Sierpinski Simple

Sierpinski Simple preview image

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Uri_dolphin3 Uri Wilensky (Author)

Tags

mathematics 

Tagged by Reuven M. Lerner about 9 years ago

Model group CCL | Visible to everyone | Changeable by group members (CCL)
Model was written in NetLogo 5.0.4 • Viewed 193 times • Downloaded 22 times • Run 0 times
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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 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

HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

COPYRIGHT AND LICENSE

Copyright 1998 Uri Wilensky.

CC BY-NC-SA 3.0

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2001.

Comments and Questions

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Click to Run Model

turtles-own [ modulus ]

; create a turtle and set its initial location and modulus

to setup
  clear-all
  crt 1
  [
    set heading 0
    setxy 0 -3
    set modulus 0.5 * max-pycor
    pen-down
  ]
  reset-ticks
end 

; ask the turtles to go forward by modulus, create a new turtle to
; draw the next iteration of sierpinski's tree, and return to its place

to grow
  hatch 1
    [ fd modulus
      set modulus (0.5 * modulus) ]  ; new turtle's modulus is half its parent's
end 

; draw the sierpinski tree

to go
  ask turtles
  [
    repeat 3
    [
      grow
      right 120  ; turn counter-clockwise to draw more legs
    ]
    die  ; kill all the living turtles
  ]
  tick
end 


; Copyright 1998 Uri Wilensky.
; See Info tab for full copyright and license.

There are 10 versions of this model.

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Uri Wilensky over 9 years ago Updated to NetLogo 5.0.4 Download this version
Uri Wilensky almost 10 years ago Updated version tag Download this version
Uri Wilensky almost 10 years ago Updated to version from NetLogo 5.0.3 distribution Download this version
Uri Wilensky over 10 years ago Updated to NetLogo 5.0 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 12 years ago Model from NetLogo distribution Download this version
Uri Wilensky over 12 years ago Sierpinski Simple Download this version

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