The 4th dimension space

4th dimensional space

In mathematics, four-dimensional space ("4D") is a geometric space with four dimensions. It typically is more specifically four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields.

Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.

In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is not a Euclidean space.

Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space — three of dimensions of space, and one of time.^[1] In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,^[2] and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.^[3] Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates (x₁, ..., x_n). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

 Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example, a general point might have position vector a, equal to

This can be written in terms of the four standard basis vectors (e₁, e₂, e₃, e₄), given by

so the general vector a is

Vectors add, subtract and scale as in three dimensions.

The dot product of Euclidean three-dimensional space generalizes to four dimensions as

It can be used to calculate the norm or length of a vector,

and calculate or define the angle between two vectors as

Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing  actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.

The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e₁₂, e₁₃, e₁₄, e₂₃, e₂₄, e₃₄). They can be used to generate rotations in four dimensions.