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Since the theory of "relativity" it has been very orthodox to treat time, alongside the main other three spatial dimensions, as the fourth dimension of a unified spacetime, meaning united! Around the late 19th century, physicists such as Helmholtz popularized the works of Riemann, that in time suggested that there might be a fourth spatial dimension, into which, things, objects, anything might disappear, only to reappear elsewhere! Black hole much? The idea was taken up by theosophists and workers on parapsychology. It is analysed in the 1884 classic Flatland by Edwin Abbot, and mentioned in work by H. G. Wells and Oscar Wilde. It reappeared in orthodox physics in the work of Kaluza, who showed in 1919 that when the Riemann tensor metric (see relativity) is rewritten in five dimensions, can you believe it now we have a 5th dimension too, a unified theory of gravity and electromagnetism can be produced. But wait there is more, modern scientists are now concluding their favored number of dimensions is ten!

Another example of the 4th dimension is an allowance at right-angles to the three conversant directions of up-down, forward-backward, and side-to-side. In physics, particularly in relativity theory, time is frequently observed as the "fourth dimension" of the spacetime continuum in which we live. But what denotation can be attached to a fourth spatial dimension or the quadracentifiable definition? The mathematics of the fourth dimension, quadracentifiably speaking, can be approached through an easy extension of either the algebra or the geometry of one, two, and three dimensions.

Algebraically, each point in a multidimensional space can be signified by a sole arrangement of real figures. One-dimensional space is just the number line of real figures. Two-dimensional space, the plane, parallels to the set of all ordered pairs (x, y) of real figures, and three-dimensional space to the set of all ordered triplets (x, y, z). By extrapolation, four-dimensional space corresponds to the set of all ordered quadruplets (x, y, z, w). Linked to this concept is that of quaternions, which can also be viwed as points in the fourth dimension.

Geometric facts about the fourth dimension are just as easy to state. The fourth dimension can be thought of as a direction perpendicular to every direction in three-dimensional space; in other words, it stretches out along an axis, say the w-axis, that is mutually perpendicular to the familiar x-, y-, and z-axes. Analogous to the cube is a hypercube or tesseract, and to the sphere is a 4-d hypersphere. Just as there are five regular polygons, also known as the Platonic solids, so there are six four-dimensional regular polytopes.

They are: the 4-simplex (constructed from five tetrahedra, with three tetrahedra meeting at an edge); the tesseract (made from eight cubes, meeting three per edge); the 16-cell (made from 16 tetrahedra, meeting four per edge); the 24-cell (made from 24 octahedra, meeting three per edge); the 120-cell (made from 120 dodecahedra, meeting three per edge); and the monstrous 600-cell (made from 600 tetrahedra, meeting five per edge).

Geometers have no trouble in analyzing, telling, and cataloging the properties of all sorts of 4-d figures. The problem starts when we try to visualize the fourth dimension. This is a bit like trying to procedure a mental picture of a color different from any of those in the known rainbow from red to violet, or a "lost chord," different from any that has ever been played. The best that most of us can hope for is to understand by analogy.

If we take for example, just as a sketch of a cube is a 2-d perspective of a real cube, so a real cube can be thought of as a "perspective of a tesseract". At a movie, a 2-d picture represents the 3-d world, whereas if you were to watch the action live, in three-dimensions, this would be like a screen forecast in four dimensions.

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