# Useful Formulae

52 HIGHERDIMENSIONS beyond the familiar knot Many geometric forms in the twodimensional plane can be extended to three dimensions by a process of analogy,circles and spheres are a good example see page 8. Although our visual imagination gets left behind,we can easily continue this process, describing hyperspheresin 4,5 or more dimensions. In general an ndimensional space is given by the set of points with coordinates x1, x2, ..., xn. Higher dimensional distances, angles, and other quantities can then be defined by analogy. Einstein used four dimensions to model the physics of space time,and modern cosmologists today use models of ten or more dimensions in specific applications. It is important to remember, however,that an ndimensional space can be defined and studied without any necessary interpretation in terms of physical space, time or anything experiential. Certain whole numbers ndistinguish themselves in that the associated ndimensional space has some unique geometric property. A simple example is the fact that threedimensional space alone can support knots embeddings of a circle. In any other number of dimensions any such embedding can always be unknotted. Similarly, only fourdimensional space permits the mathematical curiosities known as exotic differential structures. But thats another story. 53