Nature’s Geometry
Posted 23 July 2011, by Jessica Pellien, Princeton University Press, press.princeton.edu
The geometry we learned in high school is ideal for describing “man-made” forms such as buildings, roads, fences, etc. But lines, circles, and triangles don’t seem to do justice to trees, clouds, or mountains. What about the forms of nature? Is there a geometry for them? The late mathematician Benoit Mandelbrot (1924-2010) pioneered just such a geometry; he called fractal geometry after the Latin word fractus, which means broken or irregular.
A fractal is a shape composed of smaller copies of itself (think “fractured”). For example, a cauliflower is composed of florets—little flowers—which look just like little cauliflowers. We can use this idea to draw many natural forms using precise, step-by-step methods called algorithms. In the figure below we start with a simple, three-stick tree in (a) and then repeatedly turn each branch tip into a smaller, three-stick tree. The last step (f) is a computer rendering of the fractal the shapes are converging to.
Step-by-step drawing of a fractal tree.
The close-up below illustrates one of the reasons Annalisa Crannell and I chose the striking photograph Winter Road along the Trees by Wil Van Dorp for the cover of Viewpoints: Mathematical Perspective and Fractal Geometry in Art. The fractal beauty of the trees was impossible to resist!
Detail of the cover of Viewpoints.
Nowadays computers use fractal algorithms to generate photographically real landscapes in many feature films that require special effects. However, mathematicians and computer scientists may not have been the first to follow this road. As Benoit Mandelbrot pointed out, Asian artists have employed fractal-like portrayals of natural forms for centuries. As you can see below, Japanese woodblock artists of the nineteenth century used abbreviations for natural forms that are surprisingly similar to fractals investigated by mathematicians and scientists more than a century later!
A “quadric Koch island” fractal as described by Mandelbrot.
Boats in a Tempest in the Trough of the Waves off the Coast of Choshi (detail), from the series A Thousand Pictures of the Sea, by Katsushika Hokusai (1760-1849).
Fractal generated by an iterated function system.
Shono: Driving Rain (detail), from the series The Fifty-Three Stations of the Tokaido, by Ando Hiroshige (1797-1858).
Fractal model of two-fluid displacement in a porous medium.
Short History of Great Japan (detail), by Ikkasai Yoshitoshi (1839-1892).
Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate.
Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of Writing Projects for Mathematics Courses.
This is the final installment in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.
http://press.princeton.edu/blog/2011/07/23/natures-geometry/
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Posted by BreakThru Radio on May 16, 2012 at 5:59 am
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