You’d be forgiven for falling in love with the beauty of natural phenomenon all around us, even those outside our direct line of sight, far out in the observable universe. Most agree that there is beauty, in some form or the other, in almost everything that the laws of physics help facilitate. But few people notice the underlying patterns, and the rationale of the cosmos, behind the beautiful observations we get to make.
The universe is interesting because it was born in an incomprehensible flash – what we call the Big Bang, and is still young relative to our estimates on the lifetimes of stars. Our universe is too young too reach thermodynamic equilibrium (can be vaguely compared to saturation/steady state); the universe as a whole is a non-equilibrium system because stars are young and have not yet reached thermodynamic equilibrium, the nuclear fuel in their core is yet to be entirely consumed. The ﬂux of energy from this core through the surface of the star and out into space drives the complex dynamics of the star’s plasma and magnetic ﬁeld. Similarly, because the Earth is young and its interior has not yet cooled down, the ﬂux of heat from its hot core out through its surface, together with heat received from the Sun, generates the dynamics of the atmosphere, ocean, and mantle, in a way responsible for such interesting creation and beautifully decorates clouds, waves, patterns on rocks. And it is this same ﬂux of energy from the Earth and Sun that sustains Earth’s intricate biosphere. However this explanation was not self-sufficient to give s quantitative approach towards this interesting topic. To make progress, scientists have used different softwares such as Matlab to simulate results mathematically.
The primary goal of this article is to provide a popular level approach to pattern formation, as can be gauged from my hints at a non linear system in the above paragraph. I would be happy if this article can arouse an interest about the beautiful things we see and create.
These patterns have a pleasing rhythm to them, inducing appreciation similar to how many of us appreciate music! To give it a definite structure, naturally generated patterns are discussed first and man-made are discussed just after that. We begin with phenomena at some of the largest quantities – the length and time scales of the Universe, and then descend to human length and time scales which include Brewster angle microscopy analysis.
An example of an interesting pattern on the grandest scales of the Universe is the recently measured organization of galaxies into sheets and voids. Current observations indicate that our universe is expanding in all directions, with all the constituent galaxies moving away from each other and from the Earth. The pace of this expansion is directly related to the distance between objects – the further you are, the faster you recede. Light from a galaxy that is moving away from Earth is Doppler-shifted to a longer wavelength (becomes redder) compared to the light coming from an identical but stationary galaxy. Surprisingly, the galaxies do not ﬁll space with uniformity like molecules in a gas (we can say we do have perfect row column structure) but instead are clustered in sheets and walls with large relatively empty spaces between them. Here the pattern is not a geometric structure (e.g. a lattice) but a statistical deviation from randomly and uniformly distributed points that is difficult for the human visual system to quantify. A good analogy will be that of a foam of bubble like structures in which the galaxies are concentrated on the surfaces of the bubbles. The reason for this galactic structure is not known at this time but is presumably a consequence of the details of the Big Bang (when matter ﬁrst formed), the expansion of the universe, the effects of gravity, and the effects of the mysterious dark matter that makes up most of the mass of the Universe but which has not yet been directly observed or identiﬁed.
Galaxies themselves are spectacular candidates for pattern recognition. Now, there are a lot of stars in a galaxy. A lot. How many? Approximately 1010 stars make up an average galaxy, and the galaxy itself has a net angular momentum from the way it was born by the condensation of a large hydrogen cloud. Based on these numbers and stats, you might expect galaxies to be rotating featureless blobs of stars with a mass density that varies monotonically as a function of radius from the center. Such blobs do in fact exist and are known as elliptical galaxies. However, many galaxies do have a uniform mass density in the form of two or more spiral arms.
Our own galaxy, the Milky Way, happens to be such a spiral galaxy. Why galaxies evolve to form spiral arms is poorly understood and is an important open question in current astrophysical research. Laboratory experiments show that spiral formation is common for non-equilibrium media that have a tendency to oscillate in time or that support wave propagation. Further, experiments show that a tendency to form spirals is insensitive to details of the medium supporting the spiral. So a galactic spiral may not be too surprising since there are mechanisms in galaxies that can produce wave propagation.
Let’s zoom in on to shorter scales and take a look at some exquisite patterns available in our terrestrial environment. You might have seen ripples in the riverside or the sea beach and the desert. The wind drives the sand from one place to another, making ripples and dunes. Some places fall short of sand as it moves to somewhere else, resulting in a bulging shape. After the process, it resembles a network elongated throughout the area. Not only air, water can also form such structures over sand.
Another example of terrestrial pattern formation would be snowflakes. This is a non-equilibrium system where the patterns are formed by crystalline dendrites (these are the needle-like branches of a snowﬂake) that grow into the surrounding air. The dendrite’s shape itself changes as the system grows. The non-equilibrium driving for snowﬂake formation is the presence of air that is supersaturated with water vapour. (In contrast, an equilibrium state would involve static ice crystals in contact with saturated water vapour.) Each tip of the snowﬂake grows by adsorbing water molecules onto its surface from the surrounding air, and the rate at which the tip grows and its shape are determined in a complex way by how rapidly water molecules in the surrounding air can diffuse to the crystalline tip, and by how rapidly the heat released by adsorption can be dissipated by diffusion within the air.
There are many other patterns available in nature such as the Fern tree pattern, patterns seen in shells of sea animals (mollusks), waves, rocks and most interestingly you are likely to find patterns in anything that has growth and evolved with time. The pictures below will elucidate this fact.
Growth has a lot to do with these structured fractal formations. Though each of the formations have a unique explanation, natural growth can be vaguely generalized and can be simply understood by drawing it on a blank sheet. Take 5 blocks with glue over them and the different blocks fall down from above like in a video game. The first block which falls gets attached to the 4th block in the row of five blocks. The next block which falls actually has a magical affinity to get attached to the block which has just been attached with the 4th block. Thus the structure grows and takes the network like shape which is given as a picture below. This process has even been used as algorithms in games as it creates a tricky situation for the player.
What about artificial patterns? A significant number of natural patterns are hard to understand due to the complex natural processes they have undergone. There are multiple variables – usually 3 or 4 different natural phenomena influencing the formation of these patterns. However, in artificial design, patterns are introduced via simulation in a regulated manner for the sake of simplicity and statistical analysis. It follows a hierarchy of complexity with periodicity, when it comes to experimenting via simulation. All of this obviously makes it easier for us to understand them.
An interesting example of artificial pattern formation that might interest you would be the one which intrigued me and shifted my attention into the topic. Patterns generated in the lab are basically processes induced in a particular system artificially but following the standard laws of physics. Such pattern formation is, in a number of instances, guided by the competing tendencies of integration and segregation of constituents in mixtures with long term instabilities. Hence studies of patterns on a long time scale are important. There is therefore, a need to sustain systems for extended periods in order to study changes and after a certain long time, the change reaches its saturation and the system becomes stable. Here I have explained such a system which is similar to my explanation above.
A system can be defined which consists of two-dimensional network of mixed monolayers of Stearic Acid (StA) and Dodecane-thiol(12 thiol groups) capped Gold nano particles at the air-water interface. Dipolar and amphiphilic lipid molecules absorbed on to the water surface forms stable monomolecular films and the air water interface. Whereas spherical hydrophobic AuNP (Au-Gold NP-nano particles) form a simple 2D liquid at the air water interface which is unstable in nature and buckle up on compression and on further compression it forms multilayers. A mixture of surfactant and hydrophobic AuNPs thus presents an interesting scope for investigating the manner in which the self-assembly process of the AuNP’s and the stability of the film may be influenced. This system grows under constraints of small space which instigates and plays a major role to form the self-assembled patterns. This whole system is studied by a certain process known Brewster Angle Microscopy (BAM). Some of those pictures in different parameters are given below.
The theory based explanation would definitely be supplemented nicely if we see a picture simulated using MATLAB. All the pattern formations in nature or artificial patterns follow certain “rules’’. These are all non linear equilibrium systems. Succeeding generations of such patterns are heavily dependent on non-linear partial differential equations. Some processes follow a particular equation, whereas others follow a different one. Depending on the parameters, the chaos changes its nature and makes its target field decorated in different ways. Consider below a system where the KPP equation is the main pattern generating factor and two of the pictures given below are unstable and stable formation at different time frames.
Another interesting approach to simulation involves feeding non-linearity in certain linear systems and generating a comparative picture after adding non-linear diffusion. This complete process is done using a very simple and commonly used algorithm. More easily, this can be envisioned as a system where non-linearity is nothing but an error signal which is added to a stable system to investigate the instability and disorders maintaining a time scale. This is a very vague explanation – so here’s some simplification. Below, we have a picture of polka dots which are black dots over a white surface. Treating the image on a graph, we get two axes – X & Y to play with. On introducing some noise as a function of different parameters, non-linear diffusion is added to the image. Compare the input image and output image and you will be amazed to see a network structure similar to the one seen in the supposed “organic” experiment done on air water interface.
Some interesting patterns are given below which can be simulated in MATLAB and are of great importance when it comes to research.
This post is by no means an exhaustive description of the multiple patterns available for observation in nature, but there’s no denying that this concept itself is majorly fascinating. Interestingly, pattern formation also has many industrial applications. I think the use of patterns in design and decoration is already well known. Whether it is in decorating and styling your website or for printing a certificate, it is very encouraging to see people take help of guilloche patterns. Patterns are also studied to understand the complex molecular structure of DNA. It is a proven fact that it has greatly improved the understanding of the complex process in its construction and working. Patterns and their understanding have even gone on to improve aerospace engineering and also satellite technology to better understand ocean currents. Directly or indirectly, patterns are a very important part that is deeply connected to our life but often its study has faced hardships due to a flawed approach. The topic was initially thought to be overly complicated and mundane, but with readily available sophisticated simulation, we are now on the verge of uncovering the intricate and magnificent world of patterns!