Four-dimensional space

Four-dimensional (4D) space is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.

Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers such as (x, y, z, w). For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled as x, y, and z). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible regular 4D objects, the tesseract, which is analogous to the 3D cube.

History

The idea of making time the fourth dimension began with Jean le Rond d'Alembert's "Dimensions," published in 1754 in the Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers. That physical mechanics can be viewed as occurring also in time was posited by Joseph-Louis Lagrange in 1755 and published in 1788 in his Mécanique analytique.

Mathematics of 4D commenced in the nineteenth century.