Fourth Dimension
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In physics and mathematics, a sequence of n numbers can be understood as a location in an n-dimensional space. When n=4, the set of all such locations is called 4-dimensional space, or, colloquially, the fourth dimension.
Such a space differs from the familiar 3-dimensional space that we live in, in that it has an extra dimension, an extra degree of freedom. This extra dimension may be interpreted either as time, or as a literal fourth dimension of space, a fourth spatial dimension.
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The fourth dimension as time
Usually, when a reference is made to four-dimensional coordinates, it is the temporal interpretation which is meant. In this case, the four coordinates are understood to represent 3 dimensions of space plus 1 dimension of time. Such a space is called a Minkowski space or "(3 + 1)-space", and is the space used in Einstein's theories of special relativity and general relativity.
The fourth dimension as space
Sometimes, the fourth dimension is interpreted in the spatial sense: a space with literally 4 spatial dimensions, 4 mutually orthogonal directions of movement. This is the space used by mathematicians when studying geometric objects such as 4-dimensional polytopes. To avoid confusion with the more common Einsteinian notion of time being the fourth dimension, however, the use of this spatial interpretation should be stated at the outset.
Mathematically, the 4-dimensional spatial equivalent of conventional 3-dimensional geometry is the Euclidean 4-space, a 4-dimensional normed vector space with the Euclidean norm. The "length" of a vector
expressed in the standard basis is given by
which is the natural generalization of the Pythagorean Theorem to 4 dimensions. This allows for the definition of the angle between two vectors.
Orthogonality
In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: up/down (altitude), north/south (latitude), and east/west (longitude). These pairs of directions are mutually orthogonal: they are at right angles to each other. Mathematically, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.
A space of four spatial dimensions has an additional pair of cardinal directions which is orthogonal to the other three. This additional pair of directions lies on a fourth coordinate axis perpendicular to the x, y, and z axes, usually labelled w. Attested terms for these extra directions include ana/kata (sometimes called spissitude or spassitude), vinn/vout (used by Rudy Rucker), and upsilon/delta. These extra directions lie outside (and indeed, perpendicular to) the three observable directions in our 3-dimensional world.
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Dimensionality
We can obtain a basis for a geometric object X by removing all redundant vectors from any set that spans X. Depending on which vectors one starts with, one may obtain different bases that span X; however, it can be proven that all of these bases will always have the same number of vectors. This number is called the dimension of X. In other words, X is n-dimensional if a minimum of n of vectors are needed to span it.
Intuitively, the dimension of an object may be thought of as the number of independent directions one needs to travel in order to reach every point in it.
For example, a point is a zero-dimensional object. No vectors are necessary to span it since if one starts at the point, one has already reached all of it.
A line is a one-dimensional object. Starting at some point on the line, one needs a vector that points in the direction of the line in order to reach the other points on the line. Only one vector is necessary, since scaling it by different amounts allows one to reach any other point on the line.
A plane is a two-dimensional object. Given some starting point on the plane, at least two vectors, not parallel to each other, are needed to span it. With only one vector, only points that lie on a straight line can be reached. A second vector, not parallel to the first, is needed to move "sideways" to points on the plane outside that line. Only two directions are necessary, since one can move forwards (or backwards) along the first vector by different distances and then move sideways by different distances to cover every point on the plane. One may think of the plane as a "stack" of parallel lines; to get from one point to another in the two-dimensional plane, one first travels along the line in one direction, and then travels "across" the parallel lines in a second direction.
Space, as we perceive it, is three-dimensional. In order to reach some point in space, one not only needs to move forwards or backwards, and sideways; one needs to move upwards or downwards as well. In other words, a third vector is necessary to cover all of space. One may think of space as a "stack" of parallel planes: to travel from one point to another in space, one may move forwards or backwards along one direction, and then move sideways along a second direction, and finally move upwards or downwards in the third, vertical direction.
Four-dimensional space is a space where four independent directions are needed to cover all of it. Such a space may be visualized as a "stack" of many parallel three-dimensional spaces. To understand this concept, think of putting pieces of paper side by side. The sheets do not extend into the third dimension until one puts them on top of one another. In the same way, in order to reach into four-dimensional space, it is necessary to move in a new direction, a direction outside three-dimensional space. To reach each point in four-dimensional space, not only does one need to travel forwards and backwards, left and right, up and down, but also along a new pair of directions, ana and kata.
Dimensional analogy
To understand the nature of four-dimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how (n – 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions (Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (1994, ISBN 0-19-286189-1) Part I chapter 3 The Man Who "Saw" the Fourth Dimension).
Dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as being able to remove objects from a safe without breaking it open (by moving them across the third dimension), being able to see everything that from the two-dimensional perspective is enclosed behind walls, and remaining completely invisible by standing a few inches away in the third dimension.
Another example of this concept is seen in the Wii Video Game Super Paper Mario, where the protagonist is able to shift from a two-dimensional perspective to a three-dimensional one.
By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from our three-dimensional perspective. Rudy Rucker demonstrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.
Geometry
The geometry of 4-dimensional space is much richer than that of 3-dimensional space, due to the extra degree of freedom.
Just as in 3 dimensions, one may construct polyhedra from polygons, in 4 dimensions one may construct polychora (4-polytopes) from polyhedra. In 3 dimensions, there are 5 regular polyhedra, known as the Platonic solids. In 4 dimensions, there are 6 convex regular polychora, the analogues of the Platonic solids. In 3 dimensions, there are 13 Archimedean solids, whereas in 4 dimensions, there are 58 convex uniform polychora (64 including the regular polychora).
In 3 dimensions, one may extrude a circle to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. One may extrude a sphere to obtain a spherical cylinder (a cylinder with spherical "caps"), or one may extrude a cylinder to obtain a cylindrical prism. One may also take the Cartesian product of two circles to obtain a duocylinder. All three can "roll" in 4-dimensional space, each with its own properties.
In 3 dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction. But 2-dimensional surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space. Because these surfaces are 2-dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle is an example of such a knotted surface. Another such surface is the real projective plane.
