NetLogo Models Library:
Sample Models/Mathematics/Fractals

(back to the library)

Sierpinski Simple

Run Sierpinski Simple in your browser
uses NetLogo 4.0.3
requires Java 1.4.1+
(system requirements)

Note: If you download the NetLogo application, every model in the Models Library (besides the Community Models) is included. If you have trouble running this model in your browser, you may wish to download the application instead.

WHAT IS IT?

The fractal that this model produces was discovered by the great Polish mathematician Waclaw Sierpinski in 1916. Sierpinski was a professor at Lvov and Warsaw. He was one of the most influential mathematicians of his time in Poland and had a worldwide reputation. In fact, one of the moon's craters is named after him.

The basic geometric construction of the Sierpinski tree goes as follows. We begin with a single point on the plane and then apply a repetitive scheme of operations to it. Grow a "spider" centered at this point by drawing three equal line segments directed to the vertices of an equilateral triangle. Then at each vertex of the triangle repeat the construction -- grow a similar "spider" only scale it down by the factor of two.

               .               Step 0: Start with a point

               |
               |
               |               Step 1: Grow a spider
              / \
             /   \
            /     \

               |
               |
              /|\
             / | \
               |               Step 2: Repeat step 1
              / \
           | /   \ |
           |/     \|
          / \     / \
         /   \   /   \

The Sierpinski tree is closely related to the class of fractals called Sierpinski Carpets which includes the famous Sierpinski Triangle or as it is usually called The Sierpinski Gasket.

The features that characterize the Sierpinski tree are self-similarity and connectedness. It is not always easy to determine if a fractal is connected. It took almost a decade to prove the connectedness of the famous Mandelbrot set. However connectedness is apparent from the way Sierpinski tree is generated; at each iteration the set is connected.

HOW TO USE IT

Push the SETUP button to clear the world and initialize globals. Press repeatedly on the GO ONCE button to perform iterations of the Sierpinski algorithm.

THINGS TO NOTICE

Notice the use of "hatch" primitive which makes it so simple to generate fractals like Sierpinski tree.

THINGS TO TRY

Try to write a program that draws other self-similar shapes. For instance try the rule below

                      .               Step 0

   
                      |
                      |
                      |
               ______________         Step 1
                      |
                      |
                      |

                      |
                    __|__
                      |
                      |
              __|___________|__       Step 2
                |     |     |
                      |
                    __|__
                      |

The resulting fractal is known in Algebraic Topology as a Universal Covering of the Figure Eight.

NETLOGO FEATURES

Notice how the curves are formed using several agents following the same rules. Also, take note of the use of the hatch command.

RELATED MODELS

L-System Fractals

CREDITS AND REFERENCES

To refer to this model in academic publications, please use: Wilensky, U. (1998). NetLogo Sierpinski Simple model. http://ccl.northwestern.edu/netlogo/models/SierpinskiSimple. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

(back to the NetLogo Models Library)

NetLogo Models Library:Sample Models/Mathematics/Fractals

Sierpinski Simple

NetLogo Models Library:
Sample Models/Mathematics/Fractals