The Fourth Dimension !

In mathematics, four-dimensional space (“4D”) is an abstract concept derived by generalizing the rules of three-dimensional space. It has been studied by mathematicians and philosophers for almost three hundred years, 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. Spacetime is thus not a Euclidean space. 

 The possibility of spaces with dimensions higher than three was first studied by mathematicians in the 19th century. In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image, and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death. 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, …, xn). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis.

The fourth dimension was popularised by Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension?; published in the Dublin University magazine. He coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension.