Lab 9: The MandelBrot Set
CS145: Images and Imagination, Spring 2010

  

Goals

• Learn about while loops and do-while loops.
• Continue to work with iterated algorithms.
• Continue to work with complex numbers.
• Understand how to generate the mandelbrot set.

Resources and Readings

• Loops:
  • Loop up "while" in the Processing online reference.
• Do a web search on "mandelbrot set images".
• Wikipedia: The Mandelbrot set
• An introduction to the Mandelbrot Set
• Wikipedia: Complex Numbers

Activities

• While Loops: The while-loop is a more general type of loop than the for-loop (a for-loop can always be replaced by a while-loop but not vice versa). There are actually two versions: "while" and the "do-while":

  • "while": the condition is checked at the beginning. The statements in the body execute 0 or more times.

    while (conditional-expression) { statements }

  • "do-while": the conditionis checked at the end. The statements in the body execute 1 or more times.

    do { statements } while (conditional-expression)

  • Any for-loop can be written as a while-loop. For example: the following for-loop

    for (int i = 0; i < 100; i = i + 5) { println(i); }

    can be written as a while-loop:

    int i = 0; while (i < 100) { println(i); i = i + 5; }

    Practice by converting some of your previous for-loops into while-loops to see if you get the same result.

  • While loops are useful when one doesn't know ahead of time how many times the loop will execute, e.g.

    int headCount = 0; while (random(1.0) < .5) { count++; } println("Number of heads in a row is " + headCount);

    The above code will "flip a coin" until a tail is encountered. It will then print how many heads in a row were tossed.

    The code below prints a list of increasingly larger positive numbers until 1000 is reached.

    float distance = 0; do { distance = distance + random(100.); println(distance); } while (distance < 1000);

• In class, we will go over how the Mandebrot set is generated by iterating on the complex numbers z = z² + c, starting at z=0. If the length of z goes above 2, the iteration is stopped, otherwise it is continued until 100 iterations have been executed. The color of a pixel is based on the number of iterations.

  The code structure is given below:

  void setup() { size(500,500); mandelbrot(); save("mandelbrot.png"); } void mandelbrot() { int max = 100; // max number of iterations // set the drawing region float minX = -2, maxX = 1; float minY = -1.5, maxY = 1.5; // loop over each pixel in window for (int i = 0; i < width; i++ ) { for (int j = 0; j < height; j++) { // Use map() to convert pixel (i,j) to a point c in the drawing region: Complex c = ... Complex z = new Complex(0,0); // z starts at 0 // iterate z = z^2 + c using do-while loop until // length(z) > 2 or the number of iterations (k) reaches max = 100 setColor(k,max); // here, k is the number of iterations point(i,j); // draw point in window } } } // Set the color based on the number of iterations void setColor(int k,int max) { // set the color based on k, max if (k<max) stroke(0) ; else stroke(255); } // Insert code for the Complex class

  Finish implementing the above code. The setColor function given above will generate the mandelbrot in black and white, as shown here:

  

  Remember that k varies from 0 to max=100. If the iteration converges (i.e. the length never exceeds 2), then we will have k=100. This corresponds to the white region. The black region corresponds to where the interation blows up. The value of k is a measure of how fast it blows up (and so k will be less than 100). To get a range of colors in this region, modify the if-statement to include more values of k. For example, we can get three colors as follows:

  void setColor(int k,int max) { if (k< 6) stroke(255,0,255) ; else if (k < max) stroke(0,0,255); else stroke(255); }

  Once can also use the final length of z to set the color. Experiment with changing colors and regions as was done here:

Deliverables

Monday, April 12, before class:

You are to turn in via Wise a single zipped file containing your Processing sketch for the mandelbrot set. Include 3 images, each using a different region and/or coloring mapping.

Before turning in any of your code, make sure that you have cleaned it up!