In four dimensions, you can untie any knot in a piece of string
Add a fourth direction at right angles to the usual three and strange things happen. A knotted string can be slipped free by nudging it along the new axis, while flat surfaces such as the Klein bottle can tie themselves into genuine knots. Mathematicians have explored this realm since the 19th century.
A point in four-dimensional space simply needs four numbers instead of three, often written x, y, z and w. Charles Howard Hinton, who popularised the idea from 1880, coined the word tesseract for the four-dimensional cube and invented the names ana and kata, from Greek for up toward and down from, for the two extra directions. In 1886 Victor Schlegel offered a way to draw such objects on paper. The rules for vectors carry over neatly, with lengths and angles computed from four coordinates, though the familiar cross product has no four-dimensional version.
The mathematics took shape piece by piece. August Möbius realised in 1827 that a solid shape could be turned into its mirror image by rotating it through a fourth dimension. William Rowan Hamilton defined quaternions, an arithmetic of four dimensions, in 1843. In 1852 Ludwig Schläfli extended Euclid's geometry to any number of dimensions and found every regular polytope, the higher analogues of the Platonic solids, though his work only appeared in 1901. Where three dimensions allow five Platonic solids, four allow six regular convex polytopes.
The idea of time as a fourth dimension goes back to d'Alembert in 1754. Hermann Minkowski's 1908 paper made it the geometric basis of Einstein's relativity, but his spacetime measures distance differently, subtracting the time part instead of adding it. The point from the origin to (1,1,1,1) is 4 units squared away in ordinary 4D space but only 2 in Minkowski's, the source of many apparent paradoxes. H. S. M. Coxeter complained in 1973 that treating Euclid's fourth dimension as time, as in H. G. Wells's The Time Machine, had misled writers about relativity.
Four-dimensional shapes multiply the menagerie: a sphere stretched into a spherinder, a cylinder into a cubinder, and a duocylinder built from two circles. The three-dimensional surface of a 4D ball, called a 3-sphere, even appears in cosmological models of an expanding universe.
Source: Four-dimensional space
