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A New Kind of Science: The NKS Forum (
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-- 256 rules and Clifford algebra (

Posted by Frank (Tony) Smith on 05-31-2004 07:06 AM:

256 rules and Clifford algebra

I was thinking about Stephen Wolfram's 256 rules for simple 2-dim 2-color cellular automata, and comparing them to my favorite real Clifford algebra Cl(8) which has 256 dimensions.
I wondered whether anything interesting would come from looking at the 256 rules (which can be represented by 256 binary numbers) as a basis for the Cl(8) Clifford algebra, and then looking at a resulting Clifford algebraic structure among the 256 rules.
Does anybody in the forum know of any prior work looking at a Clifford algebra structure of those 256 rules?
I have written up a 230K pdf file at
with my initial thoughts, in case they may be of use to anybody.
Any comments would be appreciated. Thanks.
Frank (Tony) Smith

Posted by Tony Smith on 05-31-2004 11:57 AM:

We are sure to cause some confusion

Hopefully nobody will object too much to an unavoidably off-topic post, to make sure people are aware that there really are two Tony Smiths from different ends of the earth who are independently interested in such esoteric topics as NKS and derivatives of the rhombic dodecahedron. yet who have never previously been in direct contact.

We even share the same middle initial, and, I suspect, in another place at another time a certain Norwegian gentleman failed to quite work out our degrees of separation.

Tony Smith
Complex Systems Analyst
TransForum developer
Local organiser

Posted by Lawrence J. Thaden on 06-02-2004 12:34 PM:

Frank (Tony) Smith,

I have a question on grade 4.

By self dual do you mean that the bit strings are the reflected complements of themselves?

Example: rule 105: 01101001 when reflected is 10010110.
Which when complemented is again rule 105.

L. J. Thaden

Posted by Frank (Tony) Smith on 06-02-2004 02:21 PM:

256 rules and Clifford algebra

Originally posted by Lawrence J. Thaden
Frank (Tony) Smith,

I have a question on grade 4.

By self dual do you mean that the bit strings are the reflected complements of themselves?

Example: rule 105: 01101001 when reflected is 10010110.
Which when complemented is again rule 105.

Yes, that is what I mean by self dual in that context.
Frank (Tony) Smith

Posted by Frank (Tony) Smith on 06-02-2004 10:22 PM:

Re: 256 rules and Clifford algebra

Originally posted by Frank (Tony) Smith
Yes, that is what I mean by self dual in that context.
Frank (Tony) Smith

I wrote too hastily in the above quote.
On further thought, I should make some corrections.

As to self-dual, the duality operation I mean is the
operation of changing 1s to 0s and 0s to 1s.
Therefore, no CA would be its own dual, but each CA of the
256 is one of a dual pair, such as 0010011 and 1101100.
In the diagrams in my pdf paper at
the central vertical line can be regarded as a mirror as to
which reflection to the similar position on the other side of
the line takes you from a CA to its dual.
When I say that grade-4 is self-dual, I mean that the set
of 35+35 = 70 grade-4 CAs remains invariant under that
duality operation (or, equivalently, under reflection using
the vertical line). That is, the duality takes one CA member
of the grade-4 set into another member of the grade-4 set.
Of course, by definition, any CA is the dual of its dual.

Some other corrections to some figures in my pdf paper,
mostly correcting errors due to my poor eyesight perception
of some distinctions among the 256 CA images, are:

Following a suggestion by Michael Gibbs, the SU(3) vertices
should be modified by swapping "... the two on the right
(Rule 3 and Rule 9) so that diagrams with similar character
(except left-right reflected) would be opposite one another
in the diagram.
Rules 3 and 17 seem to be gently sloped straight lines.
Rules 9 and 65 are jagged 45-degree slopes.
Rules 5 and 33 are vertical lines with stripes.
These pairs would correspond to "red-antired", "green-antigreen", and "blue-antiblue". ...".

Also, for
the cuboctahedron of SU(2,2) = Spin(2,4)
the two vertices of SU(2) line segment
there should be some swapping:
rules 6 and 66 (both jagged) should be in SU(2) not cuboctahedron
rules 130 and 144 (both smooth) should be in cuboctahedron not SU(2).
In that way,
all 12 of the cuboctahedron vertices would be smooth.

It may be that more detailed study of CA rules might lead
to further corrections, but I hope that the overall ideas of
my pdf paper are fairly clear.

Frank (Tony) Smith

Posted by Michael Schreiber on 06-03-2004 10:06 AM:

Perspectives and problems of linking Clifford algebras and NKS

An interface between NKS and Clifford algebras might help to connect discrete and continuum models of reality.

The binomial grading of elementary rules according to their numbers of ones might be a first step. Yet we might want to study whether there are other correspondences and computable properties of classes of automata and basic elements of Clifford algebras or not.

Maybe such correspondence can be found on the level of elementary automaton rules or maybe one needs larger spaces of rule numbers for more neighboring nodes of computation and more states alias colors.

Any intermediary interpretations based on similarities caused by notation of distinctions by binary digits or features of illustrations needs further research even if it looks most convincing. It is thus helpful if proponents of such theoretical identifications look for and report features which do not seem to match or present out other open problems.

Isaacson for instance points out a marginal difference and hopes for resolving it in larger rulespaces:

Isaacson, J. D. Steganogramic representation of the baryon octet in cellular automata. International Society for the System Sciences, Proceedings Series: 45th ISSS Annual Meeting and Conference 2001

An other way to establish a fundamental correspondence between formal algebras and NKS systems of rule based computation might be to reduce the complexity of rules and basic Clifford elements to one common operator like the Spencer-Brown form.

One half of that is done already. It is easy to show that the 256 elementary rules can be constructed from 26 Spencer-Brown forms by composition of three meta operations (1) exchanging black and white in inputs (2) exchanging black and white in outputs (3) permutation alias rotation of unknown inputs.

The second half is not finished yet. While it turns out that it is possible to carry out addition of infinite binary signals by Spencer-Brown Form formulas (Please see my NKS 2004 Conference presentation notebook for details) - this might not be enough for those who follow Cantor.

One of the key features of real numbers and their complex tuples is their infinite resolution. While it might be possible to model physics in discrete spaces the length of diagonals remains an issue for models of mathematics in consciousness at least. It thus remains to be shown that all the capabilities of complex, quaternion and octonation computation can be replicated by adopting discrete models based on the Spencer-Brown Form reentry operation or other minimal common sources of computational distinctions.

Michael Schreiber

Posted by jdi on 06-09-2004 06:26 PM:

[Added later:]
Sorry... I am told that the isss server is down (09-01-04)
for re-organization. Second document can be found
at Search for Patent No. 4,286,330
--jdi [09-08-04]

In as much as you mention

you may wish to consider also

which underlies the first paper mentioned above.

-- jdi

Posted by Michael Schreiber on 06-12-2004 11:26 AM:

Merits of gambit expositions and rights meeting cycles

The paper was mentioned because it includes explicit remarks on mismatches between the proposed interpretation of a simple model and physics. We want new insights so we have to expose the limits of our preconceptions like in a chess gambit which sacrifices a pawn for a bishop.

Concerning your patent referenced by hyperlink I wonder whether your model of error correcting storage processes works if there are multiple cycles. You seem to suggest storing values as initial sequences of distinctions in a string which is processed according to simple cellular automaton rules. These transformations are supposed to regenerate their earlier configurations inspite of random changes of configrations.

It is true that deterministic transformations of a finite or periodic c-color configuration of n-states must return to a previous state after at most c^n steps of evolution. Yet the number of distinct cycles is usually larger than one so a random error may cause the cycle to shift into another cycle which connects a different set of configurations. (compare "Poincaré recurrence time" 2^n for bi-colored spaces, statistical and algebraic aspects, and tables of cycle multiplicities in: S.Wolfram, "Cellular Automata and Complexity, Collected Papers", addison Wesley, 1994: 41ff., 71ff., 291ff.).

Michael Schreiber

Posted by jdi on 06-18-2004 08:44 PM:

Hi Michael,

Thanks for your interest in the "steganogramic" paper
and the patent.

[Added later:]
Sorry... I am told that the isss server is down (09-01-04)
for re-organization. Second document can be found
at Search for Patent No. 4,286,330
--jdi [09-08-04]
(independently of Murray Gell-Mann) has also used
SU(3) in the early 1960s to deduce the structure of
the baryon octet. Neeman has seen the steganogramic
paper recently and commented that it is relevant
to SU(3) and its generators.

Your comment on multiple cycles in the error-correction
scenario is apropos. However, that multiplicity
is expoited, in broader pattern recogniton and
machine-perception schemes that utilize this patent,
to creatively generate new classes of objects from
existing classes. (Too long to describe here,
and probably off-topic). Keep in mind, though,
that the patent was first written in 1975,
summarizing work done in the prior decade...

You seem to miss the more significant (actually profound...)
aspects of the patent. For one, it patents rule 129
(also its complement, rule 126) thru claims involving
recursive "triunation". For all I know, these are
the only rules (out of the 256 rules) ever granted
patent protection...

Recursive "tetracoding", also patented there, is
a fundamental process in all forms of perception,
in addition to embodying dialectical logic. The implications
are enormous.


-- jdi

Posted by Ivan Antonowitz on 07-03-2004 12:28 PM:

Originally posted by Lawrence J. Thaden
Frank (Tony) Smith,

I have a question on grade 4.

By self dual do you mean that the bit strings are the reflected complements of themselves?

Example: rule 105: 01101001 when reflected is 10010110.
Which when complemented is again rule 105.

If the 256 modes are plotted on a 16x16 matrix, then Duality as defined by Quine makes sense of the concept of Self-Duality. I like the term 'reflected complements'.

I have done a great deal of work on this matrix, and will have more to report when I get used to finding my way around this forum.

Posted by jdi on 07-08-2004 07:30 PM:

More on Dialectical CAs

As I said in one of the posts above, I thought that the actual perceptual schemes performed by these dialectical CAs may be off-topic. Yet, I observe with amusement that my ID on this NKS forum is exactly 256!... which kind of impels me to be true to this peculiar coincidence and therefore elaborate some more...

I have had some inquiries, all in private. Following excerpts are
typical: (also, some of Jon Awbrey's thoughts in his
"Variety and Regulation" notes may be relevant here).


Joel, do you have pictures of the simulations that you can send me to print and bring to the meeting. Including 2-D and 1-D for comparison.


I'll fax you a collection of figures from a technical report to ONR/SDIO of 1987.

CAUTION: These 2-D patterns are not always easy to interpret, especially at first sight . By analogy, consider bubble chamber or cloud chamber photography -- you don't expect an untrained eye to draw the necessary deductions/implications just like that... it takes some doing...

Some of the interpretations have taken me years to figure out.... of course, now that I can point my finger at what's going in there, it is a different story... except that my finger is not long enough to reach from St. Louis to Tel-Aviv...


Another issue we didnt have time to talk about is stability. How
stable are the converged patterns to added noise to the rules?


These processes are incredibly stable, or robust. There is a manifest, unusual feature that I called in the patent "autonomic error-correction". See columns 25 to 27 in the patent:

[Added later:]
Sorry... I am told that the isss server is down (09-01-04)
for re-organization. Patent document can be found
at Search for Patent No. 4,286,330
--jdi [09-08-04]

I show there three types, with examples, but there are also more. These are:

(1) next-step correction

(2) re-generated cycle correction

(3) correctness without correction

The underlying reason for these is that each of these dialectical
processes is destined to converge on a "basin of attraction" (or limit cycle) and it is generally hard to escape that 'attraction'. Now, even when escaping, a new basin is formed, which provides for a form of 'creativity'. Let me explain.

In 1-D (also in 2-D, but it is much more complex to explain here), under a vast multiplicity of recursive tetracoding (RT), the universe of all possible strings (regardless of (finite) length, and regardless of 'alphabets' they are written in -- dubbed "StringLand"; do you remember FlatLand?) is divided up into a relatively small number of equivalence classes.

Exposing a system comprised of a multiplicity of RTs to a variety of input strings 'trains' the system by establishing various cycles, each of which becoming a 'recognizer' for all external strings belonging to the same class.

Occasional noise or malfunction may alter an established cycle into
a cycle never 'experienced' before via input of external strings! So, the system now has a recognizer for strings (or events) it has never experienced before!

It is like forming a hypothesis that something 'out there' ought to exist, and be ready to confirm the hypothesis upon encountering that thing in future inputs. (Almost something like pre-cognition.) Hypothesis-formation (also called 'abduction' by C. S. Peirce) is one of the highest forms of creativity.

Without going much further, I think you may get a feel for 'creativity' here, but also for notions of creative 'random mutation' that may be inherent in these kinds of systems. A lot more in this regard to be discussed later.


(now I am on a safer ground of going from philosophy to specific questions). Have you tried to look at elements that have internal states and if so did you look at a case that the states are like quantum states namely each of the elements is in a superposition of its internal state and the rules on the number
(changing the number of the elements ) affects the expectations of the internal states of each . Or alternatively it can be viewed as virtual particles.


The question above is not sufficiently focused for me to answer directly. For one, I can't see clearly the 'internal states' and 'quantum states' point in that question. But I can say a number of things that may be related to the gist of your

First, bear in mind that I never intended to do modeling in physics, or particle physics. I had my hands full trying to figure out visual perception, awareness, cognition, and things of that sort; then build artifacts that are as close as possible to the actual *natural* thing. (Called it 'True Machine Intelligence' (TMI) in a majpr NASA project -- got some traditional AI people very upset, but was able to prevail for the duration of the project...)

The connection to the baryon octet, hence to particle physics, has been unexpected, unintentional, and purely serendipitous. I don't fancy myself a particle physicist, so, what I'll say below is based on a naive point of view, as far as physics is concerned; but these descriptions are tied directly to what can be objectively observed by others in the behavior of the processes under discussion.

We have definite things like "vacuum" with stuff popping in and out spontaneously. We have "sea quarks" and "valence quarks". We have "interferences", "oscillations", wave-like propagations. We have convoluted enfoldments that may amount to "superpositions". We also have virtual particles. And so on and on... sometimes it looks to me as though people working in theoretical particle physics are reporting on their own perceptual processes, as opposed to something 'out there'. For the similarity is so very incredible...

Now, something to note in regard to enfoldment and "hidden" domains. When you'll look at those 2-D simulations you'd need to know that these are essentially generalizations of 1-D. Effectively, thru each pixel you have 4 recursive tetracoding (RT) working simultaneously, crisscrossing vertically, horizontally, and the two diagonal. At the same time, each such RT captures all other pixels along a straight line. So, the whole image is globally tied up, while the rules are local distinction rules. Also, instead of only 4 states in 1-D, we now have 256 states in 2-D!!! These is enormous for any CA -- just humongous! But the stuff I'll fax you is implemented via neural nets. So, all features in the graphics are composed of individual neurons firing/not-firing. There is NO WAY IN THE WORLD that you (or any one else, for that matter) would be able to look at patterns of firing neurons (in the simulations, or in real nervous tissue) and read-off any connections to elementary particles -- just impossible!
Things are too complex, too enfolded, too dynamic, etc., etc.

Yet -- here one can freeze the action at will; tell exactly which
particular neuron fires when and why; where/what the interference pattern is; where are the sea quarks and where are the valence quarks; etc. etc., and bring out the goings on in terms of patterns of elementary particles dynamics. Also, fantomark strings (thru their 'streaks') can be analogous to 'guiding waves' or carriers of unexpected field effects.

Posted by jdi on 08-07-2004 07:46 AM:

Steganogramic ET messages in junk DNA(?)

Paul Davies speculates in this week's NewScientist that junk DNA may contain hidden messages from advanced civilizations.

See SEQUENCE FROM THE STARS by Paul Davies, NewScientist, August 7-13, 2004 at p. 30. [Not yet available on-line, but should be in a few days].

While it is certainly a wild speculation, it is consistent
with my steganogramic baryon octet paper, also
the "fantomark" strings that I had introduced in that
patent. [links to the paper/patent are provided
elsewhere in this thread].


Added later 08/09/04

Full Davies' paper is below.

The reason I call attention to it is that until very recently
"establishment" science journals refused, in principle, papers on "steganogramic" issues...

While Davies' paper is wildly speculative, NewScientist's promotion of steganogramic stuff may signal a certain shift in attitudes toward spookier modes of "information"
communication. Also, many interpretations of CA
behaviours are essentially thru steganogramic means
of discovery. [Ask Stephen Wolfram...;-))]

Do we have to spell it out?

New Scientist vol 183 issue 2459 - 07 August 2004, page 30

ConTACTinG humAns musT be frusTrATinG if They miss The messAGe ThAt's stArinG Them in The fACe, says Paul Davies

SEARCHING for alien messages is a wild and speculative idea. For more than 40 years, a heroic band of astronomers has been sweeping the skies with radio telescopes in the hope of stumbling across a signal. Though the silence so far has been deafening, this search is buoyed by the belief that
the truth is out there somewhere. But what if the truth isn't out there at all? What if it lies somewhere else? Now may be the time to try a radically different approach.

Even if it turns out to be a hopeless or completely misconceived quest, the search for extraterrestrial intelligence, or SETI, is worth carrying out because it forces us to think deeply about the nature of life and intelligence, and the place of humanity in the universe. The use of radio telescopes in this endeavour is predicated on several questionable assumptions. Even if we take for granted that there are intelligent aliens
who use radio technology, and that they are trying to contact us - already a big leap of faith - the problem of timing remains acute. By common consent and simple statistics, any alien civilisation in our corner of the galaxy is likely to be millions - possibly tens or even hundreds of millions - of years ahead of us technologically. They will have been waiting a very long time for earthlings to come on air. It is inconceivable that ET would beam
signals at our planet continuously for untold aeons merely in the hope that one day intelligent beings might evolve and decide to turn a radio telescope in their direction. But if ET transmits messages only sporadically, the chances of us tuning in at the right time are infinitesimal.

It would be more credible if the aliens could somehow spot the emergence of terrestrial radio technology, so that they begin blasting the airwaves at a time when they have a reasonable expectation that we might be listening. But our own radio signals, travelling across the galaxy at the speed of light, are unlikely to have reached any alien civilisations yet, even using the
most optimistic estimates of SETI enthusiasts. So at this time, ET has no idea that Earth hosts radio astronomers and so no reason to begin signalling us.

An altogether more attractive strategy from ET's viewpoint would be to plant artefacts containing messages in the vicinity of any planets that have the potential to evolve intelligent life at some unknown stage in the future. Then, if and when a technological community emerged on that planet, it would encounter the cosmic calling card on its doorstep. This is a favourite
science fiction theme: remember the obelisk in 2001: A space odyssey?

The problem with this "set-and-forget" technique of communication is that the information content of the message may have to survive for hundreds of millions of years. A conventional artefact placed on the Earth's surface
might be overlooked, and would be subject to the vagaries of tectonic activity, glaciation and other turmoil. In near-Earth orbit it would be even less conspicuous and at the mercy of cosmic radiation, meteorites and solar flares. Obviating these problems by making the artefact physically large would enormously increase the cost of sending it here.

A better solution would be a legion of small, cheap, self-repairing and self-replicating machines that can keep editing and copying information and perpetuate themselves over immense durations in the face of unforeseen environmental hazards. Fortunately, such machines already exist. They are called living cells. The cells in our bodies, for example, contain messages written by Mother Nature billions of years ago.

So might ET have inserted a message into the genomes of terrestrial organisms, perhaps by delivering carefully crafted viruses in tiny space probes to infect host cells with message-laden DNA? It's an idea that has been swirling around for a few years, and has recently been championed by
the Apollo astronaut Rusty Schweickart. But on the face of it, there is a serious problem. Living cells are not completely immune to change. Mutations introduce random errors into the stored information, and over a long enough
time span they would inexorably transform ET's message into molecular gobbledygook.

To minimise the effects of mutations, it would make sense to incorporate the message into a highly conserved segment of DNA. Such segments are normally associated with key coding regions of the genome that control the most vital
functions of the organism. They tend to be unchanged between species, suggesting an ancient origin. Mutations in such regions are invariably fatal. But unfortunately tinkering with them by inserting alien DNA would likely prove as lethal as any random mutation.

Conversely "junk" DNA - sections of the genome that seem to serve no useful purpose - can be loaded with all manner of genetic oddments without affecting the performance of the cells. Inserting a message here would almost certainly be harmless. The trouble is, junk DNA is famous for accumulating lots of mutations. So the choice seems to be between killing
the messenger and compromising the message. What is needed is a region of junk DNA that is also highly conserved.

Until recently, this would have been regarded as an oxymoron. But no more. Genomics researchers at the Lawrence Berkeley National Laboratory in California who compared human and mouse DNA have reported the discovery of
vast, highly conserved sequences of junk DNA (New Scientist, 5 June, p 18). These segments are apparently surplus to requirements. When the researchers
deleted them from the mouse DNA, the animals seemed to be perfectly normal. If ET has put a message into terrestrial organisms, this is surely where to look.

Looking for messages in living cells has the virtue that DNA is being sequenced anyway. All it needs is a computer to search for
suspicious-looking patterns. Long strings of the same nucleotides are an obvious attention-grabber. Peculiar numerical sequences like prime numbers would be a clincher and patterns that stand out even when partially degraded by mutational noise would make the most sense. A great example was given by
cosmologist Carl Sagan at the end of his novel Contact, in which the supposedly random digits of pi, when displayed as a two-dimensional array, unexpectedly contained the figure of a circle. In the same way, if a sequence of junk DNA bases were displayed as an array of pixels on a screen
(with the colour depending on the base: blue for A, green for G, and so on), and a simple image like a ragged circle resulted, the presumption of tampering would be inescapable.

Such a feature would merely serve the purpose of flagging the information. What might the message contain? One segment of DNA excised by the Lawrence
Berkeley team contained more than a million base pairs - enough for a decent-sized novel or a potted history of the rise and fall of an alien civilisation. But the message need not be the last word from ET. Rather, it could tell us how to download the entire contents of Encyclopaedia Galactica by conventional radio or optical techniques.

I am not suggesting that radio SETI be abandoned just yet. The commissioning of the long-awaited Allen Telescope Array in northern California will bring a much larger volume of the galaxy within the scope of current search techniques. Trying to second-guess alien communication strategies is fraught
with uncertainty, so we should try everything we can afford. The truth may be out there somewhere. Or it could be a lot closer to home.

Paul Davies
Paul Davies is at The Australian Centre for Astrobiology at Macquarie University, Sydney, and author of The Origin of Life

Posted by jdi on 09-08-2004 08:04 PM:

Links are dead...

Well, miracles happen! Links came alive somehow!
They seem to work as of this afternoon -- 09-08-04. -- jdi

For those who have tried to link to the two files recently:

Sorry... I am told that the isss server is down (09-01-04)
for re-organization. Second document can always be found
at Search for Patent No. 4,286,330
--jdi [09-08-04]


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