All of us are comfortable with three-dimensional space, the dimensions being height, width and depth. Four-dimensional space seems something entirely else; for example, how many corners does a four-dimensional cube have? What is a four-dimensional cube? How do we measure, or even define - in 4D-space - distance, volume, spheres etc? (To answer the question on the corners of the cube, think of it as two 3D cubes whose edges are joined one-to-one.)

It turns out that most of these questions can be answered without serious difficulties, even in higher dimensions, if one uses a suitable" non-geometric" description of space which can be easily generalised to any number of dimensions.

Although it is hard to imagine a four-dimensional world, it is often useful to model "multifactor" quantities as points in a higher dimension space. For example, the trajectory of a moving particle can be thought of as a set of points in 4D-space, three dimensions for the location of the particle, and the fourth for the time at which it was at that location. As another example, suppose you've planted tomatoes at various places in your garden. Each tomato can be seen as an observation point in a high-dimensional space, the dimensions being its size, the amount of sunlight it received throughout the season, the amount of water and various supplements, and so on. To find out which of the factors were most important, and to "quantify" their importance to the size of the tomato, one often-used approach involves finding a high-dimensional" plane" which is in some mathematical sense the closest to the set of observation points.

Aims of the project:

References:

• Gusti van Zyl (Mathematics)