We propose a method to perform the intermittency analysis of multi hadron production in up to three momentum space dimensions. The analysis of realistic (approximately) selfsimilar cascade models shows that the strongest rise of multiplicity moments with decreasing phase space intervals and the closest approach to a power law occurs in three dimensions. Moments of higher orderF(q) are related to the second momentF(2) by a power law in good approximation. These powers are found independent of the dimensions in the models and in available data; furthermore, they are only weakly dependent on the reaction type. The relevance to the quark gluon plasma search is discussed.