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 10 677 448 202 203 1.0 1 10 1 1 1 0 0 0 1 -202 202 -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. @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 airplane true 0 Polygon -7500403 true true 150 0 135 15 120 60 120 105 15 165 15 195 120 180 135 240 105 270 120 285 150 270 180 285 210 270 165 240 180 180 285 195 285 165 180 105 180 60 165 15 arrow true 0 Polygon -7500403 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box false 0 Polygon -7500403 true true 150 285 285 225 285 75 150 135 Polygon -7500403 true true 150 135 15 75 150 15 285 75 Polygon -7500403 true true 15 75 15 225 150 285 150 135 Line -16777216 false 150 285 150 135 Line -16777216 false 150 135 15 75 Line -16777216 false 150 135 285 75 bug true 0 Circle -7500403 true true 96 182 108 Circle -7500403 true true 110 127 80 Circle -7500403 true true 110 75 80 Line -7500403 true 150 100 80 30 Line 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