Dimension

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.

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.

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.

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, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension.

Although 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.

This state of affairs was highly marked in the various cases of the Poincaré conjecture, in which four different proof methods are applied.

The dimension of a manifold depends on the base field with respect to which Euclidean space is defined.

Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions.

The most intuitive way is probably the dimension of the tangent space at any Regular point of an algebraic variety.

[5] Similarly, for the class of CW complexes, the dimension of an object is the largest n for which the n-skeleton is nontrivial.

[citation needed] The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals.

In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values.

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it.

The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space.

To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions.

One well-studied possibility is that the extra dimensions may be "curled up" at such tiny scales as to be effectively invisible to current experiments.

At the level of quantum field theory, Kaluza–Klein theory unifies gravity with gauge interactions, based on the realization that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances.

However, at sufficiently high energies or short distances, this setup still suffers from the same pathologies that famously obstruct direct attempts to describe quantum gravity.

Therefore, these models still require a UV completion, of the kind that string theory is intended to provide.

In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a Calabi–Yau manifold.

D-branes are dynamical extended objects of various dimensionalities predicted by string theory that could play this role.

They have the property that open string excitations, which are associated with gauge interactions, are confined to the brane by their endpoints, whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or "the bulk".

This could be related to why gravity is exponentially weaker than the other forces, as it effectively dilutes itself as it propagates into a higher-dimensional volume.

For example, brane gas cosmology[10][11] attempts to explain why there are three dimensions of space using topological and thermodynamic considerations.

According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect.

Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set of geometric primitives corresponding to the spatial dimensions:[12] Frequently in these systems, especially GIS and Cartography, a representation of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented.

This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).

From left to right: a square , a cube and a tesseract . The square is two-dimensional (2D) and bounded by one-dimensional line segments ; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes.
The first four spatial dimensions, represented in a two-dimensional picture.
  1. Two points can be connected to create a line segment .
  2. Two parallel line segments can be connected to form a square .
  3. Two parallel squares can be connected to form a cube .
  4. Two parallel cubes can be connected to form a tesseract .
The complex plane can be mapped to the surface of a sphere, called the Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mapped to the equator, and a point at infinity mapped to the other pole.
Illustration of a Calabi–Yau manifold