Monday Exercise 4.2: A Brief Detour Through the Mandelbrot Set

Before we explore using DAGs to implement workflows, let’s get a more interesting job. Let’s make pretty pictures!

We have a small program that draws pictures of the Mandelbrot set. You can read about the Mandelbrot set on Wikipedia, or you can simply appreciate the pretty pictures. It’s a fractal.

We have a simple program that can draw the Mandelbrot set. It's called goatbrot, and you can find it on most Linux servers in /usr/local/bin/goatbrot.

Running goatbrot From the Command Line

You can generate the Mandelbrot set as a quick test with two simple commands.

  1. Generate a PPM image of the Mandelbrot set:

    [email protected] $ /usr/local/bin/goatbrot -i 1000 -o tile_000000_000000.ppm -c 0,0 -w 3 -s 1000,1000
    

    The goatbroat program takes several parameters. Let's break them down:

    • -i 1000 The number of iterations. Bigger numbers generate more accurate images but are slower to run.
    • -o tile_000000_000000.ppm The output file to generate.
    • -c 0,0 The center point of the image. Here it is the point (0,0).
    • -w 3 The width of the image. Here is 3.
    • -s 1000,1000 The size of the final image. Here we generate a picture that is 1000 pixels wide and 1000 pixels tall.
  2. Convert the image to the JPEG format (using a built-in program called convert):

    [email protected] $ convert tile_000000_000000.ppm mandel.jpg
    

Dividing the Work into Smaller Pieces

The Mandelbrot set can take a while to create, particularly if you make the iterations large or the image size large. What if we broke the creation of the image into multiple invocations (an HTC approach!) then stitched them together? Once we do that, we can run the each goatbroat in parallel in our cluster. Here's an example you can run by hand.

  1. Run goatbroat 4 times:

    [email protected] $ /usr/local/bin/goatbrot -i 1000 -o tile_000000_000000.ppm -c -0.75,0.75 -w 1.5 -s 500,500
    [email protected] $ /usr/local/bin/goatbrot -i 1000 -o tile_000000_000001.ppm -c 0.75,0.75 -w 1.5 -s 500,500 
    [email protected] $ /usr/local/bin/goatbrot -i 1000 -o tile_000001_000000.ppm -c -0.75,-0.75 -w 1.5 -s 500,500 
    [email protected] $ /usr/local/bin/goatbrot -i 1000 -o tile_000001_000001.ppm -c 0.75,-0.75 -w 1.5 -s 500,500
    
  2. Stitch the small images together into the complete image (in JPEG format):

    [email protected] $ montage tile_000000_000000.ppm tile_000000_000001.ppm tile_000001_000000.ppm tile_000001_000001.ppm -mode Concatenate -tile 2x2 mandel.jpg
    

This will produce the same image as above. We divided the image space into a 2×2 grid and ran goatbrot on each section of the grid. The built-in montage program simply stitches the files together and writes out the final image in JPEG format.

View the Image!

Run the commands above, make sure you can create the Mandelbrot image. When you create the image, you might wonder how you can view it. If you're comfortable with scp or another method, you can copy it back to your laptop to view it. Otherwise you can view it in your web browser in three easy steps:

  1. Make your web directory (you only need to do this once):

    [email protected] $ cd ~
    [email protected] $ mkdir public_html 
    [email protected] $ chmod 0711 . 
    [email protected] $ chmod 0755 public_html
    
  2. Copy the image into your web directory (the below command assumes you're back in the directory where you created mandel.jpg):

    [email protected] $ cp mandel.jpg ~/public_html/
    
  3. Access http://learn.chtc.wisc.edu/~<USERNAME>/mandel.jpg in your web browser (change to your username on the submit machine).