Game of Life with B345/S45 rules

A lot of people are familiar with British mathematician John Horton Conway’s ”Game of Life” – an algorithm to simulate cellular growth and decay, first published in Scientific American in 1970.

The concept is simple. Imagine an infinite grid of empty squares. Each of these squares can either be empty or occupied by a ”cell”. The are only two rules. One rule for empty squares (the birth rule) and one rule for squares occupied by a cell (the survival rule).

Birth: If an empty square has exactly three neighboring cells a new cell is born in that square.
Survival: An existing cell will only survive if it has either two or three neighbors.

These two rules implies a time axis along which the grid will project forward in discrete phases.

But why do we have these specific rules? Why is exactly three neighbors required to give birth to a new cell?

Conway’s classical rules are sometimes abbreviated to the code B3/S23.

The reason for these particular rules to have become so popular isn’t that strange. Anyone who has experimented with a simulation engine using these rules knows that they often create very complex and interesting situations.

But this being a fact doesn’t exclude the possibility of other rules also having the potential of creating complex and interesting scenarios.

Conway himself and others have explored a lot of other possible rules for Birth and Survival.

I played around two days ago with my own javascript-powered game of life engine where I can chose any set of rules for birth and survival. I wanted to explore an idea I had to use the game of life cellular automaton concept to create a cryptographic hashing algorithm. So, by a mere coincidence, I tried the B345/S45 rules and was fascinated by the way that an initial small colony of cells transformed during hundreds of phases only to later die out. This was almost always the fate that awaited small colonies occupying an initial eight times eight square. I tried out new random shaped colonies, one after another and very seldom a shape would pop up that wouldn’t die out but instead grow and grow and never stop growing. How come, I wondered?

I dealt with this question systematically and found out that there are only 36 symmetrical shapes out of a total of 37888 that will grow indefinitely. All the other symmetrical 8×8-shapes will eventually be either totally annihilated or get stuck in small for ever repeating loops of shapes (so called pulsars).

Here is the total set of miracle seeds – the smallest possible symmetrical shapes that I found that will grow forever. All the tens of thousands of others are destined to perish or stagnate.


I have shown that there exist no smaller symmetrical shapes, for example 7×7-shapes or 6×6-shapes, that will grow indefinitely.

Add or remove a single pixel anywhere and they will die – that’s how sensitive they are (try it out for yourself).

So these are the symmetrical ones. What about the asymmetrical ones?

I don’t know, but I guess that you can’t find an asymmetrical shape smaller that 8×8 that will grow indefinitely.

It could be tested systematically but the number of all shapes that occupy a maximum space of 8 times 8 squares are much larger than just the symmetrical ones.

Click on this link or on the picture above to go to my B345/S45 simulation engine.

Update: July 28 2015

Amazing news. After seventeen hours of automatic testing of random shaped populations eight different asymmetric 8×7 indefinite growers presented themselves!

8x7 grower 8x7 grower 2 8x7 grower 3 8x7 grower 4 8x7 grower 5 8x7 grower 6 8x7 grower 7 8x7 grower 8

I  also found three shapes that lasted a long time but in the end didn’t quite make it.

8x7 lasted 539 8x7 lasted 542 8x7 lasted 716

They lasted 539, 542 and 716 (!) rounds respectively.



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