A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Abstract

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

Non-standard analysis for fractal calculus

“…Essential topological characteristics such as connectivity, ramification, and loopiness were exhibited by fractals and can be quantified using six independent dimension values. However, some fractal types may reduce the count of these dimensions due to their unique traits [29]. Fracture network modeling was proposed, with a specific focus on accentuating fractal attributes within geological formations.…”