Introduction

The following describes iterated function systems, a particular implementation on the Macintosh, and presents some examples.

One of the many ways of describing the affine transformations for iterated functions systems (IFS) is as follows:

Generally there are a number of these mappings (different values of r, s, theta, phi, h, k) which are chosen at random at each iteration.

How does the program work

The fundamental iterative process involves replacing rectangles with a series of rectangles called the generator. The rectangles are replaced by a suitably scaled, translated, and rotated version of the generator.

For example consider the generator below

It consists of three rectangles, each with its own center, dimensions and rotation angle. The initial conditions usually consist of a single square, the first iteration then consists of replacing this square by a suitably positioned, scaled and rotated version of the above generator. The result would just be the generator above. The next iteration involves replacing each of the rectangles in the current system by suitabled positioned, scaled, and rotated versions of the generator resulting in the following

The next iteration replaces each rectangle above again by the initial generator as shown below

and so on, three iterations later

IFS allows the user to explore the result of applying different generators. The generator rectangles can be created, shifted, scaled, and rotated by the handles on each rectangle shown below.

The user may iterate forward or in reverse, colour the borders, interior, or background.

A library of interesting generators is provided for initial experimentation. Generators created by users may be saved in files and recalled at a later time.

Images so created may be copied from IFS and pasted into a more standard drawing package for incorporation into a document and printing.

Hopalong or "The Chaos Game"

A technique exists by which the resulting form after an infinite number of iterations can be derived. This is a function of the form

This gives a series of (x,y) points all which lie on the result of an infinite IFS. Although it still takes an infinite number of terms in this series to form the result the appearance can be readily appreciated after a modest number of terms (10000 say).

Note that with both methods it is possible to create the image at any scale. In many but not all cases zoomed in examples will be exhibit self similarity at all scales.

Applications

Data reduction for model files. If a generator can be found for a complex image then storing the generator and the rules of production results in a great deal of data reduction. For example the weed in the examples above might eventually contain over 2000 rectangles but is completely specified by the characteristics of 3 rectangles, only 5 numbers, center (cx,cy), scale (sx,sy), and angle

Note: it is not necessarily trivial to derive a rectangular generator for an arbitrary form, although it is possible to create a polygonal generator for any form. The following shows a maple leaf generated using the IFS program.

Some other IFS examples of plant-like structures are shown below

References

Demko, S., Hodges, L., and Naylor B. Construction of Fractal Objects with Iterated Function Systems. Computer Graphics 19, 3, July 1985, Pages 271-278

Barnsley, M. Fractals Everywhere. Academic press, 1988

Barnsley, F., and Sloan, A.D. A Better Way to Compress Images. Byte Magazine, January 1988, Pages 215-222

Oppenheimer, P.E. Real Time Design and Animation of Fractal Plants and Trees. Computer Graphics, 20, 4, 1986

Reghbati, H.K. An Overview of Data Compression Techniques. Computer, 14, 4, 1981, Pages 71-76

The following images are a selection from an infinite set of Random Iterated Function Systems (IFS) of the form

For the mapping to contractive the following must all be true.

For any particular image there are 4 different sets of values for (a,b,c,d,e,f) all on the interval [-1,1]. At each iteration step one of the sets is chosen at random to form the next term in the sequence (series).

The results fall into the following broad classes of images

Infinite or constant, these can't be drawn and are specifically tested for and ignored by the software. (These are not affine mappings)

Single blobs Often with outgoing rays.

Interesting, attractive, and often organic forms.

The third "interesting" category can be further classified as follows

Things made up of intersecting spikes They often look like Kanji text.

Things that look like blobs of seaweed on the beach.

This bush is created by iteratively choosing one of the following functions at random to form a series p_n. Each point in the series is represented as a small sphere, the final collection is rendered using PovRay. In order minimise the number of spheres to be rendered (IFS tend to require a very large number of points) the spheres are filtered such that no two spheres lie within 10% of their respective radii.

• scale by (0.1,0.2,0.1)

• translate by (0,-1,0)
  scale by (0.9,0.8,0.9)
  rotate about the y axis by 144 degrees
  translate by (0,1,0)

• rotate about the z axis by 60 degrees
  scale by 0.3
  translate by (0,0.2,0)

The IFS equations are as follows

                   set 1     set 2     set 3     set 4
             a     0.1400    0.4300    0.4500    0.4900
             b     0.0100    0.5200   -0.4900    0.0000
             c     0.0000   -0.4500    0.4700    0.0000
             d     0.5100    0.5000    0.4700    0.5100
             e    -0.0800    1.4900   -1.6200    0.0200
             f    -1.3100   -0.7500   -0.7400    1.6200

Features in the Diaorama Magazine, issue 4, May 2013.

The IFS equations are as follows

                   set 1       set 2        set 3
             a     0.787879   -0.121212     0.181818
             b    -0.424242    0.257576    -0.136364
             c     0.242424    0.151515     0.090909
             d     0.859848    0.053030     0.181818
             e     1.758647   -6.721654     6.086107
             f     1.408065    1.377236     1.568035
   probability     0.90        0.05         0.05

The IFS equations are as follows

                   set 1     set 2
             a     0.2020    0.1380
             b    -0.8050    0.6650
             c    -0.6890   -0.5020
             d    -0.3420   -0.2220
             e    -0.3730    0.6600
             f    -0.6530   -0.2770

The IFS equations are as follows

                   set 1     set 2     set 3     set 4     set 5     set 6     set 7
             a     0.0500   -0.0500    0.0300   -0.0300    0.5600    0.1900   -0.3300
             b     0.0000    0.0000   -0.1400    0.1400    0.4400    0.0700   -0.3400
             c     0.0000    0.0000    0.0000    0.0000   -0.3700   -0.1000   -0.3300
             d     0.4000   -0.4000    0.2600   -0.2600    0.5100    0.1500    0.3400
             e    -0.0600   -0.0600   -0.1600   -0.1600    0.3000   -0.2000   -0.5400
             f    -0.4700   -0.4700   -0.0100   -0.0100    0.1500    0.2800    0.3900

Derby, Western Australia

The IFS equations are as follows

                   set 1     set 2     set 3     set 4     set 5     set 6
             r     0.0500    0.0500    0.6000    0.5000    0.5000    0.5500
             s     0.6000   -0.5000    0.5000    0.4500    0.5500    0.4000
         theta     0.0000    0.0000    0.6980    0.3490   -0.5240   -0.6980
           phi     0.0000    0.0000    0.6980    0.3492   -0.5240   -0.6980
             e     0.0000    0.0000    0.0000    0.0000    0.0000    0.0000
             f     0.0000    1.0000    0.6000    1.1000    1.0000    0.7000

The IFS equations are as follows

                   set 1     set 2     set 3     set 4
             a     0.0100   -0.0100    0.4200    0.4200
             b     0.0000    0.0000   -0.4200    0.4200
             c     0.0000    0.0000    0.4200   -0.4200
             d     0.4500   -0.4500    0.4200    0.4200
             e     0.0000    0.0000    0.0000    0.0000
             f     0.0000    0.4000    0.4000    0.4000

The IFS equations are as follows

                   set 1     set 2     set 3     set 4     set 5
             a     0.1950    0.4620   -0.6370   -0.0350   -0.0580
             b    -0.4880    0.4140    0.0000    0.0700   -0.0700
             c     0.3440   -0.2520    0.0000   -0.4690    0.4530
             d     0.4430    0.3610    0.5010    0.0220   -0.1110
             e     0.4431    0.2511    0.8562    0.4884    0.5976
             f     0.2452    0.5692    0.2512    0.5069    0.0969
   probability     0.2       0.2       0.2       0.2       0.2

The IFS equations are as follows

                   set 1       set 2      set 3      set 4
             a     0.0000      0.7248     0.1583     0.3386
             b     0.2439      0.0337    -0.1297     0.3694
             c     0.0000     -0.0253     0.3550     0.2227
             d     0.3053      0.7426     0.3676    -0.0756
             e     0.0000      0.2060     0.1383     0.0679
             f     0.0000      0.2538     0.1750     0.0826

The IFS equations are as follows

                   set 1       set 2      set 3      set 4      set 5      set 6
             a     0.38200     0.11800    0.11800   -0.30900   -0.30900    0.38200
             b     0.00000    -0.36330    0.36330   -0.22450    0.22450    0.00000
             c     0.00000     0.36330   -0.36330    0.22450   -0.22450    0.00000
             d     0.38200     0.11800    0.11800   -0.30900   -0.30900   -0.38200
             e     0.30900     0.36330    0.51870    0.60700    0.70160    0.30900
             f     0.57000     0.33060    0.69400    0.30900    0.53350    0.67700

One of the very common and attractive forms generated by Iterated Function Systems (IFS) is the fern leaf shown on the right. The following will describe how to generate this form and allow the reader to experiment with other IFS generators. Iterated function systems are described by repeatedly computing terms in two series, one series describes the x coordinate and the other series the y coordinate. The equations describe translation, scaling, rotation, and shearing of points in a plane with the restriction that the transformations are "affine". The general form of the series are as follows xn+1 = a xn + b yn + e yn+1 = c xn + d yn + f A point is drawn at each pair (xi,yi) for i greater than some number, typically 10 to 100. The magic is in finding the values of (a,b,c,d,e,f) that give the desired form. In many application it is necessary to have a number of sets of (a,b,c,d,e,f). As the series is being generated a particular set is chosen at random for each term. Such IFS systems are often known as Random Iterated Function Systems.

The fern can be constructed using the table of value on the right. It turns out that if the different sets of (a,b,c,d,e,f) are chosen with appropriate probabilities then the fern will "emerge" much faster and evenly than if the sets are chosen with equal chance. The last row in the table gives the optimal probabilities. The fern above resulted from 100 thousand (105) iterations, that is, 100 thousand points are drawn (of course when drawn on the bitmap many will overlap)

- set1 set2 set3 set4 a 0.0 0.2 -0.15 0.75 b 0.0 -0.26 0.28 0.04 c 0.0 0.23 0.26 -0.04 d 0.16 0.22 0.24 0.85 e 0.0 0.0 0.0 0.0 f 0.0 1.6 0.44 1.6 p 0.1 0.08 0.08 0.74

The image is self similar at all scales, one can zoom in as far as one wishes and the fronds will continue to resolve themselves. For example, the following image is a zoom in by 50. Note however that it takes an ever increasing number of iterations to resolve the image as the zoom factor increases, this image took 100 million (10⁸) iterations.

Some straightforward C source is given here (source.c) which generates the figure shown above. Note, you will have to supply your own image drawing tools.

A slightly different set of codes gives the result on the right. set 1 set 2 set 3 set 4 a 0.0 0.2 -0.15 0.85 b 0.0 -0.26 0.28 0.04 c 0.0 0.23 0.26 -0.04 d 0.16 0.22 0.24 0.85 e 0.0 0.0 0.0 0.0 f 0.0 1.6 0.44 1.6 probability 0.01 0.07 0.07 0.85

References

Demko, S., Hodges, L., and Naylor B. Construction of Fractal Objects with Iterated Function Systems. Computer Graphics 19, 3, July 1985, Pages 271-278

Barnsley, M. Fractals Everywhere. Academic press, 1988

Barnsley, F., and Sloan, A.D. A Better Way to Compress Images. Byte Magazine, January 1988, Pages 215-222

>Oppenheimer, P.E. Real Time Design and Animation of Fractal Plants and Trees. Computer Graphics, 20, 4, 1986

Reghbati, H.K. An Overview of Data Compression Techniques. Computer, 14, 4, 1981, Pages 71-76

The IFS equations are as follows

                   set 1     set 2     set 3
             a     0.0100    0.7000    0.0000
             b    -0.4100    0.3300    0.1750
             c     0.3900   -0.3500    0.0130
             d     0.0000    0.7000    0.4600
             e    -0.2800    0.1850   -0.0950
             f    -0.1850    0.0150   -0.2850

The following is a rather unusual example of an iterated function system (IFS) that involves choosing at random one of 19 transformations of the form

A zoom into the top portion of the "A", shown in blue above, shows that it is indeed made up of smaller versions of the word "chaos".

And to labour the point, the following is a further zoom into the "A", shown in blue above.

The parameter table:

    a           b           c            d            e         f
    0           0.053      -0.429        0           -7.083     5.43
    0.143       0           0           -0.053       -5.619     8.513
    0.143       0           0            0.083       -5.619     2.057
    0           0.053       0.429        0           -3.952     5.43
    0.119       0           0            0.053       -2.555     4.536
   -0.0123806  -0.0649723   0.423819     0.00189797  -1.226     5.235
    0.0852291   0.0506328   0.420449     0.0156626   -0.421     4.569
    0.104432    0.00529117  0.0570516    0.0527352    0.976     8.113
   -0.00814186 -0.0417935   0.423922     0.00415972   1.934     5.37
    0.093       0           0            0.053        0.861     4.536
    0           0.053      -0.429        0            2.447     5.43
    0.119       0           0           -0.053        3.363     8.513
    0.119       0           0            0.053        3.363     1.487
    0           0.053       0.429        0            3.972     4.569
    0.123998   -0.00183957  0.000691208  0.0629731    6.275     7.716
    0           0.053       0.167        0            5.215     6.483
    0.071       0           0            0.053        6.279     5.298
    0          -0.053      -0.238        0            6.805     3.714
   -0.121       0           0            0.053        5.941     1.487

The IFS equations are as follows

                   set 1       set 2
             a     0.824074    0.088272
             b     0.281428    0.520988
             c    -0.212346   -0.463889
             d     0.864198   -0.377778
             e    -1.882290    0.785360
             f    -0.110607    8.095795
   probability     0.8         0.2

The IFS equations are as follows

                   set 1       set 2      set 3
             a     0.387       0.441     -0.468
             b     0.430      -0.091      0.020
             c     0.430      -0.009     -0.113
             d    -0.387      -0.322      0.015
             e     0.2560      0.4219     0.4
             f     0.5220      0.5059     0.4
   probability     1/3         1/3        1/3

The IFS equations are as follows

                   set 1       set 2      set 3
             a     0.0         0.0        0.5
             b    -0.5         0.5        0.0
             c     0.5        -0.5        0.0
             d     0.0         0.0        0.5
             e     0.5         0.5        0.25
             f     0.0         0.5        0.5
   probability     1/3         1/3        1/3

This beast is created by iteratively choosing one of the following functions at random to form a series p_n. Each point in the series is represented as a small sphere, the final collection is rendered using PovRay. In order minimise the number of spheres to be rendered (IFS tend to require a very large number of points) the spheres are filtered such that no two spheres lie within 10% of their respective radii.

• scale by 1/3
  
• translate by (a,a,a)
  scale by b
  
• translate by (a,a,-a)
  scale by b
  
• translate by (a,-a,a)
  scale by b
  
• translate by (a,-a,-a)
  scale by b
  
• translate by (-a,a,a)
  scale by b
  
• translate by (-a,a,-a)
  scale by b
  
• translate by (-a,-a,a)
  scale by b
  
• translate by (-a,-a,-a)
  scale by b
  

where a = (1 + 1/3) / 2 and b = (1 - 1/3) / 2

This beast is created by iteratively choosing one of the following functions at random to form a series p_n. Each point in the series is represented as a small sphere, the final collection is rendered using PovRay. In order minimise the number of spheres to be rendered (IFS tend to require a very large number of points) the spheres are filtered such that no two spheres lie within 10% of their respective radii.

• scale by 1/3
  
• translate by (a,0,0)
  scale by b
  
• translate by (0,a,0)
  scale by b
  
• translate by (0,0,a)
  scale by b
  
• translate by (-a,0,0)
  scale by b
  
• translate by (0,-a,0)
  scale by b
  
• translate by (0,0,-a)
  scale by b
  

where a = (1 + 1/3) / 2 and b = (1 - 1/3) / 2