Patterns In Nature: Waves and Spirals

The information here will be instructive regarding the functioning of the universe (of which the designer should have at least a rough grasp). It is useful when considering the temporal aspects of growth (i.e. how things grow over time), but is only marginally useful as a physical design template. It does, however, happen to be really fascinating.

In Bill Mollison’s Permaculture: A Designers’ Manual there is a passing reference to “Winfree’s ‘doped’ chemicals” that long ago caught my eye. The chemicals Arthur Winfree was working on were, in fact, the Belousov Zhabotinsky Reaction, which is what is known as a reaction-diffusion system. As we’ll see, such systems govern many of the biological and physical systems we see in nature.

Before we get too far ahead of ourselves, let’s start with Boris Belousov. Belousov was looking for clues on the glycolysis process (and happened to be on the right track, too). He found a reaction that would react, reverse, and repeat the process with a regular period.

In trying to get his research published, he was given the brush off by the establishment because, to the world of chemistry at that time, what he was claiming sounded akin to striking a match, then having the process reverse, then reignite, then reverse, and so on. All known reactions at the time settled linearly into an equilibrium state.

Later, Anatoly Zhabotinsky took a look at Belousov’s work, and expanded on it. The reaction the two scientists pioneered became known as the Belousov-Zhabotinsky reaction. The BZ Reaction is a catalytic reaction in which the catalyst forms out of the reactions own reactants. This autocatalysis burns out and meets with a different reaction that forms products needed to restart the autocatalysis once again.

In the reaction, travelling circular waves emerge and propagate outward. Should those waves happen to meet with an obstruction in the medium, they form spiral waves – something we will come back to several times in our explorations here. Describing it only goes so far. It’s better to see it for yourself.

Reaction-Diffusion in Nature

Reaction-diffusion systems also occur in nature – a lot. In a biological system, cells will start in a state susceptible to excitation. They become excited from the stimulus of neighbours, passing on the excited state. They then go into a period of recovery. This is embedded in the mathematical template governing the chemical automation that runs your body, other life forms, species interaction, and possibly galaxies, too.

Your heart operates this way, for instance. A wave propagates across your heart, giving the cells an instruction to beat. If it meet an obstruction – a damaged area of the heart – a spiral wave can form and propagate, as in the BZ reaction. This is what happens in ventricular fibrillation when one has a heart attack.

This same reaction-diffusion dynamic (excitation, spread, recovery) occurs in interacting species populations, as well. For instance, you see the waves temporally in pest populations in your garden. The pests appear, providing an untapped food source. Predator populations then respond with increased localized populations, reducing the initial wave of pests. The loss of food leads to a decline in predator population, allowing a recovery of the pest/prey population.

Similarly, you might also notice this model is at least prima fascia applicable to memetic social systems like propaganda, for example. True or false horror stories about the official enemy emerge, followed by outrage in the population with potentially deadly results, followed by a return to relative sanity. Repeat as necessary, nationality irrelevant. You might also imagine similar patterns emerging in economics, fashion, and so on.

Enter Chemotaxis

Let’s consider a bacterial population. One cell emits a chemoattractant that diffuses out into the medium it is in. Detecting this signal, neighbours are drawn in. The neighbours congregate where there is the greatest concentration of chemoattractant, resulting often in either circular waves, or spiral ones.

I’ve noticed the same pattern often emerges in mycelial propagation. Enter Paul Stamets who has noted the appearance of these patterns in his book Mycelium Running.

Nature tends to build on successes. The mycelial archetype can be seen throughout the universe: in the patterns of hurricanes, dark matter…. The similarity in form to mycelium may not be merely a coincidence. – Paul Stamets, Mycelium Running

I would say indeed it may not be. I would argue that these forms are mathematically predestined. (This is not to suggest, however, that hurricanes are the product of reaction-diffusion systems. They emerge out of fluid dynamics.)

Galaxies? Really? Come off it!

I had always only ever thought of galaxies coming about as a result of gravitational interactions. My education in physics being limited to undergrad studies, I did not encounter much in the way of astrophysics, unfortunately. As it turns out, the way I had envisioned galaxy formation to occur would, in fact, result in a galaxy that would quickly wind so tightly was to appear to be just a nondescript disc.

This problem of formation was mostly solved when Chia-Chiao Lin and Frank Shu proposed the Density Wave Theory, in which the density of the spiral arms of the galaxy prevents the galaxy from winding up into a disc.

It is a great theory, elegant, simple, plausible, and with backing evidence. But it doesn’t quite explain every type of spiral galaxy. Theoretical physicist Lee Smolin had a look at the problem of galaxies where the density wave doesn’t hold, and proposed a hypothesis whereby the galaxy was actually one great reaction-diffusion system. In his model, shockwaves from star formation and supernovas drive one reaction, and ultraviolet radiation from giant stars serves to inhibit it. The hypothesis isn’t perfect but just might explain some aspects of galaxy formation.

Now Available in 3D!

The BZ reaction shows a two dimensional expression of the propagation of travelling waves, or spirals, as the case may be. Taken in three dimensions, the travelling waves form expanding toroids, or, in the case of spirals, scroll rings. This form is reminiscent of the much talked about but perhaps sometimes misunderstood “core model” in permaculture (more on this in a future article).

Take the example of a jet of fluid flowing forward into a medium. The leading edge thrusts forward, and friction at the sides slow it down, creating a mushroom shape. These edges often form spirals as the following image of a portion of a von Karman vortex street shows. Keep in mind, however, that this is a characteristic of fluid dynamics, and not the product of a reaction-diffusion system. I am including it for illustration purposes only. Remember, though, that the appearance of spirals in a reaction-diffusion system is a result of fluid dynamics. Hence the relevance.

So there you have it, spirals from wave propagation. Is there some great mystical universal something going on? I believe these patterns emerge because they must. They are the mathematically prescribed result of chemical interaction in space and time. Galaxies do not form giant portraits of Homer Simpson because that is not a mathematically possible outcome. Bacteria propagating in a uniform petri dish do not form interlaced nonagons because that is not a possible outcome. What you see is what you can get.

As promised, here is a brief look at some more interesting patterns we get from the properties of surfactants.

How can you tightly pack elements in a design without resorting to the familiar and troubled row agriculture? There just so happens to be a property of surfactants that can help provide a little inspiration for that problem.

In the last pattern article, we looked at adhesion (like things being attracted to each other), and the polar/non-polar features of surfactants. We had considered the effect of surfactants on surface tension, but if we keep adding surfactants to a solution, the non-polar ends attract one another, and form spheres called micelles. The polar heads of the molecule – the ends which are attracted to water – are on the outside, and the non-polar ends are in the centre. If you keep on adding more surfactant, you will start to get long cylindrical micelles. It is here where things start to get really interesting, and really useful for us as designers. The micelles repel one another, meaning that they exert a force on each other. If you recall from the first article, applying force on a system decreases its symmetry, but increases its patterning. The repulsive force of micelle against micelle, forces the system to rearrange in such a way as to minimize the total energy.

The following image shows this process in action.

In a and b, micelles form. Notice how the hexagonal array discussed in the previous article arises here. As more squeezing occurs, a different pattern emerges, shown in c. There is fairly efficient use of space at this point, enough so to put a similar pattern into a design. With further force, the highly structured pattern in d emerges.

Why would anyone go to all this trouble? While rows are easy to build, they are not without their problems. I have seen “eco-farms” with row planting with the rows running up and down the topography. Barring thoughtlessness, I presume the intent is to promote erosion. These non-linear patterns are less conductive to runoff. Straight rows also require a sizeable enough patch of land to accept their unnatural array. This is often unsuitable in tight areas, or the rocky areas so common in the Canadian Shield. Rows are also generally set up to accommodate cultivators, which are great if your intent is to damage the soil, and increase erosion. We aim to make soil healthier, however, not degrade it. Rows, too, tend to be planted in pest-smorgasbord fashion, with grouping of like plants together, making it easier for pests to travel from preferred plant to preferred plant.

One could use these more natural arrays either for non-linear rows or as pathways for keyhole bed layout. While it would be a bit much to strictly follow such a pattern, it does give us ideas for systems a little more imaginative, and adaptive to the landscape than straight rows.

Coming up next, we’ll start to look at waves and pulses. Stay tuned!

Why is a honeycomb hexagonal, and how can that help me? How can I make a curved tensile surface like a roof with minimum tension? How can you make a structure stronger yet lighter than reinforced concrete?

Stick around, I’ll answer these questions, increase your designer’s toolbox, and even tell you how to make one of these guys (radiolaria) on the left here.

In the previous article, I asked the reader to consider a spherical raindrop. Indeed, as long as external forces do not impose upon water – gravity, air friction, sides of a container, etc. – it takes a spherical shape. But why?

Water has cohesion, which is a fancy way of saying that it is attracted to itself. To look at it another way, it is sticky to itself. When was the last time you saw the ocean just spontaneously break apart from itself into individual small droplets or a vapour? Liquids just don’t do that. The internal forces – the positive and negative charges on either end of a water molecule – hold the water together.

Within the drop, there are attractive forces all around an individual molecule. On the surface, however, there are no atoms outside the drop to attract the surface atoms. This creates a tighter bond amongst the surface atoms, which in turn is what is responsible for surface tension. In water, the surface tension is strong enough that your friendly neighbourhood water strider is able to glide across the surface of water like some sort of Gerridae messiah.

Because the surface molecules are all pulling equally around the entire surface of the droplet, it forms the only shape that equal force would allow, which is a sphere. As hard as you might try to push the droplet into a dodecahedron, as soon as you remove the force, it will snap back into a sphere.

You might wonder what all this has to do with honeybees. Don’t worry, I’m getting to that part. But first, we must look at bubbles. We’ve done half the work already, though. We can consider bubbles to just be anti-drops of water, only instead of water on the inside, we have water on the outside. The key is still the surface tension where air meets liquid.

And before we look at bubbles, we need to look at how they are made. Soap helps us make bubbles because soap molecules are amphiphiles ( meaning “likes both,” often called surfacants), which are molecules with a polar head and a non polar tail. The polar head is attracted to water, which is a polar molecule. The non-polar tail – which is the end sticking out on a bubble – is attracted to non-polar molecules like oil. This reduces the surface tension, which, counter-intuitively, perhaps, makes the bubble more resilient. (This property of being an amphiphile also lets us clean greasy dishes. The soap molecules are attracted to the oil at the non-polar end, surrounding them, and the polar end lets the water carry it.)

You’ve seen two bubbles meet before. They form a plane between them with the dividing bubble wall being just as thick as the wall of a single bubble. Even though two bubbles meet, the wall is not twice as thick. Maintaining that thickness would require more energy. Things naturally seek the lowest energy state, however, so the surplus soapy liquid distributes across the whole surface area of the two bubbles. They also join in a way that minimizes the surface area with respect to their internal pressure.

A light just went off in designer’s heads here. The bubbles form a stable shape with minimum use of material. And this is more than just interesting theory. This aspect of surface tension has been put into design as we shall soon see.

On to the honeycomb! If we take three equal-sized bubbles and touch them together, they will meet with an angle of 120° between the walls. A honeycomb is made up of hexagons, with each point of the hexagon being the centre by which three hexagons touch. There are six points in a hexagon, and six such nexuses around each cell.

It just so happens that there are only three equal-sided shapes that will stack into a plane without leaving gaps: equilateral triangles, squares, and hexagons. Humans tend to like the square because it is simpler to understand. Nature, on the other hand, only knows the principle of minimum energy. So when it comes time to stack round things tightly nature makes the honeycomb rather than the side-by-side stacking in a case of beer (time to redesign beer cases). So when you want to maximize the use of space, for example when planting, you have to mimic the pattern in nature and use a hexagonal array.

It was the minimization of surface area that architect Frei Otto employed so many times. Otto would make wire frames (shown above) and cover them with a soap film to reveal the minimum shape. Then he would apply this to his design.

Now, I had promised to tell you how to make a radiolarian. Here’s what you don’t do. You don’t try to write some sophisticated genetic code (sorry  geneticists) to synthesize proteins that are going to construct these elaborate shapes… somehow. You just write a genetic code for a little guy to blow a foam of bubbles, then inject a mineral in a slurry and allow it to precipitate between the borders of the bubbles. Presto! Marine biologists everywhere will marvel at your intricate beauty!

For our next instalment, we’ll have a little look at an interesting property of surfactants that yields intricate, tightly packed, non-repeating patterns.

The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve.
– D’Arcy Thompson