A Brief Survey of Paradigmatic Fractals from a Topological Perspective

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...

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Bibliographic Details
Published inFractal and fractional Vol. 7; no. 8; p. 597
Main Authors Patiño Ortiz, Julián, Patiño Ortiz, Miguel, Martínez-Cruz, Miguel-Ángel, Balankin, Alexander S.
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2023
Subjects
Cantor, Georg
connectedness
connectivity
Decomposition
degrees of freedom
dimension numbers
Euclidean space
Fractal geometry
Fractals
loopiness
Menger, Karl
ramification
Scale invariance
Self-similarity
Set theory
Topology
New York
Germany
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Summary: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.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract7080597