In this paper we first show that the usual three dimensionality of space, which is taken for granted, results from the spinorial behaviour of Fermions, which constitute the material content of the universe. It is shown that the resulting three dimensionality rests on two factors which have been hitherto ignored, viz., a Machian or holistic property and the stochastic underpinning of the universe itself. However the dimensionality is scale dependent in the sense that at very large scales, or at very small scales, we encounter a different dimensionality, as indeed is borne out by observation and experiment. For example the large scale structures in the universe are cellular in nature on the one hand, and we encounter fractional charges and handedness at very small scales. Finally it is shown how fractal dimensions can emerge and as an illustrative example it is shown how this could explain the magnetism of objects like Planets on the one hand and White Dwarf stars and Pulsars on the other.