ABSTRACT
One of the greatest advances ever made in geometry was achieved long ago with the discovery of its intimate connection with algebra via what we would now refer to as analytic geometry. One of the joys of fractal geometry is the opportunity to create all manner of hideous or beautiful forms. All that is needed is a basic starting shape and a rule which step-by-step makes the figure more irregular on ever smaller scales in an endless iteration. It is found that the mode dispersion and the general character of the scattered radiation change from their well known 'phonon' form to patterns that are dramatically different. The new quantized excitations are called 'fractons' and the cross-over length scale separating the phonon and fracton regimes is very clearly defined. Using these measurements of the dynamic properties of fractal lattices, it is possible to define additional fractal dimensions that characterize other aspects of the fractal nature.
