A time-developing phenomenon is called self-similar if the spatial distributions of its properties at different points in time can be obtained from one another by a similarity transformation, and identifying one of the independent variables as time.  However, this is where Dimensional Analysis goes beyond Pi Theorem into self-similarity, which has represented progress for researchers.

In recent years there has been a surge of interest in self-similar solutions of the First and Second kind.  Such solutions are not newly discovered; they have been identified and named by Zel'dovich, a famous Russian Mathematician in 1956. They have been used in the context of a variety of problems, such as shock waves in gas dynamics, and filtration through elasto-plastic materials.

Self-Similarity has simplified computations and the representation of the properties of phenomena under investigation.  It handles experimental data, reduces what would be a random cloud of empirical points to lie on a single curve or surface, and constructs procedures that are self-similar.  Variables can be specifically chosen for the calculations.