Sunday, February 27, 2011

Fractals: Am I Repeating Myself?

March 2 is Read Across America Day, the annual tribute to the pleasures and importance of reading founded in 1998 by the National Education Association. Across the U.S., schools, libraries, community organizations, and other entities will celebrate the joys of reading and observe the birthday of Dr. Seuss, whose endless creativity inspired the event. You can find resources to help celebrate the day here, and additional classroom ideas here. Happy reading!

I’ve written about my non-love affair with geometry before, but I have to admit that I have a weakness for fractals. Fractals are geometric shapes that repeat themselves, each part of which is a smaller copy of the original shape. The mathematical term applied to this notion is called self-similarity, but to a non-mathematician like me, fractals are like little halls of mirrors, where the shape decreases in size with each reiteration. They are found in abundance in nature, in the patterns of clouds, fern fronds, coastlines, and so forth. They are, in a word, cool.

Fractals are appealing on many levels. Most students are appropriately impressed (at least for the short term) when shown images of how fractals occur in nature, both on earth and in space. Yet when confronted with new or sometimes challenging material, students often ask how such material is relevant to their lives – what good is it? Aside
from creating amazing patterns (and thus used to great effect in various types of art), fractals are in fact are widely used in a host of industries, including engineering, economics, biotechnology, computer science, and entertainment. Fractals, for example, are used in the design of computer networks, most notably in how data traffic patterns are configured. They are also used in data and digital image compression, with the advantage of offering little to no pixilation difficulties that are commonly found in jpeg and gif formats. Microsoft used fractal data compression when it issued its Encarta Encyclopedia, where thousands of articles, color images, and animations were compressed into less than 600 megabytes of data.

The engineering and manufacturing industries also rely on the benefits of fractal geometry. The strength of coiled springs, for example, can be tested in several minutes as opposed to several days as a result of fractal modeling, and agricultural and civil engineers use fractal simulations to determine the best design, layout, and location of pipes and settling tanks related to ground seepage and water filtration. DNA sequences have been determined to show fractal patterns, which has led to new research for applications in biotechnology, and fractal theory, specifically the Mandelbrot set, is widely used in predicting stock market prices. Students may be most interested to learn that fractals are widely used in computer graphics and animations for the film industry. Like those rain droplets running down the dinosaurs’ skins in Jurassic Park? Those were created by using fractal models, as were planet topographies in various Star Wars and Star Trek films, among many others.

My picks this week offer a variety of lessons and activities on fractals, all products of Shodor Interactivate. A project of the Shodor Education Foundation, Interactivate offers interactive Java-based resources in math and science. All lessons are aligned to a variety of state and national education standards. Please be sure to also check our Facebook and Twitter pages throughout the week, as we’ll feature many more resources on fractals for a variety of ages.

Introduction to Fractals: Geometric Fractals
Subject: Geometry
Grade: 6-8
This lesson outlines the approach to building fractals by cutting out portions of plane figures. Students are also introduced to other classic fractals, such as the Sierpinski Triangle and Carpet as they iterate with plane figures.

Sierpinski’s Carpet
Subject: Geometry
Grade: 6-8
In this online activity, students step through the generation of Sierpinski's Carpet – a fractal made by subdividing a square into nine smaller triangles, and then removing the middle square. This activity allows students to explore number patterns in sequences and geometric properties of fractals.

Prisoners and Escapees – Julia Sets
Subject: Geometry
Grade: 9-12
One of the most famous fractals, the Mandelbrot Set, is made up of Julia Sets. Julia Sets, in turn, are known as prisoner sets. This resource provides a brief overview of the notion of prisoners and escapees as they pertain to iterative functions. A prisoner ultimately changes to a constant while escapees iterate to infinity.

~Joann's Picks - 2/25/2011~

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