Colourised Life

Colourised variants of Life are cellular automata with multiple live states represented by different colours. If the colours of the live cells are ignored, these rules behave identically to Life:

  • As in Life, a live cell dies of underpopulation if it is surrounded by fewer than two live cells in its Moore neighbourhood;
  • A cell dies of overpopulation if it is surrounded by more than three live cells;
  • A cell is born if and only if there are exactly three live neighbours in its Moore neighbourhood.

Immigration

Immigration (also called Black & White[1]) has two live states, typically represented by red and blue cells. The following rules are present in Immigration to determine the colour of the live cells:

  • Live cells retain their colour, permanently, until they die due to over- or underpopulation;
  • When a cell is born, it takes on the colour value of the majority of its three neighbours.

Certain patterns exhibit interesting behaviour in Immigration. For example, it is possible to yield oscillators with a higher period than the equivalent Life oscillators (imagine two Herschels of different colours in a p256 loop).

The following image shows how a glider can support two different types of cells, symbiotically, without one becoming dominant: Toad glider

Quadlife

Quadlife is a colourised variant of Life, with four live states. These states are traditionally represented by different colours; this article assumes that red, green, yellow and blue are used.

Any configuration consisting of cells of just two colours behaves identically to Immigration. When newborn cells are created, they take on the majority colour of their neighbours. However, there is another possibility. If the three neighbours are all different colours, the newborn cell takes on the remaining colour. For instance, a dead cell surrounded by a single red cell, blue cell and green cell will become yellow in the next generation.

As in Immigration, cells that survive do not change colour.

Niemiec's Life

The last rule of Quadlife, where a cell takes on the remaining colour, is rather artificial. Mark Niemiec created a Life rule with eight live states, which circumvents this problem. The eight states are refered to as {a,b,c,d,a',b',c',d'}. The simulation looks best where the colours are positioned on the vertices of a cube in RGB space, and each cell is diametrically opposite to its complement. It has the following rules (as quoted from Niemiec's message):

  • Three cells of the same colour produce a child of the same colour (x+x+x→x); this is compatible with mono-coloured Life, and also two-coloured Life (Immigration).
  • Two cells of any one colour plus one of any colour favour the dominant colour (x+x+y→x); this is how conflicts are resolved in Immigration. This degrades into rule 1 if both colours are the same.
  • Three cells of different colours in the same set produce the complement of the fourth colour (x+y+z→w'); This is similar to, but not identical to, the way QuadLife does it (x+y+z→w).
  • Two complementary cells cancel out (x+x'+y→y). This also degrades into rule 1 if x and y are the same. (Unfortunately, this is NOT compatible with QuadLife; here, (a+b+c)+d+a→d'+d+a→a; in QuadLife, (a+b+c)+d+a→d+d+a→d).
  • Two cells of different colours from one set plus one from the other set favour the solitary one (x+y+z'→z').

Koenig's colourised life

This rule differs from the previous rules in that the colours are continuous, rather than discrete. A newborn cell takes on the arithmetic mean colour, in terms of the HSL colour system, of its three neighbours.

Patterns generally reach equilibrium, like the oscillators in this image: Color Oscillators

Koenig chose the HSL system over the RGB system, since patterns in the RGB system generally become grey blobs.

Spectral Life

Also known as MuziksRainbow,[2] the rule contains seven cell colours; red, green, blue, cyan, magenta, yellow and white.

Similarly to most rules, two or three cells of the same colour will give birth to the majority's cell colour. Combinations of three different types are handled as follows:

  • if one of the cells are white, the result will be a white cell,
  • if the cells are each red, green and blue, the result will be a white cell,
  • if the cells are each cyan, magenta and yellow, the result will be a white cell,
  • if two of the cells are from RGB, and the last remaining one is from CMY, the resulting cell will be the opposite of the missing RGB cell as if the two existing had been mixed. Blue and green give cyan, red and blue give magenta, and red and green give yellow.
  • a similar effect happens if CMY cells dominate; magenta and yellow give red, cyan and yellow give green, and cyan and magenta give blue.
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References
  1. "Black&White". The Life Lexicon. Stephen Silver. Retrieved on May 14, 2016.
  2. muzik (February 23, 2017). Re: Rule request thread (discussion thread) at the ConwayLife.com forums
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