turtles-own [new?] ;if the turtle is newly hatched or not

globals [ 
  angle          ;the angle that the self-similar pieces are rotated to make the next image
  scale-factor   ;the scalar that the initial image is reduced by (i.e. 1/2 or 1/3, not 2 or 3)
  initial-length ;the initial length of the fractal's line segments
  pen-down?      ;whether or not to have the pen down while drawing
  memory         ;the l-system command string of the fractal
  n-rep          ;number of self-similar copies generated from each line segment
  len            ;current length of the line segments in the fractal
  cd?            ;whether or not to clear the display after each iteration
  fractal-length ;sum of all line segments in the fractal at that iteration
  fractal-name
  current-iteration ; current level at which fractal is being displayed
]

to reset
  clear-all
  reset-ticks
  set current-iteration 0
  set initial-length 100
  set len initial-length 
  set memory "" 
  set cd? true
  ask patch 0 0  [sprout 1 
    [
      pd
      set pen-size 2
      set new? false
      set color green
      set heading 0
    ]
  ]
  tick
  ask turtles [ hide-turtle ]
end

to iterate
  set current-iteration current-iteration + 1
  if cd? [cd]
  if fractal-name = "kochp" [
    ask turtles[
    ifelse(pycor > -100)[pu setxy (xcor) (ycor - 96) pd] ;moving the turtle down 100 patches so successive iterations are visible
    [stop]
    ]
  ]
  set len len * scale-factor
  ask turtles with [not new?]
  [die]
  ask turtles with [new?] 
  [set new? false]
  ask turtles with [not new? ]  [ run memory]
  ask turtles [ hide-turtle ]
  calc-length
  tick
end

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;Examples;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

to dragon-curve
 reset
 set fractal-length 1
 set fractal-name "dragon"
 set n-rep 2
 ask turtles[
   set pen-size 2
   pen-up
   setxy 0 -70
   pd fd 140 pu
   setxy 0 -70
   pd t
 ]
dragon "lt 45 t  pd fd  len rt 45 rt 45 pd fd  len lt 180 t "
end

to dragon [dragon-command-string]
  set angle 45
  set scale-factor 1 / sqrt(2)
  set memory (word memory "\n" dragon-command-string)
  set len len / 0.71
  tick
end

to levy-curve
  reset
  set fractal-length 1
  set fractal-name "levy"
  set n-rep 2
  set angle 45
  set scale-factor 1 / sqrt(2)
  ask turtles[
    pen-up
    setxy 0 -50
    pd fd 140 pu
    setxy 0 -50
    pd t
  ]
  levy "lt 45 t fd len rt 45 rt 45 t fd len"
end

to levy [levy-command-string]
  set len len / 0.71
  set memory (word memory "\n" levy-command-string)
  tick
end

to koch-prev
  reset
  set cd? false
  set fractal-length 1
  set fractal-name "kochp"
  set n-rep 4
  set angle 60
  set scale-factor 1 / 3
  ask turtles[
    pu
    setxy -150 190
    pd
    set heading 90
    fd 300
    pu setxy -150 190 pd t
  ]
  koch-p "t fd len left 60 t fd len right 60 right 60 t fd len left 60 t fd len"
end

to koch-p [Koch-command-string]
       set memory (word memory "\n" Koch-command-string)
       set len 300
  tick
end

to Cantor-set
  reset
  set fractal-length 1
  set fractal-name "cantor"
  set n-rep 2
  set len 100
  set cd? false
  ask turtles [
    pen-up 
    set heading 90
    setxy -150 170
    pd fd 300 pu
    setxy -150 170
    t pd
    ]
  set scale-factor 1 / 3
  Cantor "t right 90 pu fd 30 pd left 90 t fd len pu fd len pd t fd len"
end

to Cantor [Cantor-command-string]
  set memory (word memory "\n" Cantor-command-string)
  set cd? false
  set len len * 3
  tick
end

to Sierpinski-triangle
  reset
  set fractal-length "N/A"
  set fractal-name "sierpinski"
  set n-rep 3
  set len 270
  set scale-factor 1 / 2
  ask turtles [ 
    pu setxy 0 135 pd
    ]
  Sierpinski "t fd len / 2 t fd len / 2 left 120 fd len left 120 fd len / 2 left 120 t right 120 fd len / 2"
end

to Sierpinski [Sierpinski-command-string]
  ifelse ticks = 1
  [
    ask turtles with [not new?] [set heading 210 run Sierpinski-command-string]
    set memory (word memory "\n" Sierpinski-command-string)
  ]
  [
    cd
    set len len / 2
    ask turtles with [not new?] [die]
    ask turtles with [new?]
    [
      set new? false
    ]
    ask turtles with [not new?] [run memory]
  ]
  tick
end

to fractal-tree
  reset
  set fractal-length 1
  set fractal-name "tree"
  set cd? false
  set n-rep 2
  set scale-factor 1 / 2
  set len 100
  ask turtles[
    pu
    setxy 0 -60
    pd fd 100 t
  ]
  set len len * 1 / 2
  tree "right 15 fd len  t rt 180 pu fd len pd rt 180 lt 40 fd len t"
end

to tree [tree-command-string]
  set cd? false
  set len len * 2
  set memory (word memory "\n" tree-command-string)
  tick
end
   

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;fractal commands;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

to t
  hatch 1 [set new? true]
end

to calc-length
  if fractal-name = "tree" [set fractal-length ticks]
  if fractal-name = "sierpinski" [set fractal-length "N/A"]
  if fractal-name = "dragon" or fractal-name = "levy" [set fractal-length fractal-length * sqrt(2)]
  if fractal-name = "kochp" [ set fractal-length fractal-length * 4 / 3]
  if fractal-name = "cantor" [set fractal-length fractal-length * (2 / 3)]
end
@#$#@#$#@
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ticks
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BUTTON
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188
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Iterate
iterate
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T
OBSERVER
NIL
NIL
NIL
NIL
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BUTTON
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184
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217
Levy Curve
Levy-curve
NIL
1
T
OBSERVER
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NIL
NIL
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TEXTBOX
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Examples of Fractals
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TEXTBOX
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126
157
148
Examples
16
0.0
1

BUTTON
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Dragon Curve
dragon-curve
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T
OBSERVER
NIL
NIL
NIL
NIL
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MONITOR
684
73
741
118
N
n-rep
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11

MONITOR
685
131
742
176
M
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MONITOR
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900
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Hausdorff dimension = log(N)/log(M)
log(n-rep) 10 / log(1 / scale-factor) 10
3
1
11

TEXTBOX
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14
833
36
Fractal Dimension
16
0.0
1

BUTTON
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152
247
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Cantor Set
Cantor-set
NIL
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T
OBSERVER
NIL
NIL
NIL
NIL
1

BUTTON
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Sierpinski Triangle
Sierpinski-triangle
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1
T
OBSERVER
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NIL
NIL
NIL
1

BUTTON
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Tree
fractal-tree
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
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TEXTBOX
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936
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N is the number of smaller copies of the previous iteration generated at each new iteration.
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TEXTBOX
751
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901
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M is the factor by which each line segment shrinks at the next iteration.
11
0.0
1

TEXTBOX
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973
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With each iteration, a fractal uses a number of smaller copies of itself to construct a new pattern.
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0.0
1

MONITOR
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801
317
Length of fractal
fractal-length
4
1
11

BUTTON
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152
118
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Koch Curve
koch-prev
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

TEXTBOX
806
265
1008
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Fractal length is the sum of segment lengths at the current iteration. Assume initial line segment is of length 1.
11
0.0
1

TEXTBOX
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50
250
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First click on an example to see the fractal generator.  Then repeatedly click \"Iterate\" to see the next iteration. 
13
0.0
1

PLOT
687
327
1004
447
Fractal Length over Time
Time
Length
0.0
10.0
0.0
5.0
true
false
"" ""
PENS
"default" 1.0 0 -10899396 true "" "if(fractal-length != \"N/A\" and ticks > 1) [plot fractal-length]"

TEXTBOX
686
246
836
266
Fractal Length
16
0.0
1

MONITOR
60
342
188
387
Current Iteration
current-iteration
17
1
11

@#$#@#$#@
## WHAT IS IT?
This model illustrates how fractals work. In this model, you can experiment with a variety of different fractal patterns to see how fractals recursively create smaller copies to produce larger patterns which have increasing length. Additionally, the fractional dimension of fractals is showcased, along with the formula for Hausdorff dimension.

## HOW IT WORKS

In the interface window, click on one of the examples.  This will display the "generator" (initial pattern) for the chosen fractal.  Then click repeatedly on "Iterate" to show each successive iteration of the fractal.  For example, each iteration of the Koch Curve produces four new copies, each three times smaller than the original, which lie on the original four segments. The number of copies generated at each iteration is denoted N and the shrinking factor for each segment is denoted M. The Hausdorff dimension is defined as log(N)/log(M). Note that the base of this log is irrelevant, which is explained further below. One of the key features of fractals is that they can have fractional dimension. Additionally, the length of a fractal (the sum of the length of all segments) can increase with each iteration.  For example, in the Koch Curve, a straight line between two points (by definition the shortest possible length) is continually replaced with a more indirect, longer path.

## THINGS TO NOTICE

Look for how the shrinking factor, M, and the number of copies, N, play into the pattern that emerges with successive iterations. You can also observe the increase in fractal length that occurs with each step: try to find a pattern between the fractal's properties (N and M) and the successive changes in curve length.

### A NOTE ON LOGS

The equation for the Hausdorff dimension is log(N)/log(M). Note that this follows the change of base formula:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;log<sub>a</sub>x = log<sub>b</sub>x / log<sub>b</sub>a
where the value of b, the base of the log in the fraction, can be any real value. Thus, you could also write the equation for the Hausdorff dimension as:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;d = log<sub>M</sub>N
This is the reason that the base of the log doesn't matter in the fraction. In this code, a common log (base 10) is used, although any base would give the same result.

## CREDITS AND REFERENCES

This model is part of the Fractals series of the Complexity Explorer project.  
 
Main Author:  Vicki Niu

Contributions from:  John Driscoll, Melanie Mitchell

Some of the code for this model was adapted from the L-System Fractals model in the Netlogo Models Library:  Wilensky, U. (2001).  NetLogo L-System Fractals model.  http://ccl.northwestern.edu/netlogo/models/L-SystemFractals.  Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

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


## HOW TO CITE

If you use this model, please cite it as: "Examples of Fractals" model, Complexity Explorer project, http://complexityexplorer.org

## COPYRIGHT AND LICENSE

Copyright 2016 Santa Fe Institute.  

This model is licensed by the Creative Commons Attribution-NonCommercial-ShareAlike  International  ( http://creativecommons.org/licenses/ ). This states that you may copy, distribute, and transmit the work under the condition that you give attribution to ComplexityExplorer.org, and your use is for non-commercial purposes.
@#$#@#$#@
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