Lab 8: The Mandelbrot Set
CS 145: Images and Imagination, Spring 2014

Summary of Goals

The main goals of this lab are to:

• Learn about while-loops.
• Learn about using the Map function to rescale number ranges.
• Generate images of the Mandelbrot Set.

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 condition is 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; // initialization of loop variable while (i < 100) { // condition for continuing to loop println(i); i = i + 5; // update of loop variable }

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

• While-loops are much more general than for-loops, e.g. they 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) { headCount++; } println("Number of heads in a row is " + headCount);

  The above code shows how to use a while-loop to "flip a coin" until a tail is encountered. It will then print how many heads in a row were tossed.

  The code below shows how to use a do-while loop to print a list of increasingly larger positive random numbers until 1000 is reached.

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

  How do you modify the above do-while loop so that it keeps looping only as long as the distance is less than 1000 AND you have called random no more than 20 times? This is the type of loop you will need to use in the Mandelbrot program below.

The Map Function

Read about the map function in the online Processing reference. The map function is very useful for rescaling numbers. For example, suppose you want to rescale pixel values (range 0 to 400) so that these values instead fall in the range of x= 1 to 2. (This is needed later to convert a pixel value to a value in the complex plane.) For example, what would the corresponding x value be for pixel 230? The image to the right, shows the relationship between the different map parameters. In this case, the x value would be 1.575

Program 1: Using the Map Function
float xMin = 1; float xMax = 2; int pixMin = 0; int pixMax = 400; int pix = 230; float x = map(pix, pixMin, pixMax, xMin, xMax); println(x);
Program output: 1.575

You will need to use the map function in the Mandelbrot program to transform a pixel in the Processing window (as defined by the width and height) to the corresponding point in the complex plane as defined by minX, maxX, minY, and maxX. The picture is shown below. Imagine these to rectangles sitting on top of one another. Given an (i,j), you want to know what the corresponding value of (c_real, c_imag). You need to apply the map function twice, once to get c_real from i and again to get c_imag from j.

Before you proceed, try writing down these two map functions. Use Program 1 as a guide.

The Mandelbrot Set

• The Mandelbrot Set: Below is the code for drawing the Mandelbrot Set. Some sections are missing. You also need to add the tab for the Complex Number class (see Program 2 in Lab 7.) We will go over in class what needs to be put in these sections. When completed, the code should give the image on the right.

  Program 2: The Mandelbrot Set

  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: float c_real = ... ; // COMPLETE USING MAP FUNCTION float c_imag = ... ; // COMPLETE USING MAP FUNCTION Complex c = new Complex(c_real,c_imag); Complex z = new Complex(0,0); // z starts at 0 int k=0; // must declare outside loop // Add a do-while loop that *keeps* looping as long as k < max // AND the length of z does not exceed 2. // Inside the do-while loop, do the following: z = ... ; // update z = z^2 + c k = ... ; // update k 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); }

• Changing the region: The region of the complex plane that is drawn is given by minX,maxX, minY, and maxY. You can change these to zoom into specific regions of the plane. Try changing these values.

• Setting the color: 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 iteration 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); }

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

To Be Submitted

At the beginning of class on Friday, April 25:

• On CS145/Lab8/ProcessingProjects, place your Mandelbrot Processing project which includes your different versions of setColor (e.g. you can call them setColor1, setColor2, etc).

• On gorr-classes, in the folder CS145/Lab8/FinalImages, place the basic black and white Mandelbrot image as shown Program 2 as well as at least 4 uniquely different images generated by changing the region and method of coloring (setColor).

• Be prepared to demonstrate your program in class.