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# Fourth dimension

The concept of a fourth dimension is one that is often described in considering its physical implications, that is, we know that in three dimensions, we have dimensions of length, width, and height. The fourth dimension is said to be at right angles to these three, and is often described as time.

## Concepts

In treating space to be akin to a vector space, that is, a set of vectors which we can think of as arrows, fixed from some single place in space which we call the origin (geometric vectors ), that point to other places in space, we can look at the following intuitive concepts to build up a definition of dimension.

A point is a zero-dimensional object. It has no extension in space, and no properties. If we were to think of this point as a geometric vector, like an arrow, it would have no length. This vector is called the zero vector, and on its own, is the simplest vector space.

A line is a one-dimensional object. If we pick some nonzero vector in some direction, this vector, has some definite length. That vector has a head at some point in space, and a tail at the origin. If we think of stretching that vector so it is twice as long, three times as long, and so on, and even stretching it backwards so it takes all possible lengths it can (even zero length, to get the zero vector), we get a single line, with one dimension of length. All the vectors that describe points on this line are said to be parallel to each other. Even though this line has no thickness, if we are to draw a line, it has some small thickness, so we can see it.

A plane is a two-dimensional object. It has length and breadth but no thickness — somewhat like a sheet of paper (but paper too has some thickness). Thinking of a plane in terms of vectors is a little more difficult. If we think of taking one vector and moving it so that it's tail is touching the head of the first, and forming a vector with its tail at the origin and the head at the head of the repositioned second vector, we have a reasonable way of talking about adding vectors. If we have two vectors that are not parallel, we can talk about all the points we can reach by only stretching either, one, or none of the vectors, and adding these vectors together, these points form a plane.

Space, as we perceive it, is three-dimensional. We can think of putting a line together with a plane. These planes are "stuck together" like a sandwich. To get to some point in space, we can imagine travelling up the line and then moving across the plane to the point. We then have three vectors to think about, one to travel some distance up the line, and two to get to some point in space.

For one dimension, we think of stretching only one nonzero vector (in this case, stretching here is really only adding parallel vectors). For two dimensions, we need to stretch and add together two nonzero vectors. For three dimensions, we need to stretch and add together three nonzero vectors. Notice that the dimension of the object we think about matches the number of vectors we need to form that object. The vectors we need are called a basis, and this idea of dimension (in mathematics, more than one is useful!) is called the Hamel dimension.

A projection is a way or representing an n-dimensional object in n − 1 dimensions. For instance, the screen you are looking at is two-dimensional, and all the photographs of three-dimensional people, places and things are represented in two dimensions by removing information about the third dimension. The retina of the eye is a two-dimensional array of receptors but can perceive the nature of three-dimensional objects using clues. Artists use perspective to give three-dimensional depth to two-dimensional pictures.

## Reality and the fourth dimension

The universe that we inhabit seems to be three-dimensional. We can move ourselves and other objects in three dimensions — up/down, left/right and forwards/backwards. We have great difficulty in imagining the existence of a fourth dimension — if you have three axes at right angles to each other, how would you add a fourth at right angles to the existing three?

From our previous discussion, visualizing such a four-dimensional space may be difficult, but analyzing mathematically a four-dimensional space is straightforward. We just think of stretching and adding together four vectors instead of three. We can perform some elementary geometry by analogy with geometry in the three dimensional case.

We can calculate the area a of a rectangle by multiplying its length l by its width w: a = lw, and can calculate the volume v of a box by multiplying its length, height, and width w, so v = lhw. Mathematicians calculate the hypervolume (call it vH say) of a four-dimensional box by multiplying the length, height, and width, plus the length along the fourth dimension, call this length z, and we have vH = lhwz. This four-dimensional object has a name — tesseract or hypercube, and one can calculate the volume of its perimeter by analogue with the three-dimensional case. One can best visualise this hypercube as a projection into three-dimensional space, where it might look like a nest of cubes inside one another, rather like a Matryoshka doll.

## String theory

One of the hypotheses of modern string theory is that our universe consists of at least ten dimensions, of which three are the familiar spatial dimensions, six compact dimensions curled up too small to be detected, and one (or more) time dimension.

03-10-2013 05:06:04