Four-dimensional space

Four-dimensional space is the mathematical extension of the concept of three-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. 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. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height. 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 s:fr:Mécanique analytique.
Mathematics of 4D commenced in the nineteenth century. 3D form rotation onto its mirror-image is possible in 4D space was realized by August Ferdinand Möbius in Der barycentrische Calcul published 1827. An arithmetic of four spatial dimensions, called quaternions, was defined by William Rowan Hamilton in 1843. Soon after, tessarines and coquaternions were introduced as other four-dimensional algebras over R. 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.
Euclidean spaces of more than three dimensions were first described in 1852, when Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods. He discovered all of the regular polytopes that exist in Euclidean spaces of any dimension, including six found in 4-dimensional space. Schläfli's work was only published posthumously in 1901, and remained largely unknown until publication of H.S.M. Coxeter's Regular Polytopes in 1947. During that interval many others also discovered higher-dimensional Euclidean space. One of the first popular 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, in which he explained the concept of a "four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. 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. 1886, Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams.
Hermann Minkowski's 1908 paper consolidating the role of time as the fourth dimension of spacetime provided the geometric basis for Einstein's theories of special and general relativity. The geometry of spacetime, being non-Euclidean, is profoundly different from that explored by Schläfli and popularised by Hinton. 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. 1967, The associative algebra of W R Hamilton was the source of the science of vector analysis in three dimensions as recounted by Michael J. Crowe in A History of Vector Analysis. The study of Minkowski space required Riemann's mathematics which is 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, a four-dimensional space is a space that needs four parameters to specify a point in it. For example, a general point might have position vector, equal to
This can be written in terms of the four standard basis vectors, given by
so the general vector is
Vectors add, subtract and scale as in three dimensions.
The dot product of Euclidean three-dimensional space generalizes to four dimensions as
It can be used to calculate the norm or length of a vector,
and calculate or define the angle between two non-zero vectors as
Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate pairing different from the dot product:
As an example, the distance squared between the points and is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between and is 4 in Euclidean space and 2 in Minkowski space; increasing decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity.
The cross product is not defined in four dimensions. Instead, the exterior product is used for some applications, and is defined as follows:
This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis. They can be used to generate rotations in four dimensions.

Orthogonality and vocabulary

In the familiar three-dimensional space of daily life, there are three coordinate axes—usually labeled,, and —with each axis orthogonal to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.
Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled. 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.
As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite velocity of light. In appending a time dimension to three-dimensional space, he specified an alternative perpendicularity, hyperbolic orthogonality. This notion provides his four-dimensional space with a modified simultaneity appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional absolute space and time cosmology previously used in a universe of three space dimensions and one time dimension.

Geometry

The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.
Just as in three dimensions there are polyhedra made of two dimensional polygons, in four dimensions there are polychora made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the Platonic solids. In four dimensions, there are 6 convex regular 4-polytopes, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.
In three dimensions, a circle may be extruded to form a cylinder. In four dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder, and a cylinder may be extruded to obtain a cylindrical prism. The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in four-dimensional space, each with its properties.
In three dimensions, curves can form knots but surfaces cannot. In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space. Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The Klein bottle is an example of such a knotted surface. Another such surface is the real projective plane.

Hypersphere

The set of points in Euclidean 4-space having the same distance from a fixed point forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:
This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where is substituted by function with meaning the cosmological age of the universe. Growing or shrinking with time means expanding or collapsing universe, depending on the mass density inside.

Four-dimensional perception in humans

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 and the angle between them. 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". In another study, the ability of humans to orient themselves in 2D, 3D, and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops. The graphical interface was based on John McIntosh's free 4D Maze game. The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D.
However, a 2020 review underlined how these studies are composed of a small subject sample and mainly of college students. It also pointed out other issues that future research has to resolve: elimination of artifacts and analysis on inter-subject variability. Furthermore, it is undetermined if there is a more appropriate way to project the 4-dimension. Researchers also hypothesized that human acquisition of 4D perception could result in the activation of brain visual areas and entorhinal cortex. If so they suggest that it could be used as a strong indicator of 4D space perception acquisition. Authors also suggested using a variety of different neural network architectures to understand which ones are or are not able to learn.