Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4" or: 4D.Īlthough the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. While these notions agree on E n, they turn out to be different when one looks at more general spaces.Ī tesseract is an example of a four-dimensional object. For example, the boundary of a ball in E n looks locally like E n-1 and this leads to the notion of the inductive dimension. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. When trying to generalize to other types of spaces, one is faced with the question "what makes E n n-dimensional?" One answer is that to cover a fixed ball in E n by small balls of radius ε, one needs on the order of ε − n such small balls.
The dimension of Euclidean n-space E nis n. This is independent from the fact that a curve cannot be embedded in a Euclidean space of dimension lower than two, unless it is a line. For example, a curve, such as a circle, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, the dimension of a point is zero the dimension of a line is one, as a point can move on a line in only one direction (or its opposite) the dimension of a plane is two etc. In other words, the dimension is the number of independent parameters or coordinates that are needed for defining the position of a point that is constrained to be on the object. In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. They may be Euclidean spaces or more general parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics these are abstract spaces, independent of the physical space. High-dimensional spaces frequently occur in mathematics and the sciences. The concept of dimension is not restricted to physical objects. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space. Minkowski space first approximates the universe without gravity the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity.
The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism.
In classical mechanics, space and time are different categories and refer to absolute space and time. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. A two-dimensional Euclidean space is a two-dimensional space on the plane. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
0 kommentar(er)
