dimension would allow a three-dimensional form to be rotated onto its mirror-image; by 1853, Ludwig Schläfli had discovered all the regular polytopes that exist in higher dimensions, including the four-dimensional analogues of the Platonic solids, but his work was not published until after his death. Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 thesis, 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. Soon after, tessarines and coquaternions were introduced as other four-dimensional algebras over R. One of the first major expositors of the fourth dimension was 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. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. In 1886, Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams. In 1908, Hermann Minkowski presented a paper consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity. But the geometry of spacetime, being non-Euclidean, is profoundly different from that explored by Schläfli and popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write: Vectors Mathematically, four-dimensional space is 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,