Fractals: They’re Everywhere!                                                                                 E. Gluckie

 

A discovery from which subject your are studying has been used to interpret everything from cosmic dust to rainforest ecology to human DNA?  Believe it or not, the subject is Mathematics!  A branch of Math that surfaced in the 1960’s has turned out be one of the ‘hottest’ areas of research for a number of scientific disciplines theses days.  This exciting branch of Mathematics is that of Fractal Geometry.

 

“Despite their complexity, especially when viewed in great detail, Fractals were very simple to describe because the rules which generated them were absurdly simple to state”  Bentiot Mandelbrot, mathematician.

 

So what is a Fractal?  Fractals are geometric figures that have two specific properties.  One property is that of self similarity: many copies of a fractal can be found in the original object, but at a much smaller size.  Most magnified images of fractals are identical to the unmagnified version. A fractal shape will look almost, or even exactly, the same no matter what size it is viewed at.  A second property of fractals is scale independence: they do not become simpler in form when magnified or reduced.  As an example, consider a mountain.  When you approach a mountain, your eyes distinguish new features in the rock, yet the smaller parts of the mountain you see are just as detailed as the whole of the mountain itself.   

 

Fractal patterns in math were recognized nearly 100 years ago and written about by mathematicians such as Gaston Julia in 1917.   However, due to the large number of calculations required, what is known as the modern day fractal could not be seen until the introduction of the computer.  It was not until 1960, when mathematician Benoit Mandelbrot studied Julia’s work (with the aid of a computer) that Fractal Geometry was born.  Mandelbrot himself stated  “Computers are essential for Fractals.  Before them, people did not believe my drawings of Fractals.”

 

Why is the mathematical discovery of fractals important?  Mandelbrot stated fractals “occur in economics with the behaviour of prices.  They occur in physiology in the growth of mammalian cells. Believe it or not ... they occur in gardens.  Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterized in fractal theory.”    Fractals are important because they make up a large part of our world.  Trees, clouds, arteries, veins, nerves, and the bronchial system of your lungs all show fractal organization. In addition, fractals can be seen in the surfaces of proteins, in landscape analysis, in physics as part of wave dynamics, the connectedness of cave systems and even the organization of cells at the back of your eyes. 

 

Because fractals are nearly everywhere in nature, their study is applicable to areas such as Physics, Economics, Art, and even Music.  Many scientists think fractals are also key in understanding Biology and Medicine, and feel a thorough understanding of Fractal Geometry will lead to breakthroughs in these areas.  It is not inconceivable that this recent discovery in Mathematics could very well lead to the next innovation in medical techniques such as magnetic resonance imaging, or even cancer cell research!

 

 

Questions for Students:

 

  1. “Science advances Technology and Technology advances Science.”  Explain how this quote is related to the story above

 

  1. Whom do you think should be credited for the discovery of Fractals?  Why?

 

  1. How often do you think a discovery from one area of Science or Mathematics affects or impacts other disciplines? Can you think of other instances in history where this has occurred?

 

  1. Many scientists keep their discoveries ‘under wraps’ (let as few people know about them as possible) until they get their results published in a science magazine or journal.  Given what you have read in the above story, do you think this ‘secrecy’ is beneficial? Why or why not? 

 

 

 

 

 

 

 

 

Fractal Images For Your Enjoyment!

 

  This fractal image is known as ‘The Mandelbrot Set’, after Bentoit Mandelbrot who coined  the term ‘fractal geometry’

 

 Another fractal image, courtesy of ‘The Fractal Microscope’

                                                           

 A third fractal image, courtesy of ‘The Fractal Microscope’

                                                            


 

References

 

  1. American Association of Petroleum Geologists.  (1991). Fractal Geometry and Its Application to the Petroleum Industry.   Dallas, TX: AAPG

 

  1. Bauer, W., Use of Fractals in Cancer Detection.  Retrieved January 22, 2002.  Available: http://www.physics.udel.edu/colloq/archive/Fall1995/112995colloq.html

 

  1. Davis, B., Sumara, D., Luce-Kapler, R.  (2000). Engaging Minds, Learning and Teaching in a Complex World.  Mahwah, NJ: Lawrence Erlbaum Associates Inc.

 

  1. Donahue, M., The Chaos Theory.  An Introduction to mathematical chaos theory and Fractal Geometry.  Retrieved January 22, 2002.  Available at http://www.duke.edu/~mjd/chaos/chaos.html

 

  1. Goldsmith, J., The Geometric Dreams of Benoit Mandelbrot.  Wired magazine (on-line).  August 1994.  Retrieved January 22, 2002.  Available at http://www.wired.com/wired/archive/2.08/mandlebrot.html

 

  1. Lapidus, M. (1993).  The Vibrations of Fractal Drums and Waves in Fractal Media.  In M. Novak (Editor).,   Fractals in the Natural and Applied Sciences: proceedings of the Second IFIP workshop on Fractals in the Natural and Applied Sciences (pp255-260). London, UK. 

 

  1. O’Connor, I., and Robertson, E.  Benoit Mandelbrot.  Retrieved January 22, 2002.  Available at: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html

 

  1. Schroder, M. (1996).  Fractals in Music.  In C. Pickover, (Editor).,  Fractal Horizons: The Future Use of Fractals.  New York, NY: St. Martins Press

 

 

Other Good Resources for Teachers

  1. Lanius, C.  A Fractals Unit for Elementary and Middle School.  Available at: http://math.rice.edu/~lanius/frac/

 

  1. National Center for Supercomputing Applications, Education Group, University of Illinois.  The Fractal Microscope.  Available at: http://archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html