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A New Kind of Science: The NKS Forum

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Posted by: Guy Birkin

Hullo,
Nice idea to set up this Artistic NKS bit. I'm doing a masters degree in Fine Art and my main theme is 'digitality'; NKS, Fredkin's Digital Philosophy, and the works of Teihard de Chardin and Jean Gebser are my main sources. I do intend to engage in some arty/NKS discussion here, but first I'm hoping someone can offer this struggling artist a bit of practical help:
At the moment, I'm working on paintings and tapestries based on block cellular automata, and have recently purchased Mathematica to save me the trouble of working out CAs with paper + pencil (as I have been doing for years).
I know from the NKS notes (p. 1023) that the basic mechanism is

BCAEvolveList[{n_Integer, rule_}, init_, t_] :=
FoldList[BCAStep[{n, rule}, #1, #2]&, init, Range[t]] /;
Mod[Length[init], n] == 0

BCAStep[{n_, rule_}, a_, d_] :=
RotateRight[Flatten[Partition[RotateLeft[a, d], n]/.rule], d]

but my attemps at writing this into a working program have yet to succeed. I'm aware that this is pretty basic stuff, but I'm new to Mathematica (and a bit lazy). Could someone please show me how it's done?
I've attached a rough image of the sort of thing I'm working on. It's part of a k2 block CA shown as a square tube, with only one cell state visible.

Thanks

guy


Posted by: Richard Phillips

The following produces a picture after having evaluated the two functions BCAEvolveList and BCAStep, although not the picture you want!


RasterGraphics[(state_)?MatrixQ, colors_Integer:2, opts___] := 
  Graphics[Raster[Reverse[1 - state/(colors - 1)]], 
   AspectRatio -> (AspectRatio /. {opts} /. AspectRatio -> Automatic), opts]



Show[RasterGraphics[BCAEvolveList[{2, {{0, 0} -> {0, 0}, {0,1} -> {1, 1}, {1, 0} -> {1, 1}, {1, 1} -> {0, 0}}}, ReplacePart[Table[0, {40}], 1, 20], 40]]]


Do you then want to produce a picture like the GIF you attached?
I assume the non-white colors are added in after by hand. Is that true?
Also, do you know the description of the rule you want? In the above example it is given as a rule list.


Posted by: Guy Birkin

Thanks Richard,

That's just what I needed to get me going. Yes, that image was done by hand,as part of a rather lengthy process: Firstly working out the evolution of the CA on graph paper, next (if it's a complex one) making a small tracing-paper model, then drawing a 'black=visible, white=invisible' image on isometric paper (GIF!) which can then be transferred to a canvas before finally being rendered in paint. And yes, ultimately I do hope to create images like that using Mathematica, but for now I just need to play around with little programs like that: With it, I'll be able to do lots of experiments, save a lot of time, and hopefully learn a bit more about Mathematica in the process.

I'm concentrating on one simple rule that conserves the number of black cells (#39 in the Wolfram Atlas of Simple Programs: 11-->00, 10-->10, 01-->01, 00-->11) as well as the k>2 rules of which this is an element. I like it because of its 'visual grammar' - the combination of numerous simple shapes and the larger, more complex threads, together with its apparent lack of vertical/horizontal distinction. But for me, the specific rule and its visual representation is of lesser significance than its digital basis and its incorporation of time.
Hmm - I'm rambling towards territory that should probably be reserved for another thread.

Thanks again - it's a great help.

guy


Posted by: Richard Phillips

Pleased to be of help. Do say if you have any problems.

By the way, the results are very nice.

On a technical and aesthetic note, I haven't before seen the geometric representation of the block CAs using triangles that you used. I'm sure anyone interested in block CAs for any purpose could use this.




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