The beautiful and intricate forms of nature have long been studied by mathematicians dating as far back as the era of classical antiquity, but it was not until the late 20th century that one mathematician would revolutionize the way we think about the world around us.
When viewing examples of continuous, self-similar iterations in nature, the unconventional mathematician Benoit Mandelbrot opened up a whole new dimension of possibilities in mathematics and science in 1975 by defining what he called "fractals."
These irregular, repeating forms are all around us and can be perceived in weather systems, river networks, coastlines, clouds, mountain ranges, ocean waves, snowflakes, plants and animals and even in the complex systems of the human body.
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"Mathematicians knew of fractals long before Mandelbrot's work, although they did not have a specific name for them," Pennsylvania State University Department of Mathematics Assistant Professor Jan Reimann said.
While Mandelbrot defined the term "fractal" in his book, "The Fractal Geometry of Nature," many today feel the definition should be broader, according to Reimann.
"There is no hard and fast definition, but just a list of properties characteristic for fractals," Reimann said, citing Kenneth Falconer's book ‘Fractal Geometry.'
Self-similarity is among the defining characteristics of a fractal.
"For example, if the whole set, or object, is geometrically similar to a proper part of itself, like a fern branch is similar to any of its sub-branches," Reimann said. "Another one would be that the methods of classical, or traditional, geometry are unsuitable to study fractals."
According to Reimann, these forms were often thought to be outside the mainstream discipline of traditional geometry before Mandelbrot's work gained acceptance.
"Mandelbrot saw fractals as rough objects as opposed to the smooth objects of classical geometry, such as circles," Reimann said. "He found that many objects in nature, mountains, for example, are not smooth but fractured through and through. He showed that one can geometrically describe or build mountains as random fractals."
"In the whole of science, the whole of mathematics, smoothness was everything. What I did was to open up roughness for investigation," Mandelbrot said in a televised interview featured on the 2011 PBS NOVA program Fractals: Hunting the Hidden Dimension.
According to Reimann, his work finally established the theory of fractals as a proper mathematical discipline and as a part of geometry that requires different methods than the study of classical, regular objects, such circles, squares, triangles, etc.
Mandelbrot also realized that fractals, and the dynamical systems that produce them, play an important role in nature, he added.
"He popularized fractals further by showing that there exists very simple functions, or rules, whose iteration creates beautiful yet extremely complicated objects," Reimann said.
Another example of patterns in nature is the occurrence of spirals. From the raging fury of a churning tropical cyclone to the small sea shells lying along a windswept beach, spirals appear again and again.
"Some fractals have logarithmic spirals present at many different scales," Reimann said, adding that not all spirals found in nature are fractals.
According to AccuWeather.com Expert Meteorologist Dan Kottlowski, the spiral nature of storms and galaxies share a similar link in gravity.
"It all has to do with rotation and gravitational attraction," Kottlowski said. "Any cyclone forming within the Earth's atmosphere will be governed by the Earth's rotation."
While the spirals of galaxies are much different from storms here on Earth, Kottlowski said, gravity is also key to their unique form.
"In the rest of the universe, rotation caused by the evolution of the universe, combined with gravitational attraction influencing the motion of both visible and dark matter is causing the spiral appearance," Kottlowski said.