| The Rainbow Life Game allows some rather unexpected aesthetic experiences. General advice on how to achieve beautiful patterns is not easy to give since these things are basicly subjective, but a starting suggestion could be: Choose an initial pattern similar to one of the patterns here to the right. At least people in the knitting or glazed tiling business should get some inspiration here. | |
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One of the first observations to make when trying out the Rainbow Life Game is that the degree of assimilation, i.e. mixing of the colours, in general seems to depend on the initial pattern. As a rule a well-mixed initial pattern, e.g. with regularly alternating colours, will quickly blend into different gray shades.(I will from now on suppose that the initial colours are black and white.)
In contrast, a 'segregated' initial pattern with two differently coloured unconnected components most often will remain segregated. (I will make it perfectly clear here that I have no racial or other similar concealed interests in these studies). A typical life story that starts from such a segragated situation is that the two factions clash and get some gray offspring which soon afterwards perish, often due to choking, after which the white and black parties continue their growth at some distance from each other. These events may then be repeated a number of times.
| The development that follows from the initial pattern to the right conforms well with the above scenario. A grayish middle part appears immediately but vanishes already in generation 5. After that the white and black parties develop independently (in fact, of the five surviving 'white' cells in generation 5 actually four contain some barely visible black genes.) Finally the 'black' and 'white' factions form three stable blinkers each, which prevents any further gene interchange. |  |
| The mentioned segregating tendency is however apparently reduced if the initial patterns become larger. In those cases there often appear gray factions and even dark gray and light gray groups that seem to fight each other. This latter scenario develops from the pattern to the right here in spite of its relatively small size. |  |
The concept of genetic impact is an interesting feature that the present program makes it possible to study . This concept can be defined in a quantitative way by means of a colour index varying between 0 (white) and 1(black). The genetic impact of a certain cell or a certain group of cells in a given initial pattern, at a certain generation, can then be defined as the total colour index of the pattern that results from a Rainbow Life Game with the given initial pattern where the cell (or group of cells) in question are coloured black and the rest of the initial pattern white. This concept may of course also be applied to a certain cell (or cells) in the given generation in such a way that a given cell A in pattern P has genetic impact g on the cell B in generation n etc... Of special interest is of course the study of genetic impact of cells on stable or cyclic final patterns. The three spaceships (mentioned and pictured in the discussion of the Black&White Life Game) evidently have some cells with very strong impact, since they manage to win B&W games in spite of large numerical disadvantage initially. This can be studied more closely by means of the present game.
The cyclic patterns, moving or not, here show a peculiar behaviour. If you start a blinker or a glider or one of the spaceships with one black cell in order to study the genetic impact of the cell, the whole pattern rapidly attains a rather uniform gray hue. The differences between two generations become invisible soon enough but one may suspect that each new generation induces a small decreasing change and that the gray colours of each cell tends to a certain asymptotical limit, common to all the cells of the pattern. Furthermore, this limit colour depends on the initial black cell in the sense that more influential, i.e. more strategically situated, cells induce darker gray shades than others.
| The case of the toad (and the clock) is however completely different. These oscillators seem to pale down to white when the initial black cell is a moving cell (in contrast to the two non-moving cells of both patterns). But if the black cell is placed as a quiet non-moving cell, as here to the right, it works as a stable source of blackness and induces not less than four intermediate shades of gray to which the colours of the four moving cells seem to converge. |  |
| The toad after some generations |
It turns out that these behaviours have a rather elementary mathematical (linear algebraic) explanation.
The colour changes in the patterns are described by simple linear expressions. The colour of a newborn cell is given by the expression (a+b+c)/3 where of course a, b and c
are the colour indices of the three parents. By iterating the above formula one can achieve linear expressions that define the colours of each cell in each generation in terms of colours of cells in earlier generations. The totality of expressions that relates colours of two given generations, say n and n+1, can be described by a formula of type
| vn+1 = Avn |
All the information on the transformation between these two generations is thus collected in the transformation matrix A.
One can observe that the sum of each row of the matrix A is 1, since the rows always describe iterated transformations of type (a+b+c)/3. Hence A is a so called 'stochastic matrix'. A consequence of this is that the sum of the genetic impact of all cells in a given pattern is equal to the total possible impact (i.e. the impact of an all black version of the given pattern).
Short explanation:
The sum of the genetic impacts is equal to the sum of the column sums of the matrix, which in turn is equal to the sum of the row sums,
which (in view of the observation above) is equal to N (the number of rows of A= the number of cells in the generation n+1)), i.e. the total possible impact.
Since cyclic patterns always return to similar figures after a fixed number (say m) of generations, corresponding cells in generations n and n+m can be identified and we are able to express the transformation from generation n to generation n+m by a quadratic matrix B:
Suppose that the colour vector vN converges to the asymptotic limit v. Then v must satisfy v=Bv, or
One can show in this manner that the glider gets the following asymptotic colour indices (the order of the cells are given to the right):
from initial black cells in corresponding positions, which agrees well with empiric evidence. | 1 2 543 |
If however the rank of B is less than M-1, the system (ii) has other solutions than (iii) and the asymptotic behaviour is not necessarily uniform. This case applies to the toad and the clock, which both have rank 4 when M=6. (The toad and the clock are in fact algebraicly equivalent since they have the same transformation matrix B from generation n to n+2). In this case the system (ii) can also be solved, this time by means of the eigenvectors of BT corresponding to the eigenvalue 1. (The number of independent eigenvectors of this type is r if rank(B) = M-r.)
It turns out that if cell 1 is black initially and the rest are white in the toad, the cells (in the order given to the right) get the following different asymptotic colour indices respectively:
| 1, 6/7, 5/7, 2/7, 1/7, 0. | 
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The reader must excuse the lack of detailed explanations here, but this is not a mathematical research article. Instead the reader is encouraged to carry on his/her own studies which, since this field is absolutely new, certainly will lead to new interesting discoveries. I will gladly receive comments and results on this address: gunnarj@math.kth.se
| Gunnar Johnsson |