Why are scientists interested in cellular automata?


It would be very impractical to build a computer this way, but if we had enough time and the life pattern were large enough, we could run every program that runs on your PC on the life PC.

Such life computers have already been constructed for special purposes, e.g. for calculating prime numbers.

I'm talking about a computer whose logical gates consist of the interaction of gliders and spaceships, i.e. of patterns in the life universe.


One could even think of a pattern that could produce oneself and other complex living beings in the life universe. Nobody has done it yet because this pattern would be very, very large. But it has been shown that it is basically possible.

It means that there are life patterns that can reproduce themselves. Doesn't that remind you of computer viruses? These life patterns can combine their blueprints and mutate their genes.

Since we can represent logical circuits with gliders and spaceships and their interaction, any artificial intelligence can theoretically be generated with a sufficiently large output pattern. One could also breathe intelligence into one's living beings.

Don't you get dizzy? Are you sure you are not a pattern? I always say math is real life.


The study of life patterns has and will lead to many discoveries in all areas of mathematics and other sciences.

You can better understand the behavior of cells and animals if you can trace them back to simple rules.

Often we suspect intelligent behavior where maybe a few simple rules apply. Think of an ant colony. Nobody has yet discovered the rules by which such a population lives. But I can offer you virtual termites that pile up pieces of wood according to 2 simple rules:

Click here

Unfortunately the site is in English. Note: if you can speak English, you get more from the Internet.

There seems to be intelligent behavior there.

Please close the old applet window first.

What does this tell us about the nature of intelligence?

The mathematical tools that studying life patterns have given us could also help us analyze traffic problems.

Computer viruses are also examples of cellular automata. Studying this simple game could mean that we might find a cure for it one time.

Human diseases could also be cured if we better understand how cells live and die.

We could explore the galaxies if we could build machines that could reproduce themselves.

All of this is theoretically possible, but it has not yet been invented. Don't you want to participate?

Math magic garden with fun


What is the Game of Life? It's not a game in the traditional sense. You can't win here, and you can't lose either. Once you have determined the initial situation of "life", the rules determine what happens next. It is a self-organizing system. Nonetheless, "life" is full of surprises. In most cases it is impossible to predict from your initial conditions with the help of the rules how "life" will develop in the future.

Rules of the "Game of Life"

"Life" is played on a grid that forms square cells - like a chessboard. You only have to imagine this grid, you can also understand it as a pixel network, extended to infinity on all sides. You know that from the drawing level at school. So that I don't organize a dry swimming course here, you can first play the "Game of Life" with a mini game board.


If you now click cells on the left in the applet, they will turn red and you will bring them to life, so to speak. Your cells need neighborhood to survive, but too much neighborhood means overpopulation and they die. Too little neighborhood means they die of loneliness from a lack of partners.

How your cells can produce offspring, I'll leave you below. You should simply bring a cluster of cells to life with a click, set the slider to "perm" (slide all the way to the right), and see how your cell population develops.

I think I forgot to tell you to click start.


Each new generation is determined according to a rule of survival (I), a rule of death (II) and a rule of birth (III).

A cell a is either occupied by a living being or not. What condition it will have in the next generation depends on the occupation of the eight neighboring cells.

1st case: Cell a is busy.

I. The living being in this cell survives if it has 2 or 3 neighbors.
II. The living being dies when it has 0, 1, 4, 5, 6, 7 or 8 neighbors.

With none or one of the neighbors it dies from loneliness, with 4 to 8 from overpopulation.

2nd case: Cell a is not occupied.

III. If there are exactly 3 living beings in the neighboring cells for this cell, a new living being is created here. In all other constellations it remains empty.

You should just try out different patterns and see how your "creation" behaves in "Life". But I also want to draw your attention to a few initial situations that cause certain phenomena in "Life". If you google it a little on the web, you will find a lot of people, I want to call them "pattern hunters" or "pattern hunters", who are looking for initial situations that cause certain effects.

If you click on the button, you will find a life applet that offers a much larger "living space" than the applet above.

A pattern with 5 cells has already been entered here. It's called R-Pentomino. Click on Go and see what happens. It was the first pattern that Conway found and that defied his attempts to simulate it by hand. In fact, the "pattern of life" seems to be becoming stable, or at least easily predictable. But that only happens after 1103 steps (generations). Some of the earliest computer programs for Life were written to explore the fate of this little pattern. Back in 1970, that was still a major challenge for many computers. Today a modern PC calculates even more complicated sequences in one second.

If you observe the fate of the R-Pentomino, you will see some characteristic populations that have also been named, e.g. gliders and blinkers. In the end, some stable populations and blinkers are left behind. Conway called the stable populations "still lifes". Here are some of the most famous still lifes, they only contain creatures with 2 or 3 neighbors and don't change anymore.

Still life


How did these populations get their name? Quite simply, whoever found her was allowed to name her. Here are a few other noticeable populations.

By the way, if you click on "Clear" in the applet, you can experiment with your own patterns. Or you go to "Open" in the menu and choose a pattern that others have thought up before you. Maybe you can do something like that too.


It belongs to the period 2 oscillators. The period indicates after how many generations the picture repeats itself.

Another noticeable population is the blinker. It consists of three living beings that lie alternately on top of each other or next to each other.
You can see the toad very briefly in the R-Pentomino. It appears to the far right after 737 paces. But after just 14 steps, it is destroyed by a nearby explosion.

Both belong to the period 2 oscillators. The period indicates after how many generations the picture repeats itself.


There are strange structures that move diagonally across the field. They bring an interesting moment in the succession of generations. The "gliders" consist of 5 living beings in each phase and have the "period" 4.

The Queen Bee Shuttle

Translated it is called Queen Bee's Shuttle, but I like the English term better. This very important pattern also appears in the R-Pentomino for a short time after 774 steps. But it doesn't last long in this stormy environment. Just enter it in the applet for yourself. First it moves to the right, creating a still life, namely a beehive. Then it moves to the left and creates a beehive there as well and moves to the right again. Because of this, Conway called it the Queen Bee Shuttle. Unfortunately, it now crashes into the previously created beehive. But that can be prevented.
Here you can see a beehive and behind it a block. Try this pattern out. What is happening? You will see the beehive go away and the block stays. If you now place a block on both sides of the queen, the beehive will be destroyed and the Queen Bee Shuttle will run back and forth forever. Where does the block have to go? Well, just find out for yourself.

It's a period-30 oscillator because it moves 15 steps in either direction.


The orthogonal spaceships do not move diagonally like the gliders, but to the right, left, up or down. They are available in 3 sizes. They are called lightweight spaceship, middleweight spaceship and heavyweight spaceship, abbreviated to LWWS, MWSS and HWWS. The populations are more reminiscent of birds than spaceships, both in their static state and in motion.


Other interesting objects



A series of 10 living cells becomes a period-15 oscillator.


This pattern turns into a very nice period-3 oscillator.


The last two starting populations are symmetrical. If you specify a symmetrical figure, the symmetry is retained in all subsequent generations. This makes it attractive in every phase. Try a window with a crossbar.

In the early days of life research, it was assumed that all populations would eventually become stable. None had yet been found that would continue to grow forever. The first ever growing output pattern that was discovered was "Run Gun 30".


You can find other interesting output patterns in the applet itself under the menu item "Open". But actually you should find interesting starting patterns yourself.

So what practical use for mathematics and science does this game by Mr. Conway have? I'll leave a little on the edge of that.

 Source math.com 
This page was last modified on Wednesday September 16, 2009 19:50.
© 2002 Wolfgang Appeal

This "game" was invented by the American mathematician Conway. The Game of Life became known when it was introduced in 1970 in the science magazine Scientific American (the German edition of which is now called Spectrum of Science). Today the web is full of it and it is really very serious math. But you're only supposed to play with it.

Before Conway laid down the rules as described in the main part, he tried many other possibilities. But either the "cells" died too quickly or too many were born. "Life" balances these tendencies. With the rules in the main part, it is difficult to predict whether a "pattern" will die out completely, a stable population will develop, or whether it will continue to grow.

"Life" is just one example of what mathematicians call "cellular automata".

A playing field that consists of many small pixels always serves as the basis for all cellular automata. Each of these pixels can assume different states. In the simplest case there are two (white or black, dead or alive, 0 or 1). The situation becomes more complex when a pixel can assume 4 or more states. The more properties a pixel can have, the more variations of the mirror rules are open. With 4 states, a pixel can be dead and alive, e.g. also permanent, which results in a more complex evaluation, but opens up completely new possibilities for cellular automata.

The interesting thing about such little black dots is that each one can represent a kind of life. Depending on the immediate environment, pixels can come to life, continue to live, or die. The criteria for this are provided by the direct neighboring pixels (4 on the sides, 4 in the corners), which explains that only continuous and no erratic expansions of figures can occur.

Playing now consists in recognizing interesting variants, i.e. the conditions for survival, death orSelecting the creation of a pixel skillfully so that the entire playing field does not turn black or white within a few cycles. Playful applications of such cellular machines that simulate primitive life are used, for example, as screen savers.

But of course that's not all why scientists or mathematicians are interested in such "self-organizing systems". Studying such systems can help us understand how the patterns on a petal or the stripes on a zebra are made. Again, it is a pattern of living cells. Yes, it may even explain the diversity of life on earth itself.

But I don't want to drive you to college, I just want to introduce you to an interesting area of ​​modern mathematics. And you don't need any special knowledge.

The 3 simple rules in the main part are all you need to make discoveries with "Life". In most computer games, the programmers create the complex game situations. There are 3 simple rules here. Still, "Life" is probably the most programmed game in the world.


Could living creatures evolve in a sufficiently large life universe if we only waited long enough? As easy as describing Life with 3 rules, Life develops a complexity that we find similarly in our own universe.

It's an exciting question to see what would happen if we were to sow tons of different random patterns in an infinitely large life universe. Probably the complexity of the patterns and what is happening would be far greater than here on your computer. Even in our own universe there is a huge difference between the history of the earth or the universe and the time frame of our own existence. About what is theoretically possible, I have already commented on the left margin.

But unlike our own universe, life is very limited. Nevertheless, it is a simple example to depict the forces of evolution.

How complex can life get?

Without knowing the main part, you will not understand anything in the margin.

You can build a computer into the life universe. A computer created from "living beings" or cells that live or die according to the 3 rules of life.

I can only hint at it here briefly. If you are really interested, I have given the American source at the bottom of the main part. Here you will find a lot of links and information.

You can create currents from gliders and spaceships that can be used to send information, much like the electrical signals in real computers. These streams of gliders and spaceships can be made to react with one another in such a way that all the logical functions on which the work of a computer is based can be represented.

Here you will only understand train station if you do not know what logical functions are. But I'm not starting to explain daaas here as well.


It continues under the period-3 oscillator on the left!