The number of dimensions can change through quantum fluctuations, which were large at the time of the Big Bang but are now so small and rare that for all practical purposes they can be ignored. It is interesting to note in this connection that the number of dimensions does not have to be an integer: it is possible to construct mathematically a space of dimension 1/3 or 9/8, for instance. An object of dimension greater than 1 and smaller than 2 is obtained by taking a planar curve and stretching and folding it innumerable times, but still in such a way that it does not fill entirely any finite region of the plane. An object of this kind is called a fractal curve, and more generally a set of points with a non-integer dimensionality is called a fractal set. According to Linde and other proponents of "chaotic inflationary cosmology", the relatively peaceful universe that nurtures us was born from a fractal and wildly fluctuating "quantum foam": it is a "bubble" of this foam that has greatly expanded in three of the space dimensions and has settled down to nine space dimensions overall.

Mathematical spaces with infinite dimensionality

Spaces with an infinite number of dimensions have been extensively studied by mathematicians and have found an application in quantum theory to represent in an abstract way the infinite possible configurations of a quantum system such as an atom. These abstract spaces of quantum theory where introduced by Hilbert, the great German mathematician who actually obtained the equations of General Relativity a few days before Einstein did, by a different method. In quantum gravity, each point of physical space has an infinite-dimensional Hilbert space associated with it. That's how complicated the theoretical picture has become, and even stranger ideas are under investigation.

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Note: A Hilbert space can have a finite or infinite number of dimensions. Finite-dimensional Hilbert spaces correspond to very simple quantum systems: for instance, the spin of the electron, or any other spin-1/2 particle, can be represented in a Hilbert space of 2 dimensions.