Clouds are not spheres, mountains are not cones, coastlines are not circles.

– Benoit Mandlebrot (mathematician)

You have you eaten them, sat underneath them or played with them. Or, you may have encountered a Jackson Pollock “drip painting” in a museum, starred at it for minutes, wondering what is it about those abstract patterns that seem so mesmerizing and soothing.

I’m talking about fractals, intricately beautiful patterns that show up profusely in nature and sometimes in art. Fractals are objects that exhibit a feature known as “self-similarity”. That is, they exhibit the same patterns over and over again at different scales and sizes that goes infinitely deep. A tree is an example of a fractal object. Its trunk forks into two or more branches, each of which then forks into two or more thinner branches and so on ad infinitum. Another example is a snowflake. Viewed under a microscope, we see that a snowflake consists of a central hexagon from which springs six smaller hexagons, which in turn produce still more hexagons in a pattern called six-fold symmetry that gives them their intricate shapes.

The term ‘fractal’ was first coined by the maverick mathematician Benoit Mandlebrot (1924 – 2010) in 1975 to describe shapes that are neither spheres, cones or circles. The fractal mathematics that Mandelbrot pioneered, along with the related field of chaos theory, lifted the veil on the hidden beauty of the world, in phenomenon as diverse as rivers, coastlines, snowflakes, seashell markings, mountains and trees.

The Mandelbrot Set pictured above discovered when Mandelbrot was a researcher at the IBM Thomas J. Watson Research Center in the 1960s. Part of the charm of this intricate object is that behind its apparent complexity is a very simple, deterministic equation, a feature shared by chaos theory.

Here are more examples of fractal patterns in nature, science and art.

Fractal Snowflake

The six-fold symmetry of this snowflake is due to the microscopic crystal structure of ice repeated several times. The central hexagon sprouts six hexagons, and the outer corners of those hexagons produce still more hexagons in a seemingly endless process of hexagonal branching.

Fractal Cauliflower

Known as Romanesco cauliflower or Romanesco broccoli, this relative of more common brassicas has a strikingly fractal appearance. The self-similar conical protrusions are composed of spirals and more spirals of tiny buds.

Fractal Rivers

Fractal Trees

In many trees, such as this sycamore, a central trunk forks into two or more branches which in turn fork again and again into thinner and thinner branches before terminating in tiny twigs. Seven or more levels of branching can be counted in this image.

Fractal Leaves

Fractal Nautilus

The logarithmic or equiangular spiral is one of the most beautiful forms in nature. One finds it in a wide variety of objects, from tiny seashells to the enormous scale of galaxies. The nautilus shell beautifully illustrates the principle of a repetitive fractal process that creates the spiral form: the organism keeps expanding its “home” by adding sections to its shell, each section a little bigger than the one before. All the while, the scaling factor and the rotation angle remain the same. Simple to model mathematically, this simplicity nonetheless leads to the breathtaking beauty of the Nautilus spiral that has become an icon in art and science.

Fractals in Science

For the first time, scientists at MIT have recently discovered the repeating patterns of a fractal in the magnetic configurations of a quantum material, neodymium nickel oxide. This is a rare earth element that can act paradoxically as both an electrical conductor and insulator, depending on its temperature.

Fractals in Art

The American “drip painter”, Jackson Pollock rose to fame at about the same time as Mandelbrot’s mathematical forays into fractals. Pollock’s claim to fame is his style of painting in which he pours paint directly from a can onto a large horizontal canvas laid across the studio floor. Although art critics never cease debating the meaning of his splattered pictures, virtually all of them agree that Pollock’s drip paintings have a certain organic, seemingly natural feel to them. They just can’t put a finger on it – until recently.

Richard Taylor, a professor at the University of Oregon and Director of the Materials Science Institute and Professor of Physics there, has analysed Pollock’s paintings in great details using computer pattern recognition techniques. His research shows Pollock’s paintings are as fractal as the patterns found in nature, which may be why many museum goers ‘freeze’ in front of a Pollock canvas, seemingly enthralled by its fractal beauty [1]

Fractals and Our Mental Well-being

Fractals are not merely eye-candy, it turns out they have sublime healing powers too. There is abundant scientific evidence, for example, that looking at nature’s patterns help to reduce anxiety and stress. A famous study done in the 1980s [2] showed that patients in a hospital recovered more quickly from surgery when given rooms with windows that look out on nature. Other studies since then have confirmed that nature’s aesthetics changes positively the way our autonomic nervous system responds to stress.

Notes

[1] In Pollock’s case, his fractal aesthetics resulted from an intriguing mixture of his body motions and his constant attempt at refining his pouring technique to increase the visual complexity of these fractal patterns. Pollock isn’t the only artist known to embed fractals in their work; fractal patterns have resonated in art across the ages and many cultures. Fractals can be found, for example, in Roman, Egyptian, Aztec, Incan and Mayan works as well as the drawings of Leonardo da Vinci (see his ‘Turbulence’ of 1500) and the woodblock prints of Japanese artist, Hokusai (notably his famous ‘Great Wave’ of 1830).

[2] Roger Ulrich et al. (1984) “View through a window may influence recovery from surgery”, Science  27 Apr 1984, volume 224, Issue 4647, pp. 420-421. https://science.sciencemag.org/content/224/4647/420