Four-dimensional space
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.
The general concept of Euclidean space with any number of dimensions was fully developed by the Swiss mathematician Ludwig Schläfli before 1853.
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.
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.
[6] By 1853 Schläfli had discovered all the regular polytopes that exist in higher dimensions, including the four-dimensional analogs of the Platonic solids.
[9][10] 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.
Higher dimensional non-Euclidean spaces were 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 (x1, ..., xn).
In 1908, Hermann Minkowski presented a paper[11] consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.
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: Little, if anything, is gained by representing the fourth Euclidean dimension as time.
Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively.
[9]: 160 As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite velocity of light.
The hyper-volume of the enclosed space is: This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where R is substituted by function R(t) with t meaning the cosmological age of the universe.
Growing or shrinking R with time means expanding or collapsing universe, depending on the mass density inside.
[15] Research using virtual reality finds that humans, despite living in a three-dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them.
[16] The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments".
[18] The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point.
It also pointed out other issues that future research has to resolve: elimination of artifacts (these could be caused, for example, by strategies to resolve the required task that don't use 4D representation/4D reasoning and feedback given by researchers to speed up the adaptation process) and analysis on inter-subject variability (if 4D perception is possible, its acquisition could be limited to a subset of humans, to a specific critical period, or to people's attention or motivation).
Researchers also hypothesized that human acquisition of 4D perception could result in the activation of brain visual areas and entorhinal cortex.
A circle gradually grows larger, until it reaches the diameter of the sphere, and then gets smaller again, until it shrinks to a point and disappears.
For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places, and things are represented in two dimensions by projecting the objects onto a flat surface.
The retina of the eye is also a two-dimensional array of receptors but the brain can perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.).
The shadow, cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections.
The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension.
(Note that, technically, the visual representation shown here is a two-dimensional image of the three-dimensional shadow of the four-dimensional wireframe figure.)
The 4-volume or hypervolume in 4D can be calculated in closed form for simple geometrical figures, such as the tesseract (s4, for side length s) and the 4-ball (
Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence.
Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions."
Another reference is Madeleine L'Engle's novel A Wrinkle In Time (1962), which uses the fifth dimension as a way of "tesseracting the universe" or "folding" space to move across it quickly.
"[23] "Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologist Gustav Fechner under the pseudonym "Dr. Mises".
[26] Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract";[27] and the Russian esotericist P. D. Ouspensky.
