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
@#$#@#$#@
GRAPHICS-WINDOW
262
10
675
426
-1
-1
1.0
1
10
1
1
1
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202
-203
203
0
0
1
ticks
30.0
BUTTON
60
271
188
328
Iterate
iterate
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
BUTTON
4
184
118
217
Levy Curve
Levy-curve
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
TEXTBOX
4
10
214
54
Examples of Fractals
18
95.0
1
TEXTBOX
7
126
157
148
Examples
16
0.0
1
BUTTON
4
216
118
249
Dragon Curve
dragon-curve
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
MONITOR
684
73
741
118
N
n-rep
0
1
11
MONITOR
685
131
742
176
M
1 / scale-factor
3
1
11
MONITOR
686
189
900
234
Hausdorff dimension = log(N)/log(M)
log(n-rep) 10 / log(1 / scale-factor) 10
3
1
11
TEXTBOX
683
14
833
36
Fractal Dimension
16
0.0
1
BUTTON
118
152
247
186
Cantor Set
Cantor-set
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
BUTTON
118
184
247
217
Sierpinski Triangle
Sierpinski-triangle
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
BUTTON
118
216
247
249
Tree
fractal-tree
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
TEXTBOX
751
73
936
114
N is the number of smaller copies of the previous iteration generated at each new iteration.
11
0.0
1
TEXTBOX
751
130
901
175
M is the factor by which each line segment shrinks at the next iteration.
11
0.0
1
TEXTBOX
684
39
973
81
With each iteration, a fractal uses a number of smaller copies of itself to construct a new pattern.
11
0.0
1
MONITOR
687
272
801
317
Length of fractal
fractal-length
4
1
11
BUTTON
4
152
118
185
Koch Curve
koch-prev
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1
TEXTBOX
806
265
1008
321
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
5
50
250
121
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:
logax = logbx / logba
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:
d = logMN
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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