Here are some results of recent tests with 6-colour cellular automata. These are 6-state relations of the simple 2-state block cellular automaton that I use a lot, rule 39 in Wolfram’s notation which translates the output states of the rule specification into a decimal number. This rule family uses a 2-cell block, offset by one cell each generation, so it tiles in a brickwork pattern. Rule 39 of the 2-state CA translates to: “if both cells are in the same state, change state on the next step; if cells are in different states, remain in the same state.” In the program run in Mathematica, this is applied as a set of transformation rules operating on lists of numbers. In full, these are: {0,0}→{1,1}; {0,1}→{0,1}; {1,0}→{1,0}; {1,1}→{0,0}. The rule 39 specification is visualised below, with one part for each of the transformation rules. The top half of each part represents the current states of the two cells in a block; the bottom half represents the states of those cells in the next generation.
The rule specification has one part for each possible configuration of states in a block. The number of different possible configurations with a block size n and with k colours is equal to k^n, so a k=2, n=2 CA such as rule 39 has 2^2 = 4 parts. A 6-colour CA rule has 6^2 = 36 parts. The number of different rules is equal to k^(n×(k^n)). With 2 colours, there are 256 possible rules. With 6 colours, there are 106387358923716524807713475752456393740167855629859291136. This is too many to explore exhaustively. I explored a subset by using constraints: to incorporate the k=2 rule 39 pattern within the k=6 rule with at least one pair of colours, and to generally conserve the colours by limiting the rule specifications to simple transformations. Rule 39 conserves the total number of different colours and is reversible, which means that any evolved state can be run backwards, by applying the rules in reverse, and it will return to its initial conditions. In general, a block CA is reversible if its rule permutes all possible blocks. Due to the constraints, many of these k=6 rules meet this criterion and are therefore reversible. This is rule 39 with random initial conditions, 32 cells in size, run for 32 generations, where time runs downwards from the top:
For the purpose of these tests, the 6 states are represented by the colours red, orange, yellow, green, blue and purple (R,O,Y,G,B,P). The final colours might be different, depending on how printing tests turn out, but will have a similar spread, being equally-spaced on the trichromatic colour wheel. My aim is to find combinations of rules and colours that induce perception of secondary colours through optical mixing.
Below are some results of 6 different rules tested with three sets of initial conditions:
- Random choice of all colours (R/O/Y/G/B/P)
- Three pairs of complementary colours picked randomly (R/G, O/B, Y/P)
- Three pairs of neighbouring colours picked randomly (R/O, Y/G, B/P)
I generated one of each type of initial condition, and used this set of 3 for all rules shown here, to make the differences more visible. Each CA is 240 cells in size, run for 240 generations.
Rule 20774264240371331443939583648662020862331869895110587700
This k=6 rule incorporates 2 instances of the k=2 rule 39 pattern: instead of black and white (B/W), it’s R/O and B/P. You can see these in the top-left and bottom-right corners of the rule specification above. Random initial conditions (top) produces a bit of a jumble, but a pleasing combination of colours. Complementary pairs of colours (middle) makes a nice gradient out of reticulated patterns. Neighbouring colours (bottom) produces a patchwork structure, but not the kind I like.
Rule 20796116639275282756883592732765805970567318290529775700
This rule incorporates 3 of the k2 rule 39 pattern with 3 pairs of neighbouring colours: R/O, Y/G and B/P. The random conditions produce a similar thread-like patterns. The patches of colour in the other two blend and interact nicely.
Rule 103541839025819083223383708563702745805808655480680346680
This rule also incorporates 3 of the k2 rule 39 pattern, with different pairs of colours: R/P, O/B, Y/G. Randomness makes mostly vertical structures jostling with each other. In the structured initial conditions, the rule makes for clear areas with rule 39 behaviour that are gradually encroached by tendrils from the neighbouring areas.
Rule 62556620991588118164093337622114268271511526829240714310
This rule incorporates 3 of the k2 rule 39 pattern, with pairs of complementary colours: R/G, O/B, Y/P. In this rule specification, if the colours aren’t one of these pairs then instead of behaving like rule 39, each cell takes on the state of its neighbour, so the colours swap places. The images below are different to the previous ones because the random initial condition produces the most orderly image, comprising vertical repeating patterns. This example is the best of the vertical type patterns yet. In contrast, the other two images are more chaotic than those in the rules shown above.
Rule 62312062993845184628174649112617059049519000489975636340
This rule is quite similar to the previous one but with only 2 rule 39 pairs(R/G, Y/B). I like the squiggly patterns produced by random initial conditions.
Rule 62556620991588118164089965300297552193934650966268789830
Although this rule is very similar to the previous one, it is unlike all the others because it doesn’t have the property of permuting all possible configurations. Two configurations instead of the usual one lead to a configuration of two yellow cells (YY): BB and PP. With all the other rules so far, if you count the total number of cells of each colour in the output parts of the rule specification (the lower half of each part), they are equal, with 12 cells of each colour. In this rule, there are 14 yellow and 10 orange, plus 12 of every other colour. For this reason, it is not reversible. It makes some lovely patterns.
Rule 20774264240899287841879128096849235332307474604780057880
This rule differs from all the others because it doesn’t incorporate the rule 39 pattern. In this rule, colours rotate. Whenever a block has 2 cells of the same colour, the colour changes to the next along in the spectral sequence (rainbow order). This property seems to make a mess of any structure in the initial conditions. The rule essentially dissipates information into the same kind of chaotic pattern; any structure in the initial conditions is soon lost. Interesting to try the rule, but uninteresting results.
Rule 62556585801747746489648698620876988974236430593535788615
What’s different about this – easily visible in the rule spec – is that some same-colour blocks stay as they are (O,Y,B,P). Only R and G are like rule 39. Like most of the other rules, it conserves the colours and is reversible. Not the most interesting results, because this rule doesn’t mix things up as much as the others.
There are a few different types of pattern produced within each image, and small variations in rule spec can produce quite different combinations of patterns. Given that most of these rules conserve the total number of different colours in the initial conditions, and therefore each image has roughly the same number of each colour pixel, it’s surprising how much they differ in terms of overall hue. With the right set of colours, these images can produce the effect of seeing colours that aren’t there, through a combination of optical mixing and colour juxtaposition. This visual effect was explored by Sydney Harry, whose paintings I’ve seen at the colour museum in Bradford. I’d like to see if the effect can work with these CA patterns in print instead of on screen, so I’ll be doing some print tests soon. It would be interesting to see what they look like in large scale – wall-sized, where they can be viewed from both near and far.
Aesthetic Complexity 