globals[ pos_vector;;position vector of the point being plotted prob_a ;;probabilities of the a/1 or b/2 or c/3 prob_b ;;translation matrices and vectors being prob_c ;;used which ;;which transformation was undergone (numbered 1 through 3) a_11 a_12 a_21 a_22 b_11 b_12 b_21 b_22 c_11 c_12 c_21 c_22 ] to setup ca reset-ticks ask patch 0 0[ sprout 1[ hide-turtle ] ] set pos_vector [0 0] repeat 1000 [ transform-mat translate-vector] end to go transform-mat translate-vector lay-seed tick end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;Examples;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to sierpinski-triangle setup set rot1_x 0. set rot1_y 0. set rot2_x 0. set rot2_y 0. set rot3_x 0. set rot3_y 0. set scale1_x 0.5 set scale1_y 0.5 set scale2_x 0.5 set scale2_y 0.5 set scale3_x 0.5 set scale3_y 0.5 set t1_x 0 set t1_y 0 set t2_x 30 set t2_y 0 set t3_x 15 set t3_y 27 end to crystal setup set rot1_x -90. set rot1_y -90. set rot2_x 90. set rot2_y 90. set rot3_x 0. set rot3_y 0. set scale1_x 0.5 set scale1_y 0.5 set scale2_x 0.5 set scale2_y 0.5 set scale3_x 0.5 set scale3_y 0.5 set t1_x 40 set t1_y 0 set t2_x 40 set t2_y 40 set t3_x 20 set t3_y 40 end to spiral setup set rot1_x 0. set rot1_y 42.13 set rot2_x -30. set rot2_y -30. set rot3_x 0. set rot3_y 0. set scale1_x 0.143 set scale1_y 0.556 set scale2_x 0.944 set scale2_y 0.944 set scale3_x 0. set scale3_y 0. set t1_x 30 set t1_y 17 set t2_x 30 set t2_y 1 set t3_x 0 set t3_y 0 end to snowflake setup set rot1_x 180. set rot1_y 180. set rot2_x 0. set rot2_y 180. set rot3_x 180. set rot3_y 0. set scale1_x 0.5 set scale1_y 0.5 set scale2_x 0.5 set scale2_y 0.5 set scale3_x 0.5 set scale3_y 0.5 set t1_x 100 set t1_y 100 set t2_x 0 set t2_y 100 set t3_x 100 set t3_y 0 end to starfish setup set rot1_x 0. set rot1_y 0. set rot2_x -40. set rot2_y -40. set rot3_x 0. set rot3_y 0. set scale1_x 0.25 set scale1_y 0.25 set scale2_x 0.95 set scale2_y 0.95 set scale3_x 0. set scale3_y 0. set t1_x 50 set t1_y 50 set t2_x 30 set t2_y 0 set t3_x 0 set t3_y 0 end to dragon setup set rot1_x -45. set rot1_y -45. set rot2_x -45. set rot2_y -45. set rot3_x 0. set rot3_y 0. set scale1_x 0.707 set scale1_y 0.707 set scale2_x 0.707 set scale2_y 0.707 set scale3_x 0. set scale3_y 0. set t1_x 50 set t1_y 0 set t2_x 0 set t2_y 20 set t3_x 0 set t3_y 0 end ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;Linear Transformations;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;Modifies the elements of the transformation matrix using the information from the rotation and scale coordinates to make-mat set a_11 precision (scale1_x * cos(rot1_x))3 set a_12 precision(scale1_y * sin(rot1_y))3 set a_21 precision(-1 * scale1_x * sin(rot1_x))3 set a_22 precision(scale1_y * cos(rot1_y))3 set b_11 precision(scale2_x * cos(rot2_x))3 set b_12 precision(scale2_y * sin(rot2_y))3 set b_21 precision(-1 * scale2_x * sin(rot2_x))3 set b_22 precision(scale2_y * cos(rot2_y))3 set c_11 precision(scale3_x * cos(rot3_x))3 set c_12 precision(scale3_y * sin(rot3_y))3 set c_21 precision(-1 * scale3_x * sin(rot3_x))3 set c_22 precision(scale3_y * cos(rot3_y))3 end ;;Selects a transformation to perform based on the probabilities calculated ;;in setup. Then, executes the corresponding matrix multiplication on the ;;position vector, modifying the values to the transformed values. to transform-mat make-mat calculate-probability let ran_num random-float 1 let x-cor 23 let y-cor 52 ifelse ran_num < prob_a[ set x-cor (a_11 * (item 0 pos_vector) + a_12 * (item 1 pos_vector)) set y-cor (a_21 * (item 0 pos_vector) + a_22 * (item 1 pos_vector)) set which 1 ] [ ifelse ran_num < (prob_a + prob_b)[ set x-cor (b_11 * (item 0 pos_vector) + b_12 * (item 1 pos_vector)) set y-cor (b_21 * (item 0 pos_vector) + b_22 * (item 1 pos_vector)) set which 2 ] [ set x-cor (c_11 * (item 0 pos_vector) + c_12 * (item 1 pos_vector)) set y-cor (c_21 * (item 0 pos_vector) + c_22 * (item 1 pos_vector)) set which 3 ] ] set pos_vector (list x-cor y-cor) end ;;Performing the translation based on the inputs from the ;;interface, corresponding to the matrix transformation ;;previously performed (indicated by which) to translate-vector let x-cor 0 let y-cor 0 if which = 1[ ;;translation vector 1 set x-cor (item 0 pos_vector) + t1_x set y-cor (item 1 pos_vector) + t1_y ] if which = 2[ ;;translation vector 2 set x-cor (item 0 pos_vector) + t2_x set y-cor (item 1 pos_vector) + t2_y ] if which = 3 [ set x-cor (item 0 pos_vector) + t3_x ;;translation vector 3 set y-cor (item 1 pos_vector) + t3_y ] set pos_vector (list x-cor y-cor) end ;;plotting a point at the coordinates indicated by ;;pos_vector to lay-seed ask turtles [ set size 0.5 ] ask patch (item 0 pos_vector) (item 1 pos_vector)[ sprout 1[ if which = 1[ set color green ] if which = 2[ set color blue ] if which = 3 [ set color red ] set size 5 ] ] end ;;Calculating the probabilities that each of the ;;transformations is chosen based on the proportion ;;of the matrix's determinant versus the sum to calculate-probability let det_a abs (a_11 * a_22) - (a_21 * a_12) let det_b abs (b_11 * b_22) - (b_21 * b_12) let det_c abs (c_11 * c_22) - (c_21 * c_12) let det_sum det_a + det_b + det_c set prob_a det_a / det_sum set prob_b det_b / det_sum set prob_c det_c / det_sum end @#$#@#$#@ GRAPHICS-WINDOW 240 19 718 413 -1 -1 3.0 1 10 1 1 1 0 1 1 1 -55 100 -20 100 0 0 1 ticks 30.0 TEXTBOX 1128 32 1278 50 Transformation Matrix 11 0.0 1 INPUTBOX 1048 50 1098 110 t1_x 50 1 0 Number INPUTBOX 1048 110 1098 170 t1_y 0 1 0 Number TEXTBOX 1048 32 1198 50 Translation 11 0.0 1 INPUTBOX 1048 174 1098 234 t2_x 0 1 0 Number INPUTBOX 1048 234 1098 294 t2_y 20 1 0 Number INPUTBOX 1048 300 1098 360 t3_x 0 1 0 Number INPUTBOX 1048 360 1098 420 t3_y 0 1 0 Number BUTTON 57 369 174 402 Go forever go T 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 56 74 185 107 Sierpinski Triangle sierpinski-triangle NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 56 105 185 138 Crystal crystal NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 56 136 185 169 Spiral spiral NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 56 167 185 200 Square Snowflake snowflake NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 56 199 185 232 Starfish starfish NIL 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 56 46 206 68 Fractal Examples 16 0.0 1 TEXTBOX 60 279 210 301 Procedures 16 0.0 1 TEXTBOX 729 27 879 49 Create your own 18 0.0 1 TEXTBOX 727 66 895 178 To make your own IFS fractal, enter the values for the rotation and scale, along with the translation vectors. then click \"Setup New IFS\" below, and click one of the \"Go\" buttons on the lower right. 11 0.0 1 BUTTON 56 231 185 264 Dragon Curve dragon NIL 1 T OBSERVER NIL NIL NIL NIL 1 INPUTBOX 908 50 958 110 rot1_x -45 1 0 Number INPUTBOX 908 110 958 170 rot1_y -45 1 0 Number INPUTBOX 975 50 1034 110 scale1_x 0.707 1 0 Number INPUTBOX 975 110 1034 170 scale1_y 0.707 1 0 Number INPUTBOX 907 174 957 234 rot2_x -45 1 0 Number INPUTBOX 907 234 957 294 rot2_y -45 1 0 Number INPUTBOX 977 174 1034 234 scale2_x 0.707 1 0 Number INPUTBOX 977 234 1034 294 scale2_y 0.707 1 0 Number INPUTBOX 907 300 957 360 rot3_x 0 1 0 Number INPUTBOX 907 360 957 420 rot3_y 0 1 0 Number INPUTBOX 977 300 1034 360 scale3_x 0 1 0 Number INPUTBOX 977 360 1034 420 scale3_y 0 1 0 Number TEXTBOX 910 32 956 50 Rotation 11 0.0 1 TEXTBOX 978 32 1017 50 Scale 11 0.0 1 MONITOR 1130 50 1187 95 NIL a_11 3 1 11 MONITOR 1187 50 1244 95 NIL a_12 3 1 11 MONITOR 1130 93 1187 138 NIL a_21 3 1 11 MONITOR 1187 93 1244 138 NIL a_22 3 1 11 MONITOR 1131 172 1188 217 NIL b_11 3 1 11 MONITOR 1188 172 1245 217 NIL b_12 3 1 11 MONITOR 1131 216 1188 261 NIL b_21 3 1 11 MONITOR 1188 216 1245 261 NIL b_22 3 1 11 MONITOR 1133 298 1190 343 NIL c_11 3 1 11 MONITOR 1188 298 1245 343 NIL c_12 3 1 11 MONITOR 1133 343 1190 388 NIL c_21 3 1 11 MONITOR 1188 343 1245 388 NIL c_22 3 1 11 BUTTON 57 337 173 370 Go 100 iterations repeat 100 [go] NIL 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 11 10 247 39 Iterated Function Systems 18 95.0 1 TEXTBOX 729 273 888 390 Points generated by the first transformation are colored green, those from the second transformation are colored blue, and points from the third transformation are red. 11 0.0 1 BUTTON 57 306 173 339 Go 1 iteration repeat 1 [go] NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 726 184 842 217 Setup New IFS setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 TEXTBOX 728 251 902 283 Note on the color scheme: 13 0.0 1 @#$#@#$#@ ## WHAT IS IT? This model demonstrates how to use iterated function systems to represent fractals. An iterated function system, or IFS, consists of a set of affine transformations that are performed on points. There are four kinds of affine transformations possible in this model:
1. Translation 2. Scaling 3. Reflection 4. Rotation
A translation is performed on a point by adding the point to a translation vector. Then, a point can be scaled if it is multiplied by a scalar constant. Finally, a point can be reflected and rotated via matrix multiplication. In our representation, the scaling of the point is incorporated into the matrix multiplication, so that a single operation scales, reflects, and rotates the point. With each iteration, there are any number of different transformations that could be performed, each with an associated probability of occurring. This model supports a maximum of three different transformations. A fractal is generated after a few thousand iterations, which are successive transformations of the original point. ## HOW IT WORKS The model starts with a turtle placed on a patch in the world. The matrices (a, b, c) and translation vectors (t1, t2, t3) located on the right-hand side have inputs where you can type in the component values. At a given iteration, the probability of each transformation occurring is equal to the proportion of that matrix's determinant over the sum of all determinants (see below for more details). A transformation is chosen according to this probability and is performed on the point, producing a new point to be transformed at the next iteration. After a few thousand iterations, a coherent image will emerge in the worldview. ## HOW TO USE IT Start out by selecting some of the examples. Press the button corresponding to the fractal you wish to view, and then select "Go". After you have investigated these, you can try creating your own fractal. Simply enter the desired values for the scaling, rotation, and translation into the input boxes on the right-hand side. If you only want to deal with two different transformations, enter only 0s into one matrix and its corresponding vector. ## THINGS TO NOTICE While working through the examples, try to notice patterns with how the transformation-matrix values relate to the patterns visible on the image. How does the image change when you modify the rotations? Multiply all of them by -1? Switch the x and y? Do the same with the translation vectors. How does the image change when you multiply all of the translation values by a number? ## THINGS TO TRY After playing around with the preset IFS fractals, try designing your own. You could make a new image just by changing the values preset for the Sierpinski triangle, or the spiral. With IFS, the fractal possibilities are much greater than with the L-systems model, which was simpler and more constrained. ## NOTE ON PROBABILITIES At each iteration, the probability that a given transformation will be chosen is equal to the determinant of that transformation's matrix divided by the sum of all transformations' matrices.
Given a matrix:
a b c d
the determinant is equal to a*d - b*c. To calculate the probability, we compute the determinants of all the matrices, then add their absolute values. The probability of one transformation is its determinant over the sum of all matrices' determinants. To explore this further, create monitors that display each transformation's probability of being chosen (hint: they're all global variables). Then, calculate the determinant of each matrix to verify that the probability calculation is correct. ## CREDITS AND REFERENCES This model is part of the Fractals series of the Complexity Explorer project. It was inspired in part by the IFS Construction Kit, developed by Larry Riddle (http://ecademy.agnesscott.edu/~lriddle/ifskit/index.htm). Main Author: Vicki Niu Contributions from: 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: Iterated Function Systems 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/). 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