Mini Project 2: Computational Art

Due: 10:50 am, Mon 1 Oct



This assignment has been adapted from Harvey Mudd Professor Chris Stone’s assignment posted at the Stanford Nifty Assignments collection.

Skills Emphasized

Introduction: Computational Creativity

The idea of this assignment is to create art using computation as an artistic medium. As you saw in class, there are a ton of interesting ways to combine computation and art. In this assignment you will be exploring one particular means of using computers to generate images, and for the “going beyond” portion you will have the opportunity to extend your work by exploring topics such as computational generation of movies and real-time visualization of auditory signals.

At some point in your mathematical career you have probably seen a figure that looks like this:

This is a graphical representation of the function f(x, y) = 2x + y where the brightness of each position in the figure above represents the value of the function evaluated at a particular (x, y) pair (note the bar on the side that translates between brightness and the numerical value of the function).

From a visual point of view, this figure doesn’t look particularly rich. However, for more complex functions things start to get more interesting. Here is a plot of the function f(x, y) = sin(10pi*xy)

Things get even more interesting when we use the power of computation to generate functions by randomly composing elementary functions with each other. Here is a plot of the randomly generated function f(x, y) = cos(pi*((cos(pi*sin(pi*(((x+y)/2*cos(pi*y))+cos(pi*cos(pi*y)))/2))*((((cos(pi*y)+(x*y))/2*(cos(pi*y)*sin(pi*x)))+sin(pi*((x+x)/2+cos(pi*y))/2))/2*sin(pi*sin(pi*sin(pi*(y+y)/2)))))+sin(pi*(cos(pi*cos(pi*cos(pi*sin(pi*x))))+sin(pi*sin(pi*((x*x)*sin(pi*y)))))/2))/2)

Cool! Next, we bring in the idea of color. Instead of generating one random function, we generate three (one for each of the color channels red, green, and blue). Here are 3 figures representing randomly generated functions for each of the color channels. (Note: in a slight change of notation we are now using the function avg(x, y) instead of (x + y) / 2.)

red(x, y) = sin(pi * avg((((cos(pi * (sin(pi * cos(pi * y)) * avg(avg(x, x),
sin(pi * y)))) * avg(sin(pi * (sin(pi * y) * (y * x))), cos(pi * cos(pi * (y *
y))))) * sin(pi * (sin(pi * (sin(pi * y) * sin(pi * y))) * cos(pi * ((y * y) *
sin(pi * y)))))) * sin(pi * avg(cos(pi * avg(((y * x) * (x * x)), sin(pi * (y
* x)))), sin(pi * avg(avg(sin(pi * x), avg(x, x)), sin(pi * avg(x, y))))))),
cos(pi * cos(pi * avg(sin(pi * sin(pi * avg((x * x), (x * x)))), sin(pi *
sin(pi * sin(pi * sin(pi * y)))))))))

green(x, y) = sin(pi * ((avg(avg(cos(pi * (cos(pi * cos(pi * x)) * (cos(pi * x)
* avg(y, x)))), ((cos(pi * cos(pi * y)) * (cos(pi * x) * (x * y))) * sin(pi *
sin(pi * avg(y, y))))), cos(pi * (avg(sin(pi * sin(pi * x)), sin(pi * sin(pi *
x))) * sin(pi * sin(pi * (x * y)))))) * avg((avg(cos(pi * sin(pi * cos(pi *
x))), avg((sin(pi * x) * cos(pi * y)), avg(cos(pi * x), cos(pi * x)))) *
avg(avg(sin(pi * cos(pi * x)), sin(pi * sin(pi * x))), (avg(cos(pi * x),
avg(y, x)) * avg(sin(pi * y), sin(pi * x))))), (cos(pi * cos(pi * (avg(y, y) *
(y * x)))) * cos(pi * cos(pi * sin(pi * avg(x, x))))))) * sin(pi *
avg(avg(sin(pi * cos(pi * sin(pi * cos(pi * x)))), avg(sin(pi * cos(pi *
cos(pi * y))), ((sin(pi * y) * (x * y)) * cos(pi * (y * y))))), cos(pi *
avg(((cos(pi * y) * (y * y)) * avg(sin(pi * y), cos(pi * y))), (((x * x) *
avg(y, x)) * cos(pi * sin(pi * x)))))))))

blue(x, y) = avg(sin(pi * (avg(cos(pi * avg((cos(pi * (x * x)) * cos(pi * (x *
y))), avg(avg((x * x), avg(y, y)), avg(cos(pi * y), cos(pi * x))))),
avg(avg(avg((sin(pi * y) * (x * y)), sin(pi * (x * x))), avg(((x * x) * sin(pi
* y)), (avg(x, x) * sin(pi * y)))), avg((cos(pi * sin(pi * y)) * cos(pi *
avg(x, x))), sin(pi * avg(sin(pi * y), sin(pi * y)))))) * cos(pi *
avg(avg(avg(sin(pi * (x * x)), avg(sin(pi * y), sin(pi * x))), cos(pi *
avg(cos(pi * y), avg(y, x)))), (((avg(x, y) * cos(pi * x)) * cos(pi * avg(y,
x))) * avg(cos(pi * (y * x)), ((x * x) * (y * x)))))))), avg(((((sin(pi *
sin(pi * avg(x, x))) * avg(avg(sin(pi * y), sin(pi * y)), avg(avg(x, x),
cos(pi * y)))) * sin(pi * sin(pi * sin(pi * (y * y))))) * avg(cos(pi *
avg(avg(avg(x, y), (y * x)), cos(pi * sin(pi * x)))), (sin(pi * sin(pi *
sin(pi * x))) * cos(pi * ((y * y) * cos(pi * x)))))) * avg(cos(pi * cos(pi *
sin(pi * cos(pi * avg(x, y))))), (sin(pi * (cos(pi * avg(y, x)) * sin(pi *
cos(pi * x)))) * ((sin(pi * cos(pi * y)) * avg(avg(x, x), cos(pi * x))) *
avg((sin(pi * x) * avg(y, x)), sin(pi * sin(pi * x))))))), ((cos(pi * cos(pi *
(sin(pi * (y * y)) * cos(pi * cos(pi * x))))) * avg(sin(pi * avg(cos(pi *
sin(pi * y)), (cos(pi * x) * avg(x, x)))), cos(pi * cos(pi * cos(pi * avg(x,
y)))))) * sin(pi * (avg((cos(pi * (y * y)) * cos(pi * sin(pi * y))), avg(((x *
x) * sin(pi * x)), cos(pi * sin(pi * y)))) * avg(sin(pi * (avg(y, x) * avg(x,
x))), cos(pi * avg((y * y), avg(y, y)))))))))

Of course the real fun is when we put all of these individual color channel images together into a single image.

Voila! Computational creativity (there are some deep philosophical issues behind whether this can be called creativity… We don’t pretend to know the answers, but we do think that these pictures look really cool!)

For an excellent read on the history of the subject with many great examples, check out Why Love Generative Art? by Jason Bailey.

Here are some other examples taken from students who have done this assignment at Harvey Mudd:

Part 0: Install and test Python Imaging Library

To get started on the assignment:

  1. Click on the invitation link at
  2. Accept the assignment and follow the instructions to get to your repository page, which looks something like
  3. Clone the repository to your computer. The starter code is in

In addition to fetching the starter code, you should also install the Pillow fork of the Python Imaging Library (“PIL”). To do so, execute the following command at the Linux terminal:

$ conda install Pillow

The starter code includes a function called test_image that uses PIL to generate an image where each pixel has a random color value. When you run the starter code, you should see several unit test failures from functions you will implement later, and it should save an image file named noise.png in your run directory. You can view noise.png using the built-in tool Image Viewer (eog on the command line).

Random pixel noise

While this looks pretty cool (and it might be fun to convince your gullible friends it is a Magic Eye picture), I wouldn’t call it art. In the rest of the assignment, you will combine randomness with structure to produce more compelling images.

Once you’ve verified that PIL is working correctly, comment out the line that calls test_image and un-comment the line that calls generate_art. If you run into errors installing Pillow or generating the test image, talk to a NINJA.

Part 1: Understanding the structure of the starter code and making your first image

First, make sure you are familiar with the starter code and how it utilizes the PIL library to generate a simple image. The relevant code is in the generate_art function:

# Functions for red, green, and blue channels - where the magic happens!
red_function = ["x"]
green_function = ["y"]
blue_function = ["x"]

# Create image and loop over all pixels
im ="RGB", (x_size, y_size))
pixels = im.load()
for i in range(x_size):
    for j in range(y_size):
        x = remap_interval(i, 0, x_size, -1, 1)
        y = remap_interval(j, 0, y_size, -1, 1)
        pixels[i, j] = (
            color_map(evaluate_random_function(red_function, x, y)),
            color_map(evaluate_random_function(green_function, x, y)),
            color_map(evaluate_random_function(blue_function, x, y)))

The arguments to this function are a string filename where the image file will be saved, and two optional arguments x_size and y_size which set the image dimensions in pixels. If the size arguments are not given, they default to 350.

Let’s look at the code line-by-line and make sure we understand what is happening

red_function = ["x"]
green_function = ["y"]
blue_function = ["x"]

For testing purposes we are going to generate the red channel with the function x, the green channel with the function y, and the blue channel with the function x. Eventually we will do more complex things as in the examples above, but this will allow us to get started.

im ="RGB", (x_size, y_size))
pixels = im.load()

In this code, we are creating a new image with the given size. The input “RGB” indicates that the image has three color channels which are in order “red”, “green”, and “blue”. The im.load command returns a pixel access object that we can use to set the color values in our new image.

for i in range(x_size):
    for j in range(y_size):

These lines create a nested loop in which we will set each pixel value based on evaluating the red, green, and blue channel functions.

        x = remap_interval(i, 0, x_size, -1, 1)
        y = remap_interval(j, 0, y_size, -1, 1)

These lines are necessary because the functions we will be generating (as in the examples above) have inputs in the interval [-1, +1] and outputs in the interval [-1, +1]. If we were to simply input the raw pixel coordinates, these would go from [0, 349]. The remap_interval function (which you will implement) transforms the pixel index (which is in [0, 349]) to the interval we want ([-1, +1]).

        pixels[i, j] = (
            color_map(evaluate_random_function(red_function, x, y)),
            color_map(evaluate_random_function(green_function, x, y)),
            color_map(evaluate_random_function(blue_function, x, y)))

This code evaluates each the red, green, and blue channel functions to obtain the intensity for each color channel for the pixel (i, j). We need to use the color_map function (which calls your remap_interval function) since the color channel function outputs are floats in [-1, +1], but PIL expects the color intensity values to be integers in the range [0, 255].

This final line saves your image to disk as the specified filename (which will be "myart.png" if you don’t change it). You can view myart.png using the built-in tool image_viewer.

The initial starter code will not work without modification. To get started and generate your first image, you will need to make the following modifications:

  1. Implement remap_interval. To help you get started we have defined several doctests and filled out the docstring.
  2. Modify the function evaluate_random_function so that if the function [“x”] is passed in, the input argument x is returned, and if the function [“y”] is passed in, the input argument y is returned. To help you understand this, we have added two doctests demonstrating the behavior your function should have.

If you have done everything properly, when you run your code you shouldn’t get any doctest failure messages and an image file will show up in your computational_art folder called myart.png. The image should look like this:

Part 2: Generating and evaluating random recursive functions

Now that you have created your first image, you should start filling out the function build_random_function. As you fill in this function, you will also want to modify your evaluate_random_function method so that it can handle the recursive functions that build_random_function is generating.

In order to be able to see the images your code is generating as you make these changes, you should overwrite the simple function values for red, green, and blue in the body of generate_art with random functions generated with minimum depth 7 and maximum depth 9 (definitely play around with other depths for really cool effects though). The modified code should look like this:

red_function = build_random_function(7, 9)
green_function = build_random_function(7, 9)
blue_function = build_random_function(7, 9)

We will be representing function composition using nested lists. Here is a scheme for representing function compositions using a list:

["elementary function name here", argument 1 (optional), argument 2 (optional)]

For instance, to represent the function f(x, y) = x we would use (you have already seen this in the first part of the assignment):


Since the function f(x, y) = x has no additional arguments, the 2nd and 3rd elements of the list have been omitted. Note, that the string “x” has no special meaning to Python, in the first part of the assignment you had to tell Python how to evaluate this function for a particular input pair (x, y).

Alternatively, consider the following list representation of the function f(x, y) = sin(pi * x)

["sin_pi", ["x"]]

In contrast to the function f(x, y) = x, here we use the second list element to specify the function that we are going to compose with sin_pi.

Next, consider the following representation of the function f(x, y) = xy

["prod", ["x"], ["y"]]

In this example we used both the second and third element to specify the two arguments to the prod function.

To create more complex functions we use more deeply-nested lists. For instance, f(x, y) = sin(pix)cos(pi*x) would be represented as:

["prod", ["sin_pi", ["x"]], ["cos_pi", ["x"]]]

Implement the function build_random_function in the provided Python file

The two input arguments to build_random_function are min_depth and max_depth. The input min_depth specifies the minimum amount of nesting for the function that you generate (the depth of the preceding examples are 2 and 3 respectively). The input max_depth specifies the maximum amount of nesting of the function that you generate. To help better understand the idea of the depth of the function the following diagram might help:

Depth tree diagram

Click image to enlarge

In this figure the nested function is represented as a tree. The elementary functions are represented in the label on the relevant box, and an arrow from box i to box j indicates that the function represented by box j is an argument to the function represented by box i. To the left of each box is the list representation of the function (notice that the list representation includes the function representation of the functions that it points to). Also to the left of the box is the depth of the function (i.e. the degree of nested-ness of the list representation). The min_depth and max_depth inputs, refer to the depths listed in the figure above.

Your build_random_function implementation should create random functions from the following required building blocks:

The key characteristic of each of these building blocks is that provided each of the inputs specified to the function is in the range [-1, 1] then the output of the function will also be in the range [-1, 1]. In addition to the required functions above, add two additional function building blocks that have the crucial property that if the inputs to the function are between [-1, 1] the output will also be in the interval [-1, 1]. As part of creating build_random_function you should add an appropriate docstring. You may also want to add additional doctests. Your implementation of build_random_function should make heavy use of the principle of recursion.

As stated before, as you modify build_random_function you will also want to modify the implementation of evaluate_random_function that you created in Part 1. In order to complete the required part of the assignment, your evaluate_random_function must be able to evaluate any random function generated by build_random_function.

Going Beyond

There are a number of interesting directions to take this assignment.

Extension 1: Lambda Functions!

Explore the use of lambda functions to represent deeply nested compositions of functions. For this extension you should replace the list representation of nested functions with lambda functions. For instance, ["cos_pi", ["prod", ["x", "y"]]] would be represented as the lambda function:

lambda x, y: cos(pi * (lambda x, y: (lambda x, y: x)(x, y) * (lambda x, y: y)(x, y))(x, y)).

Using lambda functions will allow you to return a variable of type function from your build_random_function code. In order to evaluate the function at a particular (x, y) pair all you have to do is pass (x, y) into your function variable (this obviates the need for evaluate_random_function). For instance,

green = build_random_function(7, 9)
green_channel_pixel_for_x_y = green(x, y)

Extension 2: Movies!!!

Instead of building random functions of two variables (x, y) build random function of three variables (x, y, t) where t represents the frame number in a movie rescaled to the interval [-1, +1]. The easiest way to create these movies is to output a series of image files (one for each frame of the movie) and then use a command line tool such as avconv to encode them into a movie.

For instance, if you created images frame001.png, frame002.png, …, frame100.png you could encode these images into a movie using the following command in Linux (assuming you have already installed avconv via apt-get).

$ avconv -i frame%03d.png -vb 20M mymovie.avi

Use frame%03d.png if your files are named e.g. frame001.png, frame002.png, frame003.png. Use frame%d.png if your files are named e.g. frame%03d.png, frame2.png, frame3.png.

If you want to do something cleaner, you should investigate some libraries for actually creating the movie within Python itself - e.g., moviepy (removing the need for saving the frames individually and running avconv). You can also use the Python subprocess library to call avconv from within Python, although in this case you still need to save the individual frames.

The results are pretty cool! Check out these examples that I created (Note: for these examples I ran the time variable from -1 to 1 and then in reverse from 1 to -1):

Extension 3: Pulsating Music Visualizer!!!!!

The idea here will be to have a visualization that responds in real-time to loudness changes as picked up by your computer’s microphone. This can be used to have a visual that pulses along to the music (or your voice).

Linux: In order to process audio from the microphone, check out pyalsaaudio. To use the library you will need to install both the pyalsaaudio package and some libraries that it depends on. Here are the install commands:

$ apt-get install libasound2-dev
$ pip install pyalsaaudio

Here is some simple code for getting audio from the mic and displaying the volume:

import alsaaudio
import audioop

inp = alsaaudio.PCM(alsaaudio.PCM_CAPTURE, 0)

while True:
    l, data =
    if l:
        print(audioop.rms(data, 2))

To display images in real-time (as opposed to using an image viewer to checkout the images after you have saved them to a file), I recommend using pygame. To install pygame, follow the instructions from the recipe page.

One very important thing is that your code will not be fast enough to generate the appropriate image for a given volume level in real-time. You should generate the images for various volume levels ahead of time, and have them loaded and ready to be displayed using pygame when appropriate.

Project Reflection

Please prepare a short document (~1 paragraph to 1-2 pages) that addresses:

From a process point of view, what went well? What could you improve? Other possible reflection topics: Was your project appropriately scoped? Did you have a good plan for unit testing? How will you use what you learned going forward? What do you wish you knew before you started that would have helped you succeed?

Add your Reflection to your GitHub repository. This can be in the form of:

Turning in your assignment

In order to turn in your assignment, you will need to push your code to GitHub (for the main assignment, all code will be in Submit a link on Canvas to your on GitHub. If you choose to do the going beyond portion, it is up to you how you structure your code for that portion). You should also push at least two images generated by your program using the file names example1.png and example2.png (if you don’t want to use PNG, feel free to use a different standard image format).

We will be creating an art gallery of examples from the class. To participate, submit your art files (either images or movies) by email to Ben along with a brief file called ARTIST_STATEMENT.txt that contains:

The types of things you might want to touch on in the artist statement are:

A much better and more extensive list of guidance around producing an artist’s statement can be found here.

In order to make it into the exhibition you should submit your art no later than: Sun, Sep 30.