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R.P. van der Hilst
University of Utrecht
The Netherlands

Registered: Feb 2006
Posts: 3

Rule 30, shifted

Hi all!

I've been toying around with elementary CA rule 30, and especially with its left and right shifted visualisations. Doing that, I came to several conclusions that are known already, but still interesting to emphasize, and found that the right-shifted rule is simpler to explain than the normal rule (though technically it's the same description).

First of all, I noticed there are three regions in rule 30 (not two):

1) The repeating (and growing) pattern on the left hand side,
2) The (in most aspects) random part in the middle, and
3) A nested part at the right, best visible by looking at the white triangles, following a a,b,a,c,a,b,a,d pattern (larger triangles have characters further in the alphabet), giving rise to larger triangles as soon as a specific character has appeared thrice.

I suppose most of you knew this already, and after I found this out myself I saw that there is another thread on this forum touching that right hand part 'nested-ness' from a mathematical viewpoint.
Nonetheless, this directed me towards shifting the whole picture of rule 30 right and left, to be able to emphasize either the repeating part (stop it from shifting to the left) or the nested part (stop that from shifting to the right). It's interesting to take a look at those pictures yourself.

By shifting the successive rows to the left, thus revealing the nested pattern a little bit more clear, the cellular automaton actually becomes dependent on the cell itself and its two right neighbors. It seemed interesting to take a look at the shifted rules aswell, and I think you should take a look yourself (see the attached picture; please ignore the two cells at the lower right, I'm using plain Mathematica (I'm a student; I don't have the NKS Explorer add-on).
Stephen Wolfram took 3 lines of text to explain the normal rule 30 in words, but the right-shifted version becomes quite a lot simpler:

The cell changes color except when both right neighbors are white

(Compare with page 27 of the book)

Doing this made me think that rule 30 is some kind of base ~fill in the answer~ calculator, which uses subsequent numbers on the right to come to an answer on the left: sort of like convergent sums or long division.
And that raises a lot of questions, in increasing difficulty:

- what is the number being calculated?
- what are the numbers entering the calculation?
- how does the calculation work (which equation)?
- what does the simple initial condition has to do with this all?

Anyway, the first answer is pretty simple, it only depends on the base you look in and the decimal point. In base 10 this is something like:

{0.784783, 1.56957, 3.13913, 6.27827, 12.5565} or
{0.870333, 1.74067, 3.48133, 6.96267, 13.9253}

depending on the decimal point and whether you look at even or odd steps. There seems to be a bad calculation of pi in there (3.139)...
Maybe successive need be combined to form base 3 (or more) numbers, as it seems the rows repeat every second cycle (suggesting to combine them instead of using them separately). I have no idea..

The second question is even stranger, although base 3 seems to me to be the best guess on the number system (because of the nesting in triples), which means the triangles are representing finger counting of some kind (may require alien hands ;-) ).

As for the other questions, I have no idea, my guesses are as good as anyones. I could expect the initial condition having to do with the power of the calculation or the position of the decimal point, but those are purely speculative guesses.

I wonder what you all think of this...

edit: As attaching didn't work, I suggest you look at page 27 of the book or write the rules down yourselves, putting the new cell under the leftmost cell instead of in the center.

Last edited by R.P. van der Hilst on 03-21-2006 at 03:34 PM

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Todd Rowland
Wolfram Research

Registered: Oct 2003
Posts: 116


You might be interested in the work that Eric Rowland (no relation) has done on the shifted rule 30, as
presented at the NKS 2005 Midwest conference.

He also has a paper coming out in an upcoming issue of Complex Systems.

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