Sierpinski Simple 3D

1 collaborator

Uri Wilensky (Author)

WHAT IS IT?

This is a 3D version of the Sierpinski Simple model in the NetLogo Models Library. 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 screen 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 an academic publication, we ask that you include these citations for the model itself and for the NetLogo software:

- Wilensky, U. (1998). NetLogo Sierpinski Simple 3D model. http://ccl.northwestern.edu/netlogo/models/SierpinskiSimple3D. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

In other publications, please use:

COPYRIGHT NOTICE

Permission to use, modify or redistribute this model is hereby granted, provided that both of the following requirements are followed:

a) this copyright notice is included.

b) this model will not be redistributed for profit without permission from Uri Wilensky. Contact Uri Wilensky for appropriate licenses for redistribution for profit.

This is a 3D version of the 2D model Sierpinski Simple.

This model was created 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.

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
  ca
  crt 1
  [
    setxyz 0 0 -25
    set modulus 0.5 * max-pycor
    pd
  ]
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
    ]
    tilt-up 90
    hatch 1
    [
      fd modulus
      set modulus ( 0.5 * modulus )
      tilt-down 90
    ]
    die  ; kill all the living turtles
  ]
  tick
end 


; Copyright 1998 Uri Wilensky. All rights reserved.
; The full copyright notice is in the Information tab.

There are 3 versions of this model.

Uploaded by	When	Description	Download
Uri Wilensky	almost 14 years ago	Updated from NetLogo 4.1	Download this version
Uri Wilensky	almost 14 years ago	Model from NetLogo distribution	Download this version
Uri Wilensky	almost 14 years ago	Sierpinski Simple 3D	Download this version

Attached files

File	Type	Description	Last updated
Sierpinski Simple 3D.png	preview	Preview for 'Sierpinski Simple 3D'	about 11 years ago, by Uri Wilensky	Download

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