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Mathematics
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Advances in Fractal Geometry and Applications
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Advances in Fractal Geometry and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: closed (25 August 2025) | Viewed by 1776

Special Issue Editors

Prof. Dr. Alexander S. Balankin

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Guest Editor
SEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, Mexico
Interests: fractal geometry; fractal analysis; percolation; soft mater; fracture phenomena; fluid flow in porous media
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Didier Samayoa Ochoa

E-Mail Website
Guest Editor
Instituto Politécnico Nacional, SEPI-ESIME Zacatenco, Unidad Profesional Adolfo Lopez Mateos, Mexico City 07738, Mexico
Interests: fractal analysis; fractal continuum; shear deformable beams; structural mechanics; structural health monitoring; fluid dynamics; enhanced oil recovery; elastodynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Presently, fractal geometry is widely used to model complex structures in systems with very different natures. Fractal geometry provides a quantitative description of complex patterns in real-world systems. The aim of this Special Issue is to survey recent developments and applications of the fractal geometry tools.

We look forward to receiving technical notes, full-length research articles, and review papers from different disciplines. Potential topics include, but are not limited to, the following:

  • Foundations and methods of fractal geometry;
  • Characterization of real-world fractals;
  • Fractal continuum models;
  • Physical phenomena in fractal systems;
  • Fractal approaches in material science;
  • Fractal tools for medical diagnostics;
  • Fractal mathematics in econophysics;
  • Fractal concepts in sociophysics.

Prof. Dr. Alexander S. Balankin
Prof. Dr. Didier Samayoa Ochoa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • scale invariance
  • self-similarity
  • self-affinity
  • multi-fractality
  • persistent homology
  • conformal invariance
  • noncommutativity
  • fractal topology
  • embedding
  • fractal measure
  • fractal metric
  • effective degrees of freedom
  • lacunarity
  • fractal topography
  • fractal metrology
  • calculus on fractals
  • fractal continuum
  • metamaterials
  • anomalous diffusion
  • time series
  • fractional dimensional space
  • fractal continuum

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Published Papers (2 papers)

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Research

25 pages, 782 KB  
Article
Degenerate Fractals: A Formal and Computational Framework for Zero-Dimension Attractors
by Ion Andronache
Mathematics 2025, 13(21), 3407; https://doi.org/10.3390/math13213407 - 26 Oct 2025
Viewed by 634
Abstract
This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case D=0 [...] Read more.
This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case D=0 has been little explored. Starting from the question “what happens to a fractal when its complexity collapses completely?”, Moran’s similarity equation becomes tautological (rs=1 with solution s=dimM=0) and that only the Hausdorff and box-counting definitions allow an exact calculation. Based on Banach’s fixed point theorem and these definitions, we prove that the attractor of a degenerate IFS is a singleton with dimH=dimB=0. We develop a reproducible computational methodology to visualize the collapse in dimensions 1–3 (the Iterated Line Contraction—1D/Iterated Square Contraction—2D/Iterated Cube Contraction—3D families), including deterministic and stochastic variants, and we provide a Python script 3.9. The theoretical and numerical results show that the covering box-counting retains unity across all generations, confirming the zero-dimension element and the stability of the phenomenon under moderate perturbations. We conclude that degenerate fractals are an indispensable benchmark for validating fractal dimension estimators and for studying transitions to attractors with positive dimensions. Full article
(This article belongs to the Special Issue Advances in Fractal Geometry and Applications)
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16 pages, 3838 KB  
Article
Metric Morphological Interpretation of 3D Structures by Gray–Scott Model Simulation Utilising 2D Multifractal Analysis
by Akira Takahara and Yoshihiro Sato
Mathematics 2025, 13(19), 3234; https://doi.org/10.3390/math13193234 - 9 Oct 2025
Viewed by 247
Abstract
Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure [...] Read more.
Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure of three-dimensional slices. The proposed methods were verified using Monte Carlo and Gray–Scott simulations; the pixel-existence probability (PEP)-averaging method, which averages the pixel-existence probability in the slice direction, was confirmed to be the most suitable for analysing three-dimensional structures in two dimensions. This method enables a stable quantitative evaluation, regardless of the direction from which the three-dimensional structure is observed. Full article
(This article belongs to the Special Issue Advances in Fractal Geometry and Applications)
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