Main Sequence Page 3
CellularAutomata

Suppose I have become jaded and burnt out by this big-picture cosmology, overwhelmed and weary from all the complexities of the Universe. I wring my hands and say "Why can't things just be simpler?" To the rescue comes a whole subject within mathematics and computer science: Cellular Automata.  (Folks like to abbreviate this 'CA' but I'll mostly stick to spelling it out.)

The idea of cellular automata is to proceed in three simple steps.
1. Set up a tiny self-contained universe in some initial configuration.
2. Define some simple rules for how this universe works.
2. Allow time to move forward in discrete steps, like ticks on a clock.

Then we sit back and watch our small-u universe evolve. (The Big-U Universe is defined as 'everything we can observe'; a cellular automata small-u universe is 'everything we put into a box according to the recipe.'

A couple parenthetical notes:
I think the notion of a deterministic or clockwork Universe is often associated with Aristotelian philosophy. Be that as it may, 20th century physics threw a wrench in the works of determinism but even without quantum mechanics the Universe is way too messy to be pragmatically deterministic. By which I mean predictable in any sort of practical sense beyond a limited scope of time. But this is becoming a messy digression; suffice to say that cellular automata universes are so simple that the notion of determinism moves back into the realm of the practical and this is an interesting aspect of the universe to Universe relationship.

But back to the point:

Is a Cellular Automata universe capable of producing complex surprises from simple rules?

Of course we suspect the answer to be a resounding Yes. Implementing a cellular automata universe is easily done by writing a little computer program that does all the work.  The rules may be simple but the universe usually has hundreds to thousands of locations or cells where those rules apply, so calculating what happens by hand is tedious and computers never complain, unless we consider breaking down a form of complaint.

Once a computerized universe is poised and ready we issue the 'Go!' and sit back and watch it unfold with time. As time marches along in the universe we can decide if it's interesting or not and devote our creative energy to playing around with the rules and the initial conditions.

The trick as I see it is seeking simple rules that give non-boring results. If the rules are too dull then the universe is also dull.  If the rules are too complicated, well, there is an objection to this also, but of a different sort: I started my hand wringing precisely because the real Universe is too complicated. So complex rules are undesirable from an aesthetic point of view... or mine anyway. 

Behind the ideas of cellular automata are the people who came up with them. There is a hero in the world of cellular automata and his name is... wait for it... John Conway.  There is a second hero of cellular automata named Alan Turing. The third hero is John von Neumann. There are other heros as well but Drs. Conway, Turing and von Neumann will carry the day on this particular page of the internet. This link-placeholder-page will be a very short historical sketch of these fellows. I mention them because as inventors they have given us a tremendous collection of mathematical gifts that will always be with us; so to me they are much more amazing than any athlete. (I'm proud to say I own an original von Neumann rookie card.)

1-Dimensional Cellular Automata
Here is a 1-dimensional cellular automata allowed to evolve in time, where successive time steps are showed in a downward progression. The rules encoded into this pattern-generating algorithm are precisely the same as those used in determining the even/odd status of elements of Pascal's triangle: Even + Even = Even, Odd + Odd = Even, Odd + Even = Even + Odd = Odd.

 1D CA

Here is the same (Sierpinski) triangle generated using a 2-D algorithm.

skewed serpinski

These two figures look practically identical and yet are produces via different algorithms: The first algorithm treats the "down" direction as time ticking along and plots the evolution of a 1-dimensional strip of cells. The second algorithm starts off on a two-dimensional plane at some spot, chooses one of the three corner vertices at random and hops halfway there, drawing a little dot when it lands. This process is repeated ad infinitum until the S-triangle emerges.

The Sierpinski triangle can also be generated using a "replicate and reduce" algorithm that begins with a solid triangle.

2-Dimensional Cellular Automata

N-Dimensional Cellular Automata

Chaotic Behavior
There is a whole branch of science now, 'non-linear science', that tries to address simple systems that exhibit erratic or unpredictable behavior. I don't have the wherewithal to delve into it here but I do want to provide a computer program that can be used to look at a simple and pretty example, the Lorenz attractor. The digression page for this is located here.

What Next?
There are two connections back to the real Universe from the simplified universes of cellular automatae.  The first is that we like simple rules that can account for complexity in both realms; simple rules are appealing to our limited minds. Why this is so is a nice philosophical topic to think about on a rainy Sunday afternoon.

The second connection between the Universe and the little universes arises from this observation: The laws governing the little universes must be carefully chosen to produce interesting behavior. If the laws are not carefully chosen, the universe easily falls into a state of boring uniformity. The connection to this Universe is therefore also somewhat philosophical: How is it that our rules and initial conditions have led to such non-boring results?

But beyond the philosophical this connects to an esoteric result in cosmology: In order for the universe to be this complicated and interesting it had to be infinitesimally close to a perfect sort of balance back when it began.  I do not understand this necessary balance except in very general terms; I think it has something to do with matter/anti-matter asymmetry and the time-evolution of the curvature of space. I'd like to understand this better but the subject will have to come much later on in this sequence of pages, after I figure out some more basic things which will eventually lead to the basis for cosmology. The basis for cosmology is general relativity. The basis for general relativity is special relativity. The basis for special relativity is electromagnetism. The basis for electromagnetism is the nature of light. So next I'd like to go back to experimentation concerning the nature of light.

On page 1 of the main sequence I looked into building a detector that shows the paths of particles as condensation tracks. I'll take the success of that effort as a demonstration that--at the very least--there are small things zinging around that are usually invisible. I'll also accept provisionally that many of these invisible creatures are electrons which are weakly bound to atoms and consequently "available for use" at fairly low initiation energies. This is further bolstered by my personal experience with electrical circuits, not to mention sticking hairpins into electrical outlets, which hurts quite a bit. So: We might make use of electrons by liberating them and this is easy enough to do; we can liberate electrons without having to heat up sulfur to 8000 degrees. We can just use a little battery.

But what does the availability of electrons have to do with light???

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