Genetic aspects


In the Introduction some examples were given of different behaviour of black and white patterns in the Rainbow Game.
Closer analysis reveals that these discrepancies can be understood in terms of four different types of cells.
The main distinction to make is between stator and rotor cells. The stator cells that belong to the stator part of the cyclic pattern are those that remain unchanged during the development of the pattern. In contrast the rotor part consists of the moving cells. It follows immediately from this definition that spaceships only have rotor cells. (But there are also statorless oscillators such as the phoenix, the mazing and the pentadecathlon). Also, the stator cells in the Rainbow Game have unchanging colours whereas the rotor cells always sooner or later get gray (non-white and non-black) shades.

The stator part is traditionally divided into the casing and the bushing part. The casing cells are traditionally defined as those stator cells that never get any offspring in contrast to the more active bushing cells.
[ Added 040309: Note however that some oscillators (e.g. the jam) have stator cells that get offspring but lack durable genetic impact since the offspring line soon dies out. These stator cells are therefore of casing type on the cycle level and will hereafter be classified as such. This will agree with the natural classification in terms of the cycle matrices mentioned in 'Mathematical Aspects'].

It seems that there is also a natural division among the rotor cells. It turns out that some rotor cells have no durable genetic impact at all, i.e. if such a cell is the only initial black cell the asymptotic colour of the whole pattern will be white. These rotor cells are here called secondary rotor cells. An initial primary rotor cell however always gives rise to an asymptotic non-white colour on some part of the pattern. The primary and secondary cells can also be characterized by means of transition matrices.

Examples:
All the rotor cells of the toad (and the jam, the mold, the blinker, the barber poles etc.) are secondary.
The glider and the spaceships have primary and secondary rotor cells as shown below:

Primary rotor cell: x         Secondary rotor cell: o



The two larger spaceships also have three primary rotor cells in the corresponding position.

The primary rotor cells always occur in sets of at least three cells. In these sets the asymptotic colour is uniform. Its exact shade depends on the position of the initial black cell(s).
These primary sets together with the bushing stator constitute the primary units of the pattern. The primary units function as sources of blackness and whiteness according as these colours are initially present in the units.

The asymptotic colouring of the secondary cells can be determined qualitatively by means of a simple principle reminding of diffusion or heat transfer.
First we say that a secondary cell is 'subordinate to' a primary unit (bushing stator cell or set of primary cells) if the cell has some ancestor in that unit.
Then the following holds:

The exact indices of the asymptotic colours can be computed by means of linear algebra. It turns out that these indices are always rational numbers.

More details on the Mathematical Aspects are available.