algorithmic art workshop

Viitaniemen koulu, 16...17 March, 2017

by Risto A. Paju


In this workshop, we learn to make fractal art with iterated function systems. We use Python and gnuplot to turn math into pictures. I will not teach programming from scratch; instead, we start with code examples and modify them to make different pictures.

IFS stages of the Sierpinski triangle

Timetable for both days

Getting started

  1. Download and ifs.plot into your home folder (one step "up" from the default downloads).
  2. Open a terminal (Finnish: pääte) and run the command "python". This makes a file called "ifs.dat" with the x and y coordinates of points.
  3. Run "gnuplot ifs.plot" in the terminal. This makes a graph of the points into the file "ifs.png".

Open the image in the web browser: File -> Open file... or from the terminal "firefox ifs.png &". You can keep it open and reload it by pressing F5 after you modify it.

Editing the functions

Open the file in an editor, for example "gedit &". Try modifying some of the functions s1, s2 and s3. For example, change a shift of 0.5 into something else. Try small changes at first.

To run the necessary commands at once, type "python && gnuplot ifs.plot". See how the change in functions shows up in the picture.

When you find a really nice picture, save a copy of the Python file in a different file: Save as... / Tallenna nimellä... with a descriptive name, like Then you can open the original file (e.g. and keep editing, while keeping the other file.

Extra: can you modify the triangle to make it equilateral? You may need the square root function, for example math.sqrt(3) for √3.


The first example only has shifts, but we often need rotations too. The file has a rotation function and another example. Try it with "python && gnuplot ifs.plot" and try some changes too, for example different angles of rotation.

Remember that in mathematics, a positive rotation means counterclockwise.

Hint: angles like 45, 60 or 90 degrees make sharp and blocky results. They are simple fractions of a full circle, so after a few iterations they return to the same position. For more organic shapes, use other angles like 18.69 degrees.

Colouring stages

So far, we have plotted only the final iterates. In my colour method, we plot some of the earlier stages as well, each with a different colour. The coordinates of different stages go to their own .dat files.

In the graph, you now have more points in total, so you can either use fewer initial ones (ninit = ... in Python) or smaller points (plot ... pt 0 in gnuplot).

Example: python && gnuplot ifs-stages.plot

The example uses all stages, including the initial points. You can also leave out some stages, for example use only stages 6 to 10. For this, edit the .plot file.

IFS stages of the Koch snowflake

Black and white or colour?

While multiple colours are nice, some pictures may work better in simple black-on-white — like pencil drawings. (See some of my works for example.)

Final prints and labels

After finishing your artwork, send me

  1. The PNG file
  2. The Python file (with the functions that make up the art)
  3. Your name for the label: first name / full name / no name / artist name ...
Copy them to me on a USB stick, or send them by email: I will handle the printing, and the works will be on display on April 22. The functions will be shown below the pictures.

Further help



Python and Linux


© Risto A. Paju, distributed under the GNU General Public License - Version 3, 29 June 2007.