Notes

Chapter 6: Starting from Randomness

Section 8: Structures in Class 4 Systems


[Structures in] the Game of Life

The 2D cellular automaton described on page 949 supports a whole range of persistent structures, many of which have been given quaint names by its enthusiasts. With typical random initial conditions the most common structures to occur are:

The next most common moving structure is the so-called "spaceship":

The complete set of structures with less than 8 black cells that remain unchanged at every step in the evolution are:

More complicated repetitive and moving structures are shown in the pictures below. If one looks at the history of a single row of cells, it typically looks much like the complete histories we have seen in 1D class 4 cellular automata.

Structures with all repetition periods up to 18 have been found in Life; examples are shown in the pictures below.

Persistent structures with various speeds in the horizontal and vertical direction have also been found, as shown below.

The first example of unbounded growth in Life was the so-called "glider gun", discovered by William Gosper in 1970 and shown below. This object emits a glider every 30 steps. The simplest known initial condition which leads to a glider gun contains 21 black cells. The so-called "switch engine" discovered in 1971 generates unbounded growth by leaving a trail behind when it moves; it is now known that it can be obtained from an initial condition with 10 black cells, or black cells in just a 5×5 or 39×1 region. It is also known that from less than 10 initial black cells no unbounded growth is ever possible.

Many more elaborate structures similar to the glider gun were found in the 1970s and 1980s; two are illustrated below.

A simpler kind of unbounded growth occurs if one starts from an infinite line of black cells. In that case, the evolution is effectively 1D, and turns out to follow elementary rule 22, thus producing the infinitely growing nested pattern shown on page 263.

For a long time it was not clear whether Life would support any kind of uniform unbounded growth from a finite initial region of black cells. However, in 1993 David Bell found starting from 206 black cells the "spacefiller" shown below. This object is closely analogous to those shown for code 1329 on page 287.

As in other class 4 cellular automata, there are structures in Life which take a very long time to settle down. The so-called "puffer train" below which starts from 23 black cells becomes repetitive with period 140 only after more than 1100 steps.


From Stephen Wolfram: A New Kind of Science [citation]