In materials science class, we examined wallpaper patterns for symmetries. Atoms and molecules can pack according to a variety of crystal structures. Mathematics obviously loves patterns too. There are fractal tilings and tessellations. Who doesn’t love Escher? There are probably practical applications to tiling, but more importantly they are great fun that tickles the brain. Recently I took my first stab at pattern making depicting (what else?) water polo.
Please excuse my inconsistent posting of late, I have been deep down the rabbit hole of science. Last week, I attended the Society of Industrial and Applied Math (SIAM) dynamical systems conference. What fun!
I learned about Turing Patterns, named for mathematician Alan Turing. Complex patterns can arise from the balance between the diffusion of chemicals and the reaction of those chemicals. For this reason, Turing’s model is also called the Reaction-Diffusion model. In general, these kinds of patterns can arise when there’s some kind of competition.
This sounds abstract, but suspected examples in nature abound. Have you ever wondered how the leopard got his spots or what’s behind the patterns on seashells? We often don’t know the chemical mechanisms that produce the patterns, but we can mathematically reproduce them with generic models.
Mary Silber and her grad student Karna Gowda presented research on Turing patterns in the vegetation of arid regions. When there isn’t enough precipitation to support uniform vegetation, what vegetation will you observe? If there’s too little water, their model yields a vegetation-free desert. Between “not enough” and “plenty” the model generates patterns, from spots to labyrinths to gaps. Their work expands at least two decades worth of study of Turing patterns in vegetation.
Silber and Gowda considered an area in the Horn of Africa (the bit that juts east below the Middle East). Here, stable patterns in the vegetation have been documented since the 1950s. They wanted to know how the patterns have changed with time. Have the wavelengths between vegetation bands changed? Are there signs of distress due to climate change? By comparing pictures taken by the RAF in the 1950s to recent satellite images, they found that the pattern were remarkably stable. The bands slowly travelled uphill, but they had the same wavelength and the same pattern. They only observed damage in areas with lots of new roads.
Turing patterns have even been studied experimentally in zebrafish. Zebrafish stripes might appear stationary, but they will slowly change in response to perturbations. So scientists did just. Below is a figure from the paper. The left shows the pattern on the zebrafish, the right shows the predictions of the model.
The model has been used to explain the distribution of feather buds in chicks and hair follicles in mice. Turing’s equations have even been used to explain how fingers form.
If you want to learn more, the links above are a great start. And if you want to play with the patterns yourself, check out this super fun interactive. These waves aren’t stationary like the Turing patterns I described here, but they arise from similar mathematics. The interactive can make your computer work, fyi.
Every now and again, it’s fun to rummage through your old photos and look for categories. Pictures of bicycles, pictures of cats, pictures of cats, pictures that are purple… you get the idea. But it does take a while. So today, I rummaged through my photos for pictures of patterns.
I was inspired by Flickr’s new organizational features. Under the new “camera roll” is the magic view, which allows you to look at various pre-determined categories according to Flickr. There’s various kinds of animals, landscapes, and people categories. My favorite is the style. Below is my “pattern” category. Not every image hits the nail on the head, but it’s a cool way of looking through your photos. I’ve got over 7000 on Flickr, so I can see photos with similar attributes across the years. Some of the categories were ones I already had thought of, but some were new ideas.
So, some of my favorite pattern pictures!
Viscous fingering is a fractal pattern that occurs when a less viscous (or thick) fluid spreads through a more viscous (or thick) fluid. Such systems are present in oil extraction, when we pump one fluid underground to push another one out. Fractals are common in nature even though they’re new to our mathematics, and they are beautiful.
The pictures in this post were created with basic watercolor paints using one simple principle: water containing paint is more viscous than regular water. It’s easy to try at home!
For the top picture, I laid down red paint. Before the paint dried, I added salt, then let the square dry. Water from the still-damp paper rushed to the salt (because of entropy, systems tend towards uniform distributions of things if they can help it– in this case, the lowest energy state is to have a uniform distribution of salt). But because paint molecules are larger than water molecules, they don’t move as well. The water that accumulates around the salt has less paint than the water in the rest of the paper, and thus we have a less viscous fluid spreading into a more viscous one. Try it at home! If the paint is too wet or too dry when you add the salt, the results won’t be as dramatic, so play around a bit. Larger salt crystals can be especially fun.
For the three pictures below, I simply placed a drop of water into a damp square of paint. The patterns vary depending upon the size of my drop, the wetness of the paint, and the paint color (the chemistry of which influences the viscosity of the paint).
Below are a couple of examples from the University of Alberta of viscous fingering with pentane into oil and water into oil. This particular research aims to improve the flow rate of oil during extraction. And it looks pretty similar to some humble watercolors.
Left: pentane displacing mineral oil. Right: Water displacing mineral oil (University of Alberta).