extensions [matrix] globals [ divisions ;the factor by which the length of the fractal's line segments decreases (M in the Hausdorff dimension equation) fractal-dim ;Hausdorff dimension of the fractal initial-length ;initial length of the line segments len ;current length of the line segments (after iteration) old-box-count ;number of red boxes covering parts of the fractal at the previous step new-box-count ;current number of boxes (red) covering parts of the fractal x-axis-list ;list of all the x-values for the log-log box-counting dimension plot y-axis-list ;list of all the y-values for the log-log box-counting dimension plot box-size ;current size of boxes explore? ;whether or not the boxes need to move and find a portion of the fractal to cover iteration ;counts the box-counting iteration iterations-to-go slope ;slope of the box-counting log-log plot regression line, equal to the box-counting dimension r-square ;r-squared value for the box-counting log-log plot regression line lin-reg-eq ;linear regression equation for the log-log plot any-new-boxes? x-cor ;initial x-coordinate for the first turtle y-cor ;initial y-coordinate for the first turtle fractal-example ;the name of the fractal being drawn ] ;globals ; [divisions fractal-dim initial-length ; len walk-count old-box-count ; new-box-count x-axis-list y-axis-list ; box-size explore? iteration ; iterations-to-go slope lin-reg-eq ; any-new-boxes? x-cor y-cor ; ] turtles-own ;used only for making fractal examples and custom frctals if you choose. [new?] breed ;used only for taking box counting dimension of fractal examples. [boxes box] boxes-own [past? present? future?] patches-own [fractal?] to startup setup end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;; Fractal Procedures ;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;Clears the world and sets up an initial turtle to setup ;Initializing values set explore? false set x-axis-list [ ] set y-axis-list [ ] set fractal-dim "N/A" ;the first iteration has not yet been drawn, so we don't know what the Hausdorff dimension is ;Setting the initial turtle position if (fractal-example = "koch-curve" or fractal-example = "cantor-set") [set x-cor -150 set y-cor 0 set divisions 3] if(fractal-example = "levy-curve" or fractal-example = "dragon-curve") [set x-cor -60 set y-cor 0 set divisions sqrt 2] if(fractal-example = "sierpinski-triangle")[set x-cor -100 set y-cor 0] crt 1 ;creating the initial turtle ask turtles [ ;initializing turtle properties set new? false set shape "circle" set size 1 set color green setxy x-cor y-cor set heading 90 ifelse(fractal-example != "tree") [ set initial-length 100 pd fd initial-length * divisions set len initial-length pu setxy x-cor y-cor ] [ set divisions 2 set initial-length 100 set x-cor 0 set y-cor -150 set len initial-length setxy x-cor y-cor set heading 0 walk walk ] ] reset-ticks if fractal-example = "sierpinski-triangle" [iterate] end ;Depending on the example chosen for the fractal, this will iterate the fractal to iterate ;The tree fractal requires that the previous iterations remain on the screen if fractal-example != "tree" [clear-drawing] ;Creates standard Koch curve if fractal-example = "koch-curve" [ ask patches [set pcolor black] ; clears screen ask turtles [ set new? false ] t walk l 60 t walk r 120 t walk l 60 t walk d ;l-system set divisions 3 ;the factor by which the length decreases set len (len / divisions) fractal-dimension ;calculates & updates fracal dimension tick ] ;Creates same fractal tree as in simple fractals and the L-systems if fractal-example = "tree" [ ;ask patches [set pcolor black] ask turtles[ set new? false] r 15 walk t r 180 skip len r 180 l 40 walk t ask turtles with [new? = false] [die] set divisions 2 set len (len / divisions) fractal-dimension tick ] ;Creates the Levy C Curve if fractal-example = "levy-curve" [ ask patches [set pcolor black] ask turtles[ set new? false] l 45 t walk r 90 t walk d set divisions sqrt 2 set len (len / divisions) fractal-dimension tick ] ;Creates the Sierpinski triangle if fractal-example = "sierpinski-triangle" [ ask patches [set pcolor black] ; clears screen ask turtles [set new? false pd] ;set len len / 2 t walk t walk l 120 walk walk l 120 walk l 120 t r 120 walk d set divisions 2 set len (len / divisions) fractal-dimension tick ] ;Creates the Cantor set if fractal-example = "cantor-set" [ ask patches [set pcolor black] ; clears screen ask turtles [set new? false ] t walk skip len t walk d set divisions 3 set len (len / divisions) fractal-dimension tick ] ;Creates the Heighway dragon curve if fractal-example = "dragon-curve" [ ask patches [set pcolor black] ; clears screen ask turtles [ set new? false ] r 45 t walk l 90 walk l 180 t d set divisions sqrt 2 set len (len / divisions) fractal-dimension tick ] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;Box-Counting-Dim;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to box-counting-setup set box-size initial-box-length set iteration iteration make-initial-box end to box-counting-go ask boxes ; clears screen in case there are any old boxes [die] set initial-length 100 if box-size >= initial-length ;prevents boxes from getting too big for the world [stop] set box-size box-size + increment set iteration iteration + 1 set iterations-to-go 91 - iteration set old-box-count 0 set new-box-count 1 ;eliminates an error for first round make-initial-box make-neighbors-count-patches end ;makes a starter box at the beginning of each run with present? = true. ;This box will then be used to make boxes with future? = true ;which will be used for the next run. to make-initial-box create-boxes 1 ask boxes [ set shape "square" set size box-size setxy x-cor y-cor set heading 90 set color red set past? false set present? true set future? false ] end ;makes a Moore neighborhood around the starter box and counts patches below each new box (exploit). ;If there are no new boxes with patches under them for a given run a box is sent outside the neighborhhod ;to cover non-contiguous patches (explore). If this box finds no new patches the run is complete. to make-neighbors-count-patches ask boxes with [present? = true ] [make-neighbors] ask boxes with [future? = true] [exploit] count-new-boxes if any-new-boxes? = false [explore] if any-new-boxes? = false [calculate-box-counting-dimension stop] update-past-present-future-states tick if any-new-boxes? = true [make-neighbors-count-patches] end to make-neighbors hatch 1 [fd box-size rt 90 set present? false set future? true hatch 1 [fd box-size rt 90 set present? false set future? true hatch 1 [fd box-size set present? false set future? true hatch 1 [fd box-size rt 90 set present? false set future? true hatch 1 [fd box-size set present? false set future? true hatch 1 [fd box-size rt 90 set present? false set future? true hatch 1 [fd box-size set present? false set future? true hatch 1 [fd box-size set present? false set future? true ]]]]]]]] end to exploit if count boxes in-radius (box-size / 2) > 1 ; eliminates duplicates [die] if count patches-under-me = 0 [ die ] end to-report patches-under-me report patches in-radius ( (1.4 * size ) / 2 ) with [pcolor = green] end to explore if count boxes with [present? = true] > 0 [ ask patches with [pcolor = green] [ ifelse count boxes in-radius ( box-size ) = 0 [set explore? true] [stop] ] ] if explore? [ ask one-of boxes with [present? = true] [ hatch 1 [ set present? false set future? true move-to min-one-of patches with [pcolor = green and count boxes in-radius ( box-size ) = 0 ] [distance myself]] ] ] count-new-boxes set explore? false end to count-new-boxes set old-box-count new-box-count set new-box-count count boxes ifelse old-box-count = new-box-count [set any-new-boxes? false] [set any-new-boxes? true] end to update-past-present-future-states ask boxes [ if present? = true [set past? true set present? false] if future? = true [set future? false set present? true] ] end to calculate-box-counting-dimension if count boxes >= 1 [ ; eliminates potential error message if setup is pressed during a box-counting procedure set-current-plot "Box Counting Plot" set-current-plot-pen "default" let no-boxes log (count boxes ) 10 let scale (log ( 1 / box-size ) 10 ) plotxy scale no-boxes set x-axis-list lput scale x-axis-list set y-axis-list lput no-boxes y-axis-list ] stop end to fractal-dimension if ticks = 0 [ if divisions > 0 [ ; eliminates potential error message let line-segments count turtles set fractal-dim precision(( log line-segments 10 / log divisions 10 ))3 ] ] stop end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;; Fractal Commands ;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; walk moves a turtle forward by a length (len) and leaves a trail of green patches. to walk ask turtles with [not new?] [ set pcolor green let walk-count .1 while [walk-count <= len] [ fd .1 set pcolor green set walk-count ( walk-count + .1) ] ] stop end ;hatch a new turtle for all the turtles that are not new to the current iteration and set new? to be true to t ask turtles with [not new?] [hatch 1 [set new? true]] end ;turn each turtle that wasn't created during the current iteration to the right by degree degrees to r [degree] ask turtles with [not new?] [rt degree] end ;turn each turtle that wasn't created during the current iteration to the left by degree degrees to l [degree] ask turtles with [not new?] [lt degree] end ;move each turtle that wasn't created during the current iteration forward by steps but do not leave trail of green patches. to skip [steps] ask turtles with [not new?] [ fd steps ] end ;all non-new turtles die to d ask turtles with [not new?] [die] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;Linear Reg;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;Linear regression is used to find the 'best-fit' line ;through all the tuples (box-size,number of boxes) plotted in the scatter plot. ;The slope of the line is the box counting dimension. to linear-regression if (count boxes >= 1) and (length x-axis-list > 1) [ ; eliminates potential error message if setup is pressed during a box-counting procedure ; or if there is only one point in the log-log plot let regression matrix:regress matrix:from-column-list (list y-axis-list x-axis-list) ;using the regression tool from the matrix extension ;setting y-intercept and slope (measure of goodness of fit) let y-intercept item 0 (item 0 regression) set slope item 1 (item 0 regression) set r-square item 0 (item 1 regression) ; set the equation to the appropriate string set lin-reg-eq (word (precision slope 2) " * x + " (precision y-intercept 2)) ;plotting the line on the log-log plot set-current-plot "Box Counting Plot" set-current-plot-pen "pen-4" plot-pen-reset auto-plot-off plotxy plot-x-min (plot-x-min * slope + y-intercept) plot-pen-down plotxy plot-x-max (plot-x-max * slope + y-intercept) plot-pen-up ] end @#$#@#$#@ GRAPHICS-WINDOW 270 15 799 545 -1 -1 1.0 1 10 1 1 1 0 1 1 1 -260 260 -260 260 1 1 1 ticks 30.0 PLOT 817 15 1084 270 Box Counting Plot log [1 / box-length] log number of boxes (N) -2.0 -1.0 0.0 2.0 true false "" "" PENS "default" 1.0 2 -16777216 true "" "" "pen-4" 1.0 0 -2674135 true "" "" MONITOR 817 314 1084 359 Hausdorff Dimension (log N/log M) fractal-dim 17 1 11 BUTTON 138 278 250 311 Box Counting Go box-counting-go T 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 817 358 1084 391 NIL linear-regression NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 817 269 949 314 Box-Counting Dim. precision(slope)3 17 1 11 MONITOR 91 413 173 458 Box Length precision(box-size)2 17 1 11 MONITOR 26 413 92 458 NIL iteration 17 1 11 TEXTBOX 28 251 242 275 Box-Counting Controls 18 0.0 1 TEXTBOX 824 401 1086 443 The Box-Counting Dimension is the slope of the line that best fits the points. Compare with the Hausdorff Dimension. 11 0.0 1 TEXTBOX 22 38 172 60 Fractal Examples 18 0.0 1 TEXTBOX 21 12 248 46 Box-Counting Dimension 18 95.0 1 BUTTON 73 184 187 233 NIL iterate NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 25 310 250 343 initial-box-length initial-box-length .2 100 4.0 .1 1 NIL HORIZONTAL SLIDER 24 343 250 376 increment increment 0 1 0.8 .1 1 NIL HORIZONTAL BUTTON 15 70 128 103 Koch Curve ca\nset fractal-example \"koch-curve\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 128 70 244 103 Cantor Set ca\nset fractal-example \"cantor-set\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 15 102 128 135 Levy Curve ca\nset fractal-example \"levy-curve\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 127 102 244 135 Sierpinsky Triangle ca\nset fractal-example \"sierpinski-triangle\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 15 134 128 168 Dragon Curve ca\nset fractal-example \"dragon-curve\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 127 134 245 168 Tree ca\nset fractal-example \"tree\"\nsetup NIL 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 29 376 239 418 Amount that box lenth increases per iteration of box counting 11 0.0 1 MONITOR 173 413 244 458 # of Boxes count boxes with [color = red] 17 1 11 BUTTON 25 278 140 311 Box Counting Setup box-counting-setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 1029 269 1085 314 R^2 precision(r-square)2 17 1 11 MONITOR 928 269 1029 314 Equation of Line lin-reg-eq 17 1 11 @#$#@#$#@ ## WHAT IS IT? This model investigates the box-counting dimension for a variety of fractals. ## HOW TO USE IT The interface consists of three blocks: Fractal Examples, Box-Counting Controls and Plots/Results. Start by selecting one of the fractal examples, e.g., "Koch Curve". Then click "Iterate". Notice at the first iteration the Hausdorff dimension (1.262 for the Koch Curve) appears in a monitor on the right. Click "Iterate" a few more times to get a developed fractal. Now you are ready to apply box-counting to this fractal. Under Box-Counting Controls, set "initial box length" (i.e., length of side of each box) and "increment" (i.e., amount that box length increases per iteration of box counting). Press "Box Counting Setup" and then "Box Counting Go" and you should see small red boxes covering the fractal. After the fractal has been completely covered, the log of number of boxes will be plotted versus the log of 1/box-length on the plot on the right. These iterations will continue until you press "Box Counting Go" again to stop the process. At that point you can press "linear-regression" on the right side (under the plot) to fit a line to the plotted points. The slope of this line is the box-counting dimension measured by your run; it appears in the monitor on the right, and can be compared with the Hausdorff dimension. ## THINGS TO NOTICE The box-counting dimension becomes a more accurate approximation of Hausdorff dimension as the number of iterations increase. The accuracy also can depend on the initial settings for box-length and increment (you should experiment with all of these parameters). However, you have probably noticed the iterations taking longer and longer to generate for each successive iteration. This is because the number of line segments is increasing exponentially. The Koch curve, for instance, begins with four line segments but has over a million line segments after ten iterations. ## CREDITS AND REFERENCES This model is part of the Fractals series of the Complexity Explorer project. Main Author: John Driscoll Contributions from: Vicki Niu, Melanie Mitchell 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: "Box Counting Dimension" 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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