Fuzzy Geometry and the Quantum M-Theory Super-Fivebrane

In the 19th century the impressionist painters invented a new style of painting. Actually, there were two schools of impressionist painters, with very different philosophies. One school was the `pointillists'. If you look closely at a pointillist painting you'll see that it is constructed from individual dots of colour that are bit like 'atoms' of paint. The other school of impressionist painters--let's call them the 'ondulists'--painted in a very different way. From afar, an ondulist painting may look similar to a pointillist painting, but if you look closely you see just a vague wash of brushstrokes. No dots.Any ordinary material has a structure analogous to a pointillist painting. It may appear continuous, but will betray its atomic nature under sufficiently powerful magnification. Could there be extra-ordinary materials that are analogous to an ondulist painting? Well, there is a physical analogy of the ondulists' method: holography. A laser holograph is a piece of film through which laser light is shone to produce a 3D image. An equally remarkable property of a holograph is that any small part of it can be used to produce the entire image! If the negative of an ordinary photograph is cut in half, and one half discarded, then one has lost half the image. Not so with a laser holograph; the whole image is still there, but fuzzier.The word material suggests something substantial, but modern physics, based on the ideas of quantum mechanics and Einstein's theory of 'general relativity', views even the vacuum as a kind of material--let's call it spacetime . Quantum mechanics, which tells us how the atoms of ordinary materials behave, also tells us that spacetime should have a 'quantum' structure. Until recently, we were pointillists, imagining some kind of atomic structure, but now our best theories hint that Nature is an ondulist, and that holography is the appropriate metaphor. My aim is to elucidate the fuzzy nature of quantum spacetime, and work out its consequences for the physical world, via what is called 'M-theory'.In M-theory, the basic building blocks for matter are branes , which is a generic term derived from membranes for an extended object. M theory is not yet a true theory, but rather a collection of interlocking view points, in each of which one brane is dominant. From one such view point, M theory appears to be a theory of fivebranes . I aim to understand how this perspective may lead to a theory of quantum spacetime.