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
@#$#@#$#@
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Create your own
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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.
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Dragon Curve
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Go 100 iterations
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Iterated Function Systems
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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.
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Go 1 iteration
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Setup New IFS
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OBSERVER
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Note on the color scheme:
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@#$#@#$#@
## 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/). 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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