Introduction

This section introduces two important ideas. First, phenomena that appear similar in nature can occur at completely different scales. Understanding them requires a universal theory, which is in some way scale-free. Second, most systems involve more than one scale, and are often described by models that are developed at a specific scale. Modeling complex systems across scales is a current challenge of multiscale analysis.

• Patterns across scales
• Similarity across scales
• Self-similarity around us
• Other examples of multi-scale systems

Patterns across scales

This section gives examples of pattern-forming systems. Such systems display a variety of structures (e.g. stripes or hexagons), which can occur at a wide range of scales.

There exists a universal theory of pattern formation, which explains why different systems, at different scales, may behave in a similar fashion. This theory extends concepts of dynamical systems theory, in particular bifurcation theory, to systems of partial differential equations.

Here, we only consider qualitative aspects of pattern formation.

Stripe patterns

The figures below show four stripe patterns found in nature. Taken out of context, and in the absence of color, they all look more or less the same. Can you recognize where they come from?

If you knew where these patterns came from, you would be able to estimate the period (say in cm) of each of them. Could you guess what the period is, just by looking at each image? Why or why not? Can you think of other examples of stripe patterns? At which scales do they occur?

Spiral and target patterns

Stripes are not the only patterns that you can find at different scales. Spiral and target patterns occur for instance in the Belousov-Zhabotinsky reaction (a standard oscillatory chemical reaction), in catalytic reactions at the nanoscale, in the heart, and in the aggregation of slime moulds.

Hexagonal patterns

Hexagonal structures, for instance in the form of columnar joints, can be found in the Giant's causeway, and reproduced in experiments with cornstarch.

Online resources on pattern formation

The first six links below illustrate the wide range of systems that exhibit spiral patterns. The last two links discuss hexagonal columnar joints, in nature and in the laboratory.

• The Belousov-Zhabotinsky reaction, from P. Keusch's site, Regensburg University
• Spiral and target patterns in the Belousov-Zhabotinsky reaction, from the University of Leeds
• Spatiotemporal self-organization in a surface reaction: from the atomic to the mesoscopic scale, by C. Sachs, M. Hildebrand, S. Völkening, J. Wintterlin, and G. Ertl
• Spiral patterns in the heart, from S. Panfilov's web site, University of Utrecht
• Nonlinearity in the heart, by A. Holden
• Spiral patterns in slime moulds, from F. Siegert's web site, Ludvig-Maximillians University, Munich
• The Giant's Causeway, United Nations' World Heritage web site
• Columnar joints in cornstarch, from S. Morris's lab, University of Toronto

Similarity across scales

Most of the patterns described above involve at least two "visible" scales. Can you figure out what they are?

In fact, most (if not all) systems around us involve more than one scale. This is often illustrated by movies that zoom into a given object, be it one of our cells, a distant galaxy, or our own solar system. Below we consider a particular class of multi-scale objects, which exhibit self-similarity. This means that the structure of the object looks identical, regardless of the magnification at which it is observed.

Of course, only mathematical structures, called fractals, are self-similar at all scales. Nevertheless, many systems, in nature or in the laboratory, exhibit self-similarity over a finite range of scales. Some example are given below.

Fractals

As mentioned above, a fractal is a self-similar object: it looks like itself through successive magnifications. The Julia Sets MATLAB applet allows you to explore this property.

The movie below zooms into the "filled-in" Julia set for the quadratic map f(z)=z²+c, where z is a complex variable and the parameter c is such that c = -0.52 + 0.51872 i. Colors indicate whether iterates of f starting at a point on the complex plane diverge, and if so, how fast this happens.

The fractal dimension of the Julia set for the quadratic map with parameter c is given by d = 1 + |c|²/(4 ln(2)) + ..., and is a good measure of the "complexity" of this set.

Diffusion-limited aggregation

Diffusion limited aggregation (DLA) is a classical model for fractal growth. The simulation starts with a seed, typically a single particle. Another particle is then launched from a random position far away from the seed, and performs a random walk. Either it eventually moves further away, in which case it is removed from the simulation, or it reaches the seed, and attaches to it. Then a second particle is launched, and the procedure is repeated.

In two dimensions, this process gives rise to branched clusters of fractal dimension d=1.71. Diffusion limited aggregation can also be extended to three dimensions. The result is a complex cluster, an example of which can be found on Chris Rycroft's web page (see links below).

Online resources on DLA

• Diffusion-limited aggregation, a kinetic critical phenomenon, by T.A. Witten and L.M. Sander
• Two-dimensional DLA simulation, by C.-H. Lam, Hong Kong Polytechnic University
• A three-dimensional DLA cluster, from C. Rycroft's web site, Berkeley University

Just-in-time mathematics

• Review of complex numbers

Self-similarity around us

Many examples of self-similar structures can be found in nature. Of course, these objects are self-similar only over a limited range of scales (or magnifications), but are nevertheless remarkable for exhibiting such a property.

Torn plastic sheets

When the two sides of a partially cut plastic sheet are steadily pulled apart, the shape of the tear is self-similar. Carefully controled laboratory experiments on such systems were recently performed by E. Sharon, H. Swinney, and collaborators. They identified at least five levels of self-similarity. A table-top version of these experiments is very easy to realize and provides a simple way of creating one's own self-similar (or fractal-looking) object.

Growing interfaces, cities and forests

Boundaries of domains growing by diffusion of a substance through a porous medium (e.g. ink diffusing through blotting paper) of by chemical reaction (e.g. burnt paper) are often self-similar (in fact self-affine). Similarly, the boundaries of cities, as well as of forests, are often described as self-similar objects. Based on these observations, different models have been proposed for different types of growth.

Flowers and plants

Some vegetables, such as for instance the Romanesco broccoli shown below, as well as trees, flowers and leaves, are self-similar over a few scales.

Image courtesy of A. Calini

Fractal dimension and art

In an interesting article on the aesthetics of fractals, B. Spehar and co-workers report that humans tend to find self-similar objects with a fractal dimension of 1.3 to be the most aesthetically pleasing. They also point out that clouds, plants, trees and waves that are self-similar over a range of scales, can be associated with a fractal dimension close to 1.3.

Online resources on self-similar structures in nature and man-made constructs

• Geometrically driven wrinkling observed in free plastic sheets and leaves, by E. Sharon, B. Roman, and H. Swinney
• Leaves, flowers and garbage bags: making waves, by E. Sharon, M. Marder, and H. Swinney
• Modelling urban growth patterns, by H. Makse, S. Havlin, and E. Stanley
• Universal aesthetics of fractals, by B. Spehar, C.W.G. Clifford, B.R. Newell, and R.P. Taylor
• Fractal analysis of drainage basins on Mars, by T.F. Stepinski, M.M. Marinova, P.J. McGovern, and S.M. Clifford
• Self-organized criticality in forest-landscape evolution, by J.C. Sprott, J. Bolliger, and D.J. Mladenoff

Other examples of multi-scale systems

Fluids

Hydrodynamic turbulence is a typical multiscale phenomenon observed in fluids. Two-dimensional turbulence is easily seen in soap films (see also Marten Rutgers's web site (under "Science")). These two links include photographs of giant soap films and explain how to make them.

Phenomena at the micro- and nano-scales

Below are a few examples of how phenomena at the micro- or nano-scales affect properties at the macroscopic scale.

• The glass of the Lycurgus cup contains nanocrystals of gold and silver that give it the remarkable optical property of changing color when one switches from reflected to transmitted light.
• The micro-scale filaments that cover the petals of the edelweiss protect it from damaging UV lights.
• Thanks to micro- and nano-scale hairs on its toes, the gecko can climb up walls and walk on ceilings.