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Sierpinski Simple

 If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web

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. One of the moon's craters is named after him.

HOW IT WORKS

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 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

```text . 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

HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

Please cite the NetLogo software as: