# higher dimensions

Higher dimensions are dimensions beyond the familiar
three spatial dimensions of which we are aware in every day life. Intense
speculation, both scientific and fictional, has naturally been directed
toward the possibility of a fourth
dimension. One way to think of points in four-dimensional space is as ordered sets of four numbers (see quaternion).
Clearly, this algebraic representation can be extended to arbitrary many
dimensions: *n*-dimensional space is defined as the set of the set
of points (*a*_{1}, *a*_{2}, ... , *a*_{n})
where *a*_{1} to *a*_{n} can take any real number
value.

There has been much conjecture that the universe in which we live contains many more than three spatial dimensions. This speculation began with the Kaluza-Klein theory but is now firmly embedded in modern string theory.

## References

1. Banchoff, Thomas F. *Beyond the Third Dimension: Geometry, Computer
Graphics, and Higher Dimensions*. New York: W. H. Freeman, 1990.

2. Burger, Dionys. *Sphereland: A Fantasy About Curved Spaces and
an Expanding Universe*. New York: Barnes & Noble, 1983.

3. Dewdney, A. K., The Planiverse: Computer Contact with a Two-dimensional
World. New York: Simon & Schuster, 1984.