In a Cellular Automaton, the world is divided in squares. The time as well is progressing in discrete steps. At each move,
the state of a field changes in dependency of the state of the neighbouring fields.
The computer, making it possible to carry out a gigantic number of operations within a short time, has opened a new field of mathematics. It has become possible now to inverse the usual "top down" approach by creating on the micro-level a number of elements with individual "motivations", to let them interact a great number of times and to observe the effect on the macro-level. This is called a "bottom-up" analysis.
Stephen Wolfram described Cellular Automata in his ingenious book "A New Kind of Science".
A basic representant of such a two-dimensional automaton is shown here.
The starting point is a row of squares couloured in black or in white (in the present example, I have preferred blue and yellow, because it looks nicer). Furthermore the edges are linked, so forming a ring. You may draw such a row on a piece of squared paper for making it clearer.
Now, a new row is to be generated below the first one. The colouring of each of the new squares is determined by a set of eight rules depicted at the bottom of the pushbutton as eight T-shaped entities - the crossbar of the T's representing the old row and the lone square below the new one.
The first rule top left says "if the square above me is blue, and the sqares left and right from it, too, I will become yellow." By clicking with the mouse on the lone yellow square, it will turn blue. This makes the relevant rule turn into "if..., I will become blue". In this way, all rules possible with a neighbourhood of three fields can be set.
An additional feature is the "noise" which can be introduced with the sliders. A noise of 1.0 % means that one of a hundred decisions will be carried out in the wrong sens and, instead of yellow, the square turns blue, contrary to rules. This makes the model cease to be deterministic.
The Game of Life, published in 1970 by J. H. Conway, has evolved to a tremendous field of research between then and now.
Just start a Google search at this subject, it
is amazing (and largely free of any practical use).
The rules are simple, you can also play it with a pencil and a rubber on squared paper : A certain number of cells live on a pattern. At each move, all those which have too few (0 or 1) or too many (more than 3) neighbours will die. On empty squares with exactly 3 neighbours, new cells are born.
Two things can be noticed:
First, the game is deterministic; hence, for a given initial position, the evolution will always be the same.
Second, if you start from a random initial distribution, "life" will come to an end sooner or later, only static or oscillating groupings will be left on the gameboard.
This applet offers the possibility to vary the rules a little : Some conditions can be applied with a probability of less then 100%. One interesting modification is for instance to create the additional rule that a new cell shall be born on an empty square with only two neighbours, and this with a probability of 1 %. Such a slight modification provokes a large change of the game : It is no longer deterministic. Static or oscillating groupings are no longer stable. It is also remarkable that the whole board remains in a "living" state longer than with the original rules.
Furthermore, this applet allows to introduce "immortal" (red) cells. These are always counted, like normal cells, but neither they move nor disappear. Just see what you can do with it.
In 1952, Alan Turing described this form of pattern formation in chemical
reaction-diffusion systems ("The Chemical Basis of Morphogenesis").
It also plays a role in other domains, such as for instance at the formation of coat patterns of felines
in the course of their embryonic development.
The principle is that the neighbouring fields of each square exert an influence as a function of their distance either in the direction of their own colour or in the opposite sense.
As the calculations of Mister Turing exceed by far my IQ, I have chosen instead a super-simple approach : The number of the blue neighbouring fields in the close range of each square - divided by their distance (Pythagoras) - are added (value A). The same is done for the far range (value B), defined as 2.5 to 5 square lengths. Value B is now multiplied by a factor - the one which can be set with the slider. If value A is now greater than factor x value B, the field turns blue, in the opposite case it turns yellow.
Starting from randomly distributed blue fields, a stable stationary pattern will be formed quickly after the start. This can be considered as a standing wave or an interference pattern.
Here, a lot of blue fields are varying
their brightness in function of their
neighbours' behaviour. First, the blue fields fade away, until they are
white. The scale contains 50 steps, which are passed within two
seconds. Then, the colouration runs into the opposite sense.
As the fields are defined randomly, only a flickering can be observed at the beginning. In order to create structures, interaction between the cells is needed. In this applet, you can check three variations:
In the first one, the brightness of each field is adapted to the mean value of the brightness of its neighbours. This happens at the two turning points, when a cell is white or dark blue.
In the second variation, a cell arriving at the white extreme point makes all eight neighbours fading a little away by one step (of 50). The cell itself will become eight steps darker. This is necessary to avoid the neighbour cells paralysing each other. The opposite thing happens at the blue turning point.
At the third modification, the direction of (dis)coloration is turned into the opposite sense, if the majority of the cell's neighbours do the same. When for instance a cell is fading and 5 of its neighbours are moving from white to blue, this cell will switch its direction and become darker.
Here you can paint ornaments.
At the beginning, all the fields are gray. If you click the red button "Color Seed", the central field is dyed in a random color. Since each field is continuously calculating its own colour in function of the colours of its eight - direct and diagonal - neighbouring fields, the gray boxes start to take on a colour as well. New colour seedig will affect the evolution of the ornament.
The basis of the color calculation is the RGB system, in which the components Red, Green and Blue are represented by numbers from 0 to 255.
At each cycle of 40 milliseconds, the difference of the R, G and B components of each field to its eight neighboring fields is formed. When the total difference exceeds a threshold value, each R, G and B value is modified in the direction of this average (by +1 or by -1).
Watching the mandala with patience will lead you to enlightenment.