• Sergey G. AbaimovEmail author
Part of the Springer Series in Synergetics book series (SSSYN)


The power-law dependences in the vicinity of a critical point could often be attributed to the self-similarity and fractal nature of clusters. Therefore, in this chapter, we discuss the basic formalism of fractals. We consider this chapter to be a prerequisite of fractals. Required for further discussions, we consider understanding of both the concept of fractal dimensionality and the origin of fractal power-law dependences. The reader, proficient in these concepts, can skip this chapter.

Since we consider this chapter to be a prerequisite, we only briefly discuss ideas behind the formalism of fractals, providing basic, intuitive understanding. For ­further study, we refer the reader to brilliant textbooks (Feder 1988; Vicsek 1992; Falconer 2003) and references therein.

Besides the fractals, we also discuss multifractals. Although multifractals with complex geometric support will not be applied directly in the further chapters, we encourage the reader to study their formalism in view of its similarities with the concepts of statistical physics.


Infinite Iteration Maximal Term Sierpinski Carpet Daughter Branch Logarithmic Accuracy 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Advanced Structures, Processes and Engineered Materials CenterSkolkovo Institute of Science and TechnologySkolkovoRussia

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