![]() 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. In 1886, Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams. 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. He coined the terms tesseract, ana and kata in his book A New Era of Thought and introduced a method for visualizing the fourth dimension using cubes in the book Fourth Dimension. 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. Soon after, tessarines and coquaternions were introduced as other four-dimensional algebras over R. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.Īn arithmetic of four dimensions, called quaternions, was defined by William Rowan Hamilton in 1843. : 142–143 Higher dimensions were soon put on a 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 ( x 1. In 1827, Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image : 141 by 1853, Ludwig Schläfli had discovered all the regular polytopes that exist in higher dimensions, including the four-dimensional analogs of the Platonic solids, but his work was not published until after his death. 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 dimensions of space, and one of time. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible 4D objects, the tesseract (equivalent to the 3D cube see also hypercube). ![]() It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of higher-dimensional spaces emerge. Single locations in 4D space can be given as vectors or n-tuples, i.e., as ordered lists of numbers such as ( x, y, z, w). Einstein's concept of spacetime uses such a 4D space, though it has a Minkowski structure that is slightly more complicated than Euclidean 4D space. Large parts of these topics could not exist in their current forms without using such spaces. Higher-dimensional spaces (i.e., greater than three) have since become one of the foundations for formally expressing modern mathematics and physics. ![]() The eight lines connecting the vertices of the two cubes in this case represent a single direction in the "unseen" fourth dimension. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. In 1880, Charles Howard Hinton popularized these insights in an essay titled " What is the Fourth Dimension?", which explained the concept of a " four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" being published in 1754, was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z). ![]() Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. A four-dimensional space ( 4D) is a mathematical extension of the concept of three-dimensional space (3D). ![]()
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