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Fractal Fract
Special Issues
Fractal Dimensions with Applications in the Real World
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Fractal Dimensions with Applications in the Real World

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Geometry".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 6043

Special Issue Editors

Dr. Jun Luo

E-Mail Website
Guest Editor
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Interests: fractal geometry; topology on fractals; Lipschitz equivalence; Gromov hyperbolic graphs
Dr. Xiaohao Cai

E-Mail Website
Guest Editor
School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK
Interests: computer vision; image processing; machine/deep learning; scientific computing
Special Issues, Collections and Topics in MDPI journals
Dr. Liang-Yi Huang

E-Mail Website
Guest Editor
College of Mathematics and Statistics, Chongqing University, Chongqing, China
Interests: multifractal analysis; Lipschitz equivalence; topology

Special Issue Information

Dear Colleagues,

The concept of fractals was introduced by B. Mandelbrot in the last 1970s as a class of highly irregular sets often presenting with infinite complexity, self-similarity and the nonintegral Hausdorff dimension. It has had a great impact in the development of mathematics and many other disciplines of science. In mathematics, fractal originates in chaos and dynamic systems, and soon after it was found that fractals appear in almost every area and are susceptible to systematic studies using classical and contemporary methods. In the last four decades, a large part of fractal research has been related to the dimension theories and structures of self-similar sets and measures.

The aim of this Special Issue is to present the up-to-date progress in fractal dimensions and their various applications to the real world. Topics invited for submission include, but are not limited to, the following:

  • Fractal dimensions of iterated function systems;
  • Fractal dimensions of self-similar measures;
  • Hausdorff dimension of fractal graphs;
  • Lipschitz equivalence of fractal sets;
  • Topological structures of fractal sets;
  • Diophantine approximation;
  • Fractal dimensions in dynamical systems;
  • Applications of fractal dimensions.

Dr. Jun Luo
Dr. Xiaohao Cai
Dr. Liang-Yi Huang
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. Fractal and Fractional is an international peer-reviewed open access monthly 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 2700 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

  • iterated function systems
  • self-similar sets
  • self-affine sets
  • fractal measures
  • fractal graphs
  • beta-expansions
  • Hausdorff dimension
  • box-counting dimension
  • Lipschitz equivalence
  • topological structures

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Published Papers (5 papers)

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Research

41 pages, 5963 KiB  
Article
Multiple-Stream Models for a Single-Modality Dataset with Fractal Dimension Features
by Yen-Ching Chang
Fractal Fract. 2025, 9(4), 248; https://doi.org/10.3390/fractalfract9040248 - 15 Apr 2025
Viewed by 278
Abstract
Multiple-stream deep learning (DL) models are typically used for multiple-modality datasets, with each model extracting favorable features from its own modality dataset. Through feature fusion, multiple-stream models can generally achieve higher recognition rates. While feature engineering is indispensable for machine learning models, it [...] Read more.
Multiple-stream deep learning (DL) models are typically used for multiple-modality datasets, with each model extracting favorable features from its own modality dataset. Through feature fusion, multiple-stream models can generally achieve higher recognition rates. While feature engineering is indispensable for machine learning models, it is generally omitted for DL. However, feature engineering can be regarded as an important supplement to DL, especially when using small datasets with rich characteristics. This study aims to utilize limited existing resources to improve the overall performance of the considered models. Therefore, I choose a single-modality dataset—the Chest X-Ray dataset—as my original dataset. For ease of evaluation, I take 16 pre-trained models as basic models for the development of multiple-stream models. Based on the characteristics of the Chest X-Ray dataset, three characteristic datasets are generated from the original dataset, including the Hurst exponent dataset (corresponding to a fractal dimension dataset), as inputs to the multiple-stream models. For comparison, various multiple-stream models are developed based on the same dataset. The experimental results show that, with feature engineering, the accuracy can be raised from 91.67% (one-stream) to 94.52% (two-stream), 94.73% (three-stream), and 94.79% (four-stream), while, without feature engineering, it can be increased from 91.67% to 92.35%, 93.49%, and 93.66%, respectively. In the future, the simple yet effective methodology proposed in this study can be widely applied to other datasets, in order to effectively promote the overall performance of models in scenarios characterized by limited resources. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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19 pages, 5031 KiB  
Article
Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis
by Zelin Liu, Senlin Xie, Yajun Yin and Teng Su
Fractal Fract. 2025, 9(2), 93; https://doi.org/10.3390/fractalfract9020093 - 1 Feb 2025
Viewed by 655
Abstract
Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic [...] Read more.
Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic loading–unloading tests were combined with real-time NMR monitoring to model porosity recovery, pore size evolution, and energy dissipation, while also calculating the fractal dimensions of pores in relation to stress. The results show that during the compaction phase, primary pores are compressed with limited recovery after unloading. In the elastic phase, both adsorption and seepage pores transform significantly, with most recovering post-unloading. After yield stress, new fractures and pores form, and unloading enhances fracture connectivity. Seepage pore porosity shows a negative exponential relationship with axial strain before yielding, and a logarithmic relationship afterward. The fractal dimension of adsorption pores decreases during compaction and increases afterward, while the fractal dimension of seepage pores decreases before yielding and increases post-yielding. These findings provide new insights into the flow patterns of methane in coal seams. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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12 pages, 549 KiB  
Article
Generalized Dimensions of Self-Affine Sets with Overlaps
by Guanzhong Ma, Jun Luo and Xiao Zhou
Fractal Fract. 2024, 8(12), 722; https://doi.org/10.3390/fractalfract8120722 - 6 Dec 2024
Viewed by 779
Abstract
Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we [...] Read more.
Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we define a new FTC on self-affine IFS and obtain an analogous formula on the generalized dimensions of self-affine sets. The generalized dimensions raised by He and Lau are used to estimate the Hausdorff dimension of self-affine sets. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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19 pages, 10859 KiB  
Article
Reduced Order Modeling of System by Dynamic Modal Decom-Position with Fractal Dimension Feature Embedding
by Mingming Zhang, Simeng Bai, Aiguo Xia, Wei Tuo and Yongzhao Lv
Fractal Fract. 2024, 8(6), 331; https://doi.org/10.3390/fractalfract8060331 - 31 May 2024
Viewed by 1123
Abstract
The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based [...] Read more.
The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model’s accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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18 pages, 2306 KiB  
Article
New Properties and Sets Derived from the 2-Ball Fractal Dust
by Mario A. Aguirre-López, José Ulises Márquez-Urbina and Filiberto Hueyotl-Zahuantitla
Fractal Fract. 2023, 7(8), 612; https://doi.org/10.3390/fractalfract7080612 - 8 Aug 2023
Viewed by 1436
Abstract
Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from [...] Read more.
Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from IFS fractals by adding minimal modifications to their defining rule. In this work, we propose two modifications to a fractal recently introduced by the authors: the so-called 2-ball fractal dust, which consists of a set of balls diminishing in size along an iterative process and delimited by an enclosing square. The proposed modifications are (a) adding a resizer parameter to introduce an interaction between the generator and generated ball elements and (b) a new fractal embedded into the 2-ball fractal dust, having the characteristic of filling zones not covered by the previous one. We study some numerical properties of both modified resulting sets to gain insights into their general properties. The resulting sets are geometrical forms with potential applications. Notably, the first modification generates an algorithm capable of producing geometric structures similar to those in mandalas and succulent plants; the second modification produces shapes similar to those found in nature, such as bubbles, sponges, and soil. Then, although a direct application of our findings is beyond the scope of this research, we discuss some clues of possible uses and extensions among which we can remark two connections: the first one between the parametrization we propose and the mandala patterns, and the second one between the embedded fractal and the grain size distribution of rocks, which is useful in percolation modeling. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
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