Benoît Mandelbrot

Benoît Mandelbrot, who died last week, was the man who created the term “fractal”. Mandelbrot saw the patterns in what others had only seen as chaos. He saw that if you looked closely at ferns, you saw smaller ferns … you saw a repeat of the same structure. Ferns, coastlines, smoke … were self-similar. “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” These words are the introduction of Benoît Mandelbrot’s famous book, The Fractal Geometry of Nature. One famous fractal carries his name, the Mandelbrot Set.

This is a difficult fractal to explore without sophisticated mathematics but to begin to understand how this lovely image has been created, you can check out Boston University’s Chaos club’s explanation:

http://math.bu.edu/DYSYS/FRACGEOM/node1.html Mandelbrot was born in Warsaw, Poland. His family moved to Paris in 1936 when he was 11 to avoid the Nazi threat. He attended school and university in France and California where he earned degrees in aeronautics and mathematical science. In his lifetime of mathematical work he added to our understanding of finance, informational technology, fluid dynamics, geography, statistics and natural beauty. Jonathan Coulton even wrote a song about him called “Mandelbrot Set”. To appreciate what he named and studied, let’s begin an understanding of fractals with an easy fractal, Sierpinski’s Triangle.

This figure begins as a solid triangle. The midpoints of each side of that triangle are connected and the resulting smaller triangle is removed from the figure (it becomes white in the progression above.) That action is iterated

and the figure becomes more and more delicate. If you were to try to calculate the white areas of the figure, you would find an ever-increasing total area. In this figure the white area would never exceed the area of the original triangle but it would come closer and closer to that area as you continued to iterate the process. So, the area of white triangles in Sierpinski’s Triangle is finite. The area has a limit.

If you were to measure the total perimeter of each of the white triangles, that too would become a larger and larger number. However, there is no limit to the sum of all of those tiny edges. The perimeters of those white triangles are infinite.

Another fascinating aspect of Sierpinski’s Triangle is that if you start at one of the three vertices of your original triangle and go ½ way towards one of the other vertices, you will always land on the shaded part of the figure. Continuing to repeatedly travel ½ way to any vertex and then making a dot will eventually color in a complete Sierpinski Triangle … also called Sierpinski’s Gasket. The Chaos Game is a great game that applies this phenomenon.

So, since the Sierpinski Gasket can be generated by repeatedly traveling half way to a vertex and making a dot it can be drawn with devices (computers or calculators) that don’t mind doing the same operation over and over again.

To honor Benoît Mandelbrot and to help you learn more about your calculator, I’ve listed the commands to

create a Sierpenski’s Triangle on your TI 83/84 calculator.

http://www.yummymath.com/wp-content/uploads/sierpenski.pdf

I’ve included the commands to enter and how to find each command on your calculator. Enjoy. _____________________ Sources:

http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=1&src=ISMR_HP_LO_MST_FB http://en.wikipedia.org/wiki/Benoît_Mandelbrot

http://math.bu.edu/DYSYS/applets/chaos-game.html

http://www.ticalc.org/pub/83plus/basic/math/graphing/