About IFS

How IFS-drawings were made?

IFS means Iterated Function System. This is a description of an iterated function system mainly for novices in math.

The iteration

The first thing to understand is iteration. The simplest example is the serie of numbers, where the next number is always bigger by one as the prevoius: 1, 2, 3, 4 ... The next number in the serie produced from the previous by adding one, and the this procedure is repeated to the infinit. The mathematicians describes this serie like this:

an+1 = an+1

This is called the generator-function of the iteration. The generator-function is a procedure, producing a number from an another number. If a function applied to its result again and again: this is iteration.

The function-system

The iterated function system differs from a simple iteration: it uses a function system instead of a single function. The function systems are not really interesting in this case (otherwise yes), just one is shown: which produces the drawings.

In 2D a point can be rotated, shifted and scaled. These procedures called transformations, and mathematicians uses the functions of the three transformation in a function system, just for simplicity (?). This function system called the transformation matrix. In fact a transformation matrix is a cupple of numbers (in our case 6). The matrices has many benefits, in our case just one: they can be used as generator functions, since they produces from a point an other point, and the "output" point can be an "input\ point again.

As simple example: a point shifted to east again and again produces a horizontal line, if the distance is small enough between the points. If they are also rotated, the result a spiral.

But this is not all, because one transformation matrix can produce only lines or spirals, and (although very interesting phenomenon) not enough. I use more than one transformation matrices. (In fact between 2 and 32). Let say, we have four of them. Between each iteration I randomly choose a transformation matrix (generator function). The selected matrix transforms the actual point to the next one.

This is pretty complex, but there is more. The chance in the random selection is naot the same for all matrices. Some of them has bigger chance, than others. So, that's all.

The drawings

How the drawings were made? The drawings are series of points. The first of them is always at the middle of the "sheet", and the mass of the points makes the drawings. There are tones, because a point can be 255 shades of gray, not only white or black. When the pen first touches a point, it becomes the first shade, second time, the secons shade, and so on.

The procedure is a ressource-eater one, some (not really high-tech) computers rendered them till six months mostly night and day.


The procedure also works in three dimension. The transformation matrices are just longer a bit. The points are in space, so the representation of the set is difficult (but nice) excersise. Here is to download a 3D IFS set generator and viewer software.

Online IFS generator

IFS generator
start/pause new image


Ez a szöveg a Penrose-rajzok geometriájának ismertetése. A rajzok alapja a Penrose-fedés, vagy ahogy Perneczky Géza fordította: a Penrose-parketta.