Symmetry and Fractals

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 January 2016) | Viewed by 21308

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Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
Interests: mathematical physics; Euclidean field theory; reflection positivity; representation theory; operators in Hilbert space; harmonic analysis; fractals; wavelets; stochastic processes; financial mathematics
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Dear Colleagues,

Many natural phenomena exhibit repeating patterns repeated up to scale-similarity at every scale. This leads to a new notion of symmetry. The mathematical name for this is “fractal”, and it is also used when patterns of scale-self-similarity occur nearly the same at different levels; for example, in the magnifications of the Mandelbrot set. Fractals are used when a specific and detailed pattern is seen to repeat itself.

Fractals are different from other geometric figures. In fractals, one sees one-dimensional lengths doubled, while the corresponding spatial content of the fractal scales by a power that is not necessarily an integer; a case in point is fractional Brownian motion. Such power-exponents are called fractal dimensions, or scale dimensions (akin to Hausdorff dimension), and scale-dimension is usually different from the fractal's topological dimension.

As mathematical entities, fractals are usually nowhere differentiable. Examples: (i) Infinite fractal curves winding through space differently from an ordinary line, still being a one-dimensional line yet having a fractal dimension indicating it also resembles a surface. (ii) Animation of a Sierpinski carpet, a famous two-dimensional fractal. (iii) Brownian motion in its many guises.

The term “fractal” (Latin frāctus meaning “broken” or “fractured”) was first used by mathematician Benoît Mandelbrot in 1975. In this work, he extended the more traditional concept of theoretical fractional dimensions to a host of geometric patterns in nature.

The general consensus is that theoretical fractals are infinitely self-similar, iterative constructs, i.e., detailed mathematical constructs having fractal dimensions. While there may not be an agreed upon definition, many examples and powerful applications have been formulated and studied in great depth.

In recent years, new and intriguing relationships have been discovered between symmetry and fractals. It often takes a form different from what is traditionally seen in symmetry considerations (from physics and harmonic analysis), and it is also referred to as “scale-symmetry”, “self-similarity”, “similarity, or symmetry up to scale”, “similarity in the small and in the large”. It lends itself to new and powerful multi-resolution algorithms; akin to wavelet constructions; with the function space divided up into scales of subspaces, and associated similarity operators (in these spaces). In special cases, one gets conformal self-similarity (in multi-resolutions of scale), as is seen in complex dynamics (Julia sets, etc.), or even self-similarity defined from a system of affine transformations. Iterated function systems (IFS), frames, and wavelets. This, in turn, has led to a body of interdisciplinary interaction, which includes harmonic analysis on fractals and their multiresolutions, and an associated dual setting of discrete analysis, covering, among other topics, wavelet algorithms and signal processing. The latter lends itself to methods of harmonic analysis and geometric measure theory; as well as connections to probability theory, wavelets, and frames. These themes continue to inspire striking new results within pure and applied mathematics. The power of IFS and self-similarity is relevant in numerous ways, in image processing, it allows one to encode images with a surprisingly small number of parameters, those which specify the IFS at hand. The same viewpoint yields explicit dynamical systems with chaotic behavior; as well as to new insight into dependence on initial conditions.

There are also connections to solid-state physics, statistical mechanics, and to spectral-tile results; areas, which, in turn, lead to new iterated function systems. Connections to analysis on fractals have been shown to be of relevance to yet other applications; for example, to an analysis of “big data”, which in turn often yields hidden self-similarities of the kind that are typical for fractals. They can be exploited and lead to a significant reduction in computational time. Self-similarity itself can be viewed as a key, defining feature in all three of the subjects of wavelet theory, fractals, and iterated functions systems.

While tools from such traditional areas of Fourier decompositions and wavelets use orthonormal bases in Hilbert space, the theory of frames is more flexible, not requiring orthogonality. Thus, frames allow for the kind of redundancy, typical in IFSs, used in engineering applications. Indeed, frames have been used recently in the study of compressed sensing.

Frames, and their refinement, fusion frames, are used in applications exhibiting both symmetry of scales and intrinsic redundancies; e.g., in filter bank theory, sigma-delta quantization, image processing, and wireless communications. Other applications, such as distributed processing and sensor networks in the human brain, require clever splitting of large frame systems into sets of (overlapping) smaller systems. Mathematically, this must be done in a way that allows for effective processing within each individual subsystem, leading in turn to efficient algorithms with robustness.

However, applications in a different direction include iterative algorithms for analysis of graphs and fractals.

Prof. Dr. Palle E.T. Jorgensen
Guest Editor

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Keywords

  • fractal
  • iterated function
  • multi-resolution
  • scale-similarity
  • tilings
  • invariant measures
  • probability
  • noisy signals
  • function spaces
  • discrete Laplacians
  • Markov chains
  • harmonic analysis
  • chaos
  • harmonic analysis
  • fractional Brownian motion
  • algorithms, image compression

Published Papers (4 papers)

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Article
Investigating the Performance of a Fractal Ultrasonic Transducer Under Varying System Conditions
by Euan Barlow, Ebrahem A. Algehyne and Anthony J. Mulholland
Symmetry 2016, 8(6), 43; https://doi.org/10.3390/sym8060043 - 6 Jun 2016
Cited by 4 | Viewed by 4138
Abstract
As applications become more widespread there is an ever-increasing need to improve the accuracy of ultrasound transducers, in order to detect at much finer resolutions. In comparison with naturally occurring ultrasound systems the man-made systems have much poorer accuracy, and the scope for [...] Read more.
As applications become more widespread there is an ever-increasing need to improve the accuracy of ultrasound transducers, in order to detect at much finer resolutions. In comparison with naturally occurring ultrasound systems the man-made systems have much poorer accuracy, and the scope for improvement has somewhat plateaued as existing transducer designs have been iteratively improved over many years. The desire to bridge the gap between the man-made and naturally occurring systems has led to recent investigation of transducers with a more complex geometry, in order to replicate the complex structure of the natural systems. These transducers have structures representing fractal geometries, and these have been shown to be capable of delivering improved performance in comparison with standard transducer designs. This paper undertakes a detailed investigation of the comparative performance of a standard transducer design, and a transducer based on a fractal geometry. By considering how these performances vary with respect to the key system parameters, a robust assessment of the fractal transducer performance is provided. Full article
(This article belongs to the Special Issue Symmetry and Fractals)
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1544 KiB  
Article
On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map
by Da Wang and ShuTang Liu
Symmetry 2016, 8(2), 7; https://doi.org/10.3390/sym8020007 - 26 Jan 2016
Cited by 9 | Viewed by 4288
Abstract
A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos [...] Read more.
A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009, 19:2123–2129) and (Nonlinear. Dyn. 2013, 73:1155–1163), the authors presented the two kinds of fractal sets of a class of alternated complex map and left some visually observations to be proved about the boundedness and symmetry properties of these fractal sets. In this paper, we improve the previous results by giving the strictly mathematical proofs of the two properties. Some simulations that verify the theoretical proofs are also included. Full article
(This article belongs to the Special Issue Symmetry and Fractals)
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209 KiB  
Article
On the Continuity of the Hutchinson Operator
by Michael F. Barnsley and Krzysztof Leśniak
Symmetry 2015, 7(4), 1831-1840; https://doi.org/10.3390/sym7041831 - 15 Oct 2015
Cited by 10 | Viewed by 4763
Abstract
We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded [...] Read more.
We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature. Full article
(This article belongs to the Special Issue Symmetry and Fractals)

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3091 KiB  
Letter
An Ultrasonic Lens Design Based on Prefractal Structures
by Sergio Castiñeira-Ibáñez, Daniel Tarrazó-Serrano, Constanza Rubio, Pilar Candelas and Antonio Uris
Symmetry 2016, 8(4), 28; https://doi.org/10.3390/sym8040028 - 21 Apr 2016
Cited by 8 | Viewed by 4320
Abstract
The improvement in focusing capabilities of a set of annular scatterers arranged in a fractal geometry is theoretically quantified in this work by means of the finite element method (FEM). Two different arrangements of rigid rings in water are used in the analysis. [...] Read more.
The improvement in focusing capabilities of a set of annular scatterers arranged in a fractal geometry is theoretically quantified in this work by means of the finite element method (FEM). Two different arrangements of rigid rings in water are used in the analysis. Thus, both a Fresnel ultrasonic lens and an arrangement of rigid rings based on Cantor prefractals are analyzed. Results show that the focusing capacity of the modified fractal lens is better than the Fresnel lens. This new lens is believed to have potential applications for ultrasonic imaging and medical ultrasound fields. Full article
(This article belongs to the Special Issue Symmetry and Fractals)
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