OF VEGETABLES AND POLISH TRIANGLES

     What do a fern and a snowflake have in common? If I were to tell you that they relate in the same way as DNA relates to Romanesco broccoli, would it strike you as odd? They are all cases of natural-given self-similarity. In mathematics, a self-similar object is exactly or approximately similar to a part of itself (this contradicts the popular saying: beautiful from afar, but far from beautiful).

     A couple of weeks ago I came across a creation of the Polish mathematician Warclaw Sierpinski and was amazed by its simplicity and beauty. The Sierpinski Triangle is a self-similar fractal figure created by following a very simple algorithm. In fact, the algorithm is so simple that you can draw it on a sheet of paper while reading this post:

1. First draw an equilateral triangle the size of the palm of your hand
2. Choose a random point inside the triangle and mark that point with a pen or pencil.
3. Randomly choose a corner of the triangle.
4. Mark a new point halfway from the last point you drew and the previously chosen corner.
5. Randomly choose a new corner (it may be the same one you chose before).
6. Repeat steps 4 and 5 indefinitely (do this a couple of times).

Note: You probably won´t notice anything extraordinary by doing 10 or even 50 iterations.

     I would recommend you draw a couple of points in order to understand the algorithm but leave the ‘indefinitely’ part to a machine. At Continente Siete we modelled this simple algorithm and will leave it for you to discover what emerges from the seemingly trivial steps you have followed.

     You can run the model by following these three steps:

• After starting the model, click inside the displayed triangle to select an initial point
• Repeatedly click on the ‘Add point(s)’ button located on the upper right of the window (select the ‘Auto Add Point(s)’ checkbox to do this automatically).
• Observe the emerging pattern (note: speed up the process by adding several points  at a time with the ‘Add points’ slider located on the upper right of the window).

     We welcome you to discover and experiment with the other features included in the model.

     Maybe the next time you eat broccoli you´ll take a closer look and appreciate the mathematics behind nature.