There are multiple examples of fractal patterns in nature, from peacock feathers, snowflakes, and leaves, to cloud formations and coastlines. A group of graduate students have spotted similar fractal geometries in the ponds that form at low tide on a tidal flat in Maine.

Geometric patterns are considered fractal when they repeat or look the same at every size scale. In a forthcoming paper in Physical Review E, MIT’s Barry Cael and his co-authors describe how the distribution of ponds created by the rise and fall of the tides on coastal wetlands follows a power law and exhibits corresponding fractal patterns.

It wasn’t a planned research project, springing instead from a chance observation and Cael’s natural curiosity. He was spending the summer in Maine at an Ocean Optics Summer School, and passed the tidal pool in the rural town of Damariscotta every day on his commute. One day, he passed by when the tide was low, draining out sufficient amounts of water to form striking geometric patterns in the mud.


Cael is a currently graduate student in oceanography, with particular interest in Arctic ponds, but his earlier background was in applied mathematics. He knew that prior studies had found evidence for fractal patterns in the distribution of Arctic ponds. So he immediately recognized the phenomenon and convinced two of his fellow summer school colleagues, Kelsey Bisson and Bennett Lambert, to spend weekends investigating further by taking images of the ponds and studying them at various magnifications.

It proved a much more difficult task to collect the data than expected. They initially wanted to use drones to take the footage, but “it’s hard to find a drone at the last minute in rural Maine,” Cael said.

Next they decided to suspend an iPhone inside an apparatus made of balsa wood, held aloft by helium balloons. They grossly under-estimated how many balloons it would take: even 30 helium balloons weren’t sufficient to do the trick. “We did the calculation and it turns out it takes about 50 balloons to lift an iPhone,” he said. “Little plastic coins are enough to hold a helium balloon down, so it’s not as much lift as it seems.”


In the end, they used a long piece of PVC pipe and duct-taped an iPhone onto it to snap time-lapse imagery.

Their window of opportunity was small: it had to be at just the right time of day, when the sun was at the perfect angle, and low tide only lasted a couple of hours. And the conditions weren’t exactly idyllic. “I cannot describe to you how bad the mud smells,” Cael ruefully admitted. “Plus there were horseshoe crabs mating all around us. We’re just happy that the data turned out so well.”

Cael and his friends found that, indeed, the pattern of pond formation in the tidal flat showed the telltale self-similarity across a wide range of scales that characterizes a fractal. It also followed a power law distribution: you get lots and lots of individual clusters (in this case, ponds), fewer small sets of clusters, all the way down to one big infinite cluster. The two features are closely connected. Much like fractal patterns, “A power law is the only distribution that you can rescale and yet it remains a power law,” Cael explained. “So the properties of fractals necessarily have to be distributed by power laws.”

Some mathematicians might quibble about whether or not this is a “true” fractal. In math, the self-similarity of a truly fractal pattern repeats over an infinite range, but nature is bound by physical constraints. “You can’t have the same thing repeat itself larger than the scale of the earth or below the scale of an atom,” said Cael.


In general if something shows self-similarity over 3 orders of magnitude scaling variation, it’s considered fractal — that is, it looks the same at a scale of 1 as it does at a scale of 1000. Most fractal patterns observed in nature fall into this very generous range of scale, and Cael’s tidal ponds are no exception. “It is a mathematical fractal in that you can rescale it and it has the same properties, but it happens over a finite range just like any other fractal you see in nature,” he said.

Their work could ultimately turn out to be more than a fun summer project. Cael thinks this analysis could serve as a useful model to study fractal behavior in other geophysical surfaces — Arctic melt ponds, river networks, and lakes, for example, all of which show signs of power law distributions. He is exploring funding options for a field trip to the Arctic next summer to collect imagery of the melt ponds using autonomous planes, since strapping an iPhone to a long PVC pipe won’t cut in under those conditions.


Better data will make it possible to further test the model’s theoretical predictions. “This is a fun small scale model of something that happens at larger scales,” Cael said. “These things have been hinted at it, and the hope would be to go back and use all the statistical tools in physics to see if this is actually happening, or if it just looks like it is.”


Cael, Barry B.; Lambert, Bennett; Bisson., Kelsey (2015) “Pond fractals in a tidal flat,” Physical Review E. [forthcoming]


[Via American Physical Society]

Top images: Kelsey Bisson. Bottom image: Cael et al./PRE.