. For physics, an important aspect of curvature is its connection with gravity, as described in Einstein's general theory of relativity. Basic information about this can be found in the spotlight text Gravity: From weightlessness to curvature.

Dictionary

curvature

Further Articles

Tag Cloud