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Mathematics
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Advances in Fractals
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Advances in Fractals

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 21313

Special Issue Editors

Prof. Krzysztof Gdawiec

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Guest Editor
Institute of Computer Science, Faculty of Science and Technology, University of Silesia, Katowice, Poland
Interests: fractals; computer graphics; pattern recognition
Prof. Dr. Agnieszka Lisowska

E-Mail Website
Co-Guest Editor
Institute of Computer Science, Faculty of Science and Technology, University of Silesia, Katowice, Poland
Interests: image processing; image coding; multiresolution; wavelets; fractals

Special Issue Information

Dear Colleagues,

More than forty years after the term being coined by Mandelbrot, fractals continue to fascinate the scientific community and the public with their wonderful propensity to infinitely repeat the same patterns at various (spatial and/or temporal) scales. One of the most widely known fractals applications is their use in computer graphics for generating stunning and intriguing patterns. However, fractals have also found applications in other areas, such as image processing and compression, pattern recognition, analysis of time series, medicine, geography, physics, etc. With a better understanding of fractals, completely new applications emerge.

The goal of this Special Issue is to publish a collection of interesting and novel original research papers or review articles, on a broad variety of topics related to fractals, viewed either as geometric or analytic objects, and their applications. Potential topics include but are not limited to:

  • Iterated function system and inversion fractals;
  • Fractal generation methods;
  • Fractal image compression;
  • Fractal interpolation and approximation;
  • Fractal curves and surfaces;
  • Fractal dimension;
  • Fractals and dynamical systems;
  • Fractal tilings;
  • Mandelbrot and Julia sets;
  • Superfractals;
  • Multifractals;
  • Applications of fractals, for instance, in image processing, pattern recognition, etc.

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

  • Iterated Function System
  • Fractal Dimension
  • Fractal-Based Computing
  • Self-Similarity
  • V-variable Fractal
  • Multiresolution

Published Papers (10 papers)

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Research

13 pages, 327 KiB  
Article
Fractal Newton Methods
by Ali Akgül and David Grow
Mathematics 2023, 11(10), 2277; https://doi.org/10.3390/math11102277 - 13 May 2023
Cited by 2 | Viewed by 1308
Abstract
We introduce fractal Newton methods for solving f(x)=0 that generalize and improve the classical Newton method. We compare the theoretical efficacy of the classical and fractal Newton methods and illustrate the theory with examples. Full article
(This article belongs to the Special Issue Advances in Fractals)
18 pages, 462 KiB  
Article
Iterative Schemes Involving Several Mutual Contractions
by María A. Navascués, Sangita Jha, Arya K. B. Chand and Ram N. Mohapatra
Mathematics 2023, 11(9), 2019; https://doi.org/10.3390/math11092019 - 24 Apr 2023
Cited by 4 | Viewed by 972
Abstract
In this paper, we introduce the new concept of mutual Reich contraction that involves a pair of operators acting on a distance space. We chose the framework of strong b-metric spaces (generalizing the standard metric spaces) in order to add a more extended [...] Read more.
In this paper, we introduce the new concept of mutual Reich contraction that involves a pair of operators acting on a distance space. We chose the framework of strong b-metric spaces (generalizing the standard metric spaces) in order to add a more extended underlying structure. We provide sufficient conditions for two mutually Reich contractive maps in order to have a common fixed point. The result is extended to a family of operators of any cardinality. The dynamics of iterative discrete systems involving this type of self-maps is studied. In the case of normed spaces, we establish some relations between mutual Reich contractivity and classical contractivity for linear operators. Then, we introduce the new concept of mutual functional contractivity that generalizes the concept of classical Banach contraction, and perform a similar study to the Reich case. We also establish some relations between mutual functional contractions and Banach contractivity in the framework of quasinormed spaces and linear mappings. Lastly, we apply the obtained results to convolutional operators that had been defined by the first author acting on Bochner spaces of integrable Banach-valued curves. Full article
(This article belongs to the Special Issue Advances in Fractals)
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12 pages, 1863 KiB  
Article
Estimation of Citarum Watershed Boundary’s Length Based on Fractal’s Power Law by the Modified Box-Counting Dimension Algorithm
by Michael Lim, Alit Kartiwa and Herlina Napitupulu
Mathematics 2023, 11(2), 384; https://doi.org/10.3390/math11020384 - 11 Jan 2023
Cited by 2 | Viewed by 1354
Abstract
This research aimed to estimate the length of the Citarum watershed boundary because the data are still unknown. We used the concept of fractal’s power law and its relation to the length of an object, which is still not described in other research. [...] Read more.
This research aimed to estimate the length of the Citarum watershed boundary because the data are still unknown. We used the concept of fractal’s power law and its relation to the length of an object, which is still not described in other research. The method that we used in this research is the Box-Counting dimension. The data were obtained from the geographic information system. We found an equation that described the relationship between the length and fractal dimension of an object by substituting equations. Following that, we modified the algorithm of Box-Counting dimension by consideration of requiring a high-resolution image, using the Canny edge detection so that the edges look sharper and the dimension values are more accurate. A Box-Counting program was created with Python based on the modified algorithm and used to execute the Citarum watershed boundary’s image. The values of ε and N were used to calculate the fractal dimension and the length for each scale by using the value of C=1, assuming the ε as the ratio between the length of box and the length of plane. Finally, we found that the dimension of Citarum watershed boundary is approximately 1.1109 and its length is 770.49 km. Full article
(This article belongs to the Special Issue Advances in Fractals)
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16 pages, 8253 KiB  
Article
Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography
by Lateef Olakunle Jolaoso, Safeer Hussain Khan and Kazeem Olalekan Aremu
Mathematics 2022, 10(18), 3324; https://doi.org/10.3390/math10183324 - 13 Sep 2022
Cited by 1 | Viewed by 1205
Abstract
In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s-convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of [...] Read more.
In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s-convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of the obtained polynomiographs compared to the ones from the basic family only. Moreover, the computational results reveal that the value of s in the s-convex combination of the RK iteration has a significant impact on the time taken by the iteration process for approximating the roots of the polynomials. The obtained results are interesting from an artistic and computational point of view. Full article
(This article belongs to the Special Issue Advances in Fractals)
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29 pages, 354 KiB  
Article
Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions
by Binyan Yu and Yongshun Liang
Mathematics 2022, 10(13), 2154; https://doi.org/10.3390/math10132154 - 21 Jun 2022
Cited by 11 | Viewed by 1335
Abstract
In the present paper, we try to estimate the fractal dimensions of the linear combination of continuous functions with different fractal dimensions. Initially, a general method to calculate the lower and the upper Box dimension of the sum of two continuous functions by [...] Read more.
In the present paper, we try to estimate the fractal dimensions of the linear combination of continuous functions with different fractal dimensions. Initially, a general method to calculate the lower and the upper Box dimension of the sum of two continuous functions by classifying all the subsequences into different sets has been proposed. Further, we discuss the majority of possible cases of the sum of two continuous functions with different fractal dimensions and obtain their corresponding fractal dimensions estimation by using that general method. We prove that the linear combination of continuous functions having no Box dimension cannot keep the fractal dimensions closed. In this way, we have figured out how the fractal dimensions of the linear combination of continuous functions change with certain fractal dimensions. Full article
(This article belongs to the Special Issue Advances in Fractals)
23 pages, 5857 KiB  
Article
Organization Patterns of Complex River Networks in Chile: A Fractal Morphology
by Francisco Martinez, Hermann Manriquez, Alberto Ojeda and Gabriel Olea
Mathematics 2022, 10(11), 1806; https://doi.org/10.3390/math10111806 - 25 May 2022
Cited by 9 | Viewed by 2263
Abstract
River networks are spatially complex systems difficult to describe by using simple morphological indices. To this concern, fractal theory arises as an interesting tool for quantifying such complexity. In this case of study, we have estimated for the first time the fractal dimension [...] Read more.
River networks are spatially complex systems difficult to describe by using simple morphological indices. To this concern, fractal theory arises as an interesting tool for quantifying such complexity. In this case of study, we have estimated for the first time the fractal dimension of Chilean networks distributed across the country, analysed at two different scales. These networks insert into variable environments, not only from a climatic and hydrological point of view, but also from a morphological point of view. We investigate to which extent the fractal dimension is able to describe the apparent disorganized character of landscape, by applying two methods. Striking patterns of organization related to Horton ratios and the fractal dimension are reported and discussed. This last parameter depends on the scale of the network, showing interesting groupings by tectonic and climatological factors. Our results suggest that under restricted conditions, the fractal dimension could help to capture the intricate morphology of Chilean networks and its links with the hydrological, climatic, and tectonic conditions present across the country. Full article
(This article belongs to the Special Issue Advances in Fractals)
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21 pages, 1490 KiB  
Article
Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution
by Alison K. Cheeseman and Edward R. Vrscay
Mathematics 2022, 10(5), 839; https://doi.org/10.3390/math10050839 - 7 Mar 2022
Cited by 7 | Viewed by 3028
Abstract
Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., [...] Read more.
Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log–log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions. Full article
(This article belongs to the Special Issue Advances in Fractals)
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11 pages, 317 KiB  
Article
The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals
by Mohsen Soltanifar
Mathematics 2022, 10(5), 706; https://doi.org/10.3390/math10050706 - 24 Feb 2022
Viewed by 1777
Abstract
In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff [...] Read more.
In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces. Full article
(This article belongs to the Special Issue Advances in Fractals)
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17 pages, 4138 KiB  
Article
Relationship between the Mandelbrot Algorithm and the Platonic Solids
by André Vallières and Dominic Rochon
Mathematics 2022, 10(3), 482; https://doi.org/10.3390/math10030482 - 2 Feb 2022
Cited by 1 | Viewed by 2700
Abstract
This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron), and the Firebrot (tetrahedron). In particular, we establish a geometrical classification [...] Read more.
This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron), and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the properties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so-called “Stella octangula” as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of M3, we show that one of them corresponds to a third Platonic solid: the cube. Full article
(This article belongs to the Special Issue Advances in Fractals)
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9 pages, 779 KiB  
Article
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals
by Mohsen Soltanifar
Mathematics 2021, 9(13), 1546; https://doi.org/10.3390/math9131546 - 1 Jul 2021
Cited by 4 | Viewed by 3356
Abstract
How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual [...] Read more.
How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions. Full article
(This article belongs to the Special Issue Advances in Fractals)
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