What is dimension?
  This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the generalization to multifractals, and discrete and continuous scale invariance with the emergence of complex dimensions. The idea of renormalization group flow equations can be introduced in this framework, to show how the power laws determined by dimensional analysis (engineering dimensions) get modified by extra anomalous dimensions. As an example of the RG flow equation, the scaling of conductance by disorder in the context of localization is used. A few technicalities, including the connection between entropy and fractal dimension, can be found in the appendices. 
Somendra M. Bhattacharjee
cond-mat.stat-mech cond-mat.soft