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The Mandelmorph

May 18, 2013

This semester (Spring 2013), I piloted a new self-authored textbook I've titled "A Practical Introduction to MATLAB® and Octave: First Adventures in Technical Computing for Scientists and Engineers." The contents represent a distillation of my own industry experience-- an inductive approach to learning to use the MATLAB/Octave computing environment by tackling progressively more complex calculations. Students with cash on hand may choose the commercial MATLAB package (site licensed throughout our campus), while the more penurious may select (for more than 90% of the coursework) the wonderful open-source GNU Octave package.

A capstone problem in my chapter on loops gives students an opportunity to flex their newfound silicon muscles by calculating and visualizing the venerable Mandelbrot set. This simple calculation (d=dn+c, where n = 2, iterated across the complex plane), and the profound fractal complexity it yields, help the class see the power of the computers they own. It's quite popular-- the more enthusiastic students take over our computer lab in a proliferation of fractal "art projects". Some particularly beautiful results were obtained this semester by using complex n, or by subsitution of transcendental functions in the exponent. I'll leave this to you to explore.

If a picture in a college textbook is worth a thousand words, then surely a moving picture is worth another three orders of magnitude. And once a student can render a mathematical solution in 2-D images, it's just one step to video animations. I illustrate this by demonstrating for the class how the simple addition of an outer loop, varying the exponent n, creates a "Mandelmorph" video in which the fractal character of the Mandelbrot set appears and progresses through (n-1)-fold symmetry in a fascinating way. Here's the result, varying n from 1 to 6, which positively begs for further zooming and exploration:

Articles:
>Technical Computing
>Physical Chemistry


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