Beyond Four Dimensions Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to The Fourth Dimension and Beyond

There is no limit to the number of dimensions that can be defined using the Cartesian scheme. There can be any positive whole number of dimensions. Time can be (but does not have to be) one of them.

Cartesian N -space

A system of rectangular coordinates in five dimensions defines Cartesian five-space . There are five number lines, all of which intersect at a point corresponding to the zero point of each line, and such that each of the lines is perpendicular to the other four. The resulting axes can be called v , w , x , y , and z . Alternatively they can be called x 1 , x 2 , x 3 , x 4 , and x 5 . Points are identified by ordered quintuples such as ( v , w , x , y , z ) or ( x 1 , x 2 , x 3 , x 4 , x 5 ). The origin is defined by (0,0,0,0,0).

A system of rectangular coordinates in Cartesian n -space (where n is any positive integer) consists of n number lines, all of which intersect at their zero points, such that each of the lines is perpendicular to all the others. The axes can be named x 1 , x 2 , x 3 , . . ., and so on up to x n . Points in Cartesian n -space can be uniquely defined by ordered n -tuples of the form ( x 1 , x 2 , x 3 ,. . ., x n ).

Imagine a tesseract or a rectangular four-prism that pops into existence at a certain time, does not move, and then disappears some time later. This object is a rectangular five-prism . If x 1 , x 2 , x 3 , and x 4 represent four spatial dimensions (in kilometer-equivalents or second-equivalents) of a rectangular four-prism in Cartesian four-space, and if t represents its “lifetime” in the same units, then the 5D hypervolume (call it V 5D ) is equal to the product of them all:

V 5D = x 1 x 2 x 3 x 4 t

This holds only as long as there is no motion. If there is motion, then the relativistic correction factor must be included.

Dimensional Chaos

There is nothing to stop us from dreaming up a Cartesian 25-space in which the coordinates of the points are ordered 25-tuples ( x 1 , x 2 , x 3 ,. . ., x 25 ), none of which are time. Alternatively, such a hyperspace might have 24 spatial dimensions and one time dimension. Then the coordinates of a point would be defined by the ordered 25-tuple ( x 1 , x 2 , x 3 ,. . ., x 24 , t).

Some cosmologists—scientists who explore the origin, structure, and evolution of the cosmos—have suggested that our universe was “born” with 11 dimensions. According to this hypothesis, not all of these dimensions can be represented by Cartesian coordinates. Some of the axes are “curled up” or compactified as if wrapped around tiny bubbles. Some mathematicians have played with objects that seem to be 2D in some ways and 3D in other ways. How many dimensions are there in the complicated surface of a theoretical foam, assuming each individual bubble is a sphere of arbitrarily tiny size and with an infinitely thin 2D surface? Two dimensions? In a way. Three? In a way. How about two and a half?

The examples we have looked at here are among the simplest. Imagine the possible ways in which a 4D parallelepiped might exist, or a 4D sphere . How about a 5D sphere , or a 7D ellipsoid ? Let your mind roam free. But don’t think about this stuff while driving, operating heavy equipment, cycling, or walking across a street in traffic.

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