Symmetry and Fractals

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.

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• 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