Fun with Fractals The Mandelbrot Set J Sweeney What is a fractal? Fractals are mathematical structures defined by three properties –Iterative –Self-similar.

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Presentation transcript:

Fun with Fractals The Mandelbrot Set J Sweeney

What is a fractal? Fractals are mathematical structures defined by three properties –Iterative –Self-similar –Non-integer dimension (fraction)

Iterative No, not the adjective from Webster; “marked by tedious repetition”.No, not the adjective from Webster; “marked by tedious repetition”. –(That’s my job) Iteration is the composition of a function with itself.Iteration is the composition of a function with itself. –Below is an example of an iterative function. What happens when the initial X is 0 and K is –2?What happens when the initial X is 0 and K is –2? What happens when K is -1.3 or 2?What happens when K is -1.3 or 2?

Self-similar Also described as scale- independenceAlso described as scale- independence Zooming in or out on the image reveals deep repetition of patternsZooming in or out on the image reveals deep repetition of patterns

Non-Integer Dimension Fractals on a plane only fill up a portion of the plane (between a line and a plane)Fractals on a plane only fill up a portion of the plane (between a line and a plane) The dimensional value of a fractal on a plane is always between one and twoThe dimensional value of a fractal on a plane is always between one and two There are two methods to calculate the fractal dimensionThere are two methods to calculate the fractal dimension –Box method –Geometric formula

Graphing It is possible to represent fractals graphically using Maple©It is possible to represent fractals graphically using Maple© By defining a function, selecting starting values, and choosing a scale to view we can achieve interesting resultsBy defining a function, selecting starting values, and choosing a scale to view we can achieve interesting results There are many popular fractals, the Mandelbrot set is a great exampleThere are many popular fractals, the Mandelbrot set is a great example

Mandelbrot Set

Mandelbrot Algorithm Graphic by Peter Stone

Lab Following the steps on the board, create an algorithm in Maple© to generate a Mandelbrot Set and display it.Following the steps on the board, create an algorithm in Maple© to generate a Mandelbrot Set and display it.

Homework Create a new Mandelbrot set by choosing a different center and magnificationCreate a new Mandelbrot set by choosing a different center and magnification Zoom in and out to find an interesting imageZoom in and out to find an interesting image Save the image as a graphic file along with your choices for center and magnificationSave the image as a graphic file along with your choices for center and magnification We’ll hang the images around the room for parents’ night.We’ll hang the images around the room for parents’ night.