Some Results

I’ve recently heard requests to put some concrete results on this blog, and they are splendidly welcome. When covering this topic, the temptation to cover one’s philosophical bases is almost overwhelming, but the foundations can wait. This blog entry is about some of what I’ve achieved.

It’s important to say upfront that the research I’ve been doing is not physics. At least not yet. I don’t have a grand theory for how the universe works, and I’m not trying to advertise one. What I’m trying to do instead is make a point about tools.

Physics is founded on Calculus and continuum mathematics because they are tools that deliver results. They have delivered more concrete progress than any other modeling system that the human race has ever developed. However, now they are failing. The standard model was presented in its current form in 1974. Relativity and Quantum Mechanics have resisted integration since around 1905. Thus, the most important frontier in physics has yielded only limited progress in the last hundred years, and virtually none in the last forty. String Theory, while terribly grand and clever, is so amorphous that it predicts ten to the five hundred different possible sets of physical laws and has no predictive power to speak of. In my opinion, this is because the tools in use are reaching the limits of their applicability.

Nevertheless, we cannot expect physicists to believe this, or to change the tools they use, because at this point, swapping to any other modeling system entails a massive step backwards. This step is one that only a few very brave souls are willing to take. (Frankly, they’re braver than me because I’m not a career physicist and I have nothing to lose.) Therefore, it’s very likely up to someone else–someone outside the physics community–to start producing tools that can do what continuum mathematics cannot.

The first, most important part of this task, IMO, is catching up with the last three hundred years of scientific progress. As is hopefully clear from this blog, the tools that I believe will help are those that have been developed in Computer Science. The goal then is the replication of the total set of observed symmetries of nature in a discrete, iterative system that is no more complex than strictly necessary. This includes rotational and Lorentz invariance, the wave properties of Quantum Mechanical systems that are customarily modeled through the use of Hilbert Spaces, and everything else. This includes all those symmetries employed by Gauge Theory such as SU3, if such things prove necessary under the new system. Physics hinges on symmetry. Once the symmetries can be painlessly reproduced, things will go more smoothly.

The easiest place to start seemed to me to be rotational invariance, and this is what my first paper was about. The aim was to produce a discrete medium and an iterative function that could be applied to the elements of that medium that would produce a pattern that moved equally well in all directions. For those of you familiar with Cellular Automata (CAs), the goal, if you like, was to produce a universal glider that could travel equally well in any direction, rather than just in diagonal lines. The difficulty here is that discrete systems have a limited number of degrees of freedom. That makes travel in more directions than you have degrees of freedom a challenge.

Various systems have been tried to produce such a universal glider. One such system is to use a grid as the discrete medium and to define a glider with motion described by some number of steps along each axis with each turn. For instance, to go North North East, the glider might take three steps North for every step East. One problem with this approach is that to change direction just a small amount can require enormous changes in the ratios of motion along each axis. Thus, in order to produce motion in all directions, the glider needs an effectively infinite memory in which to store what part of its movement cycle it’s in at any time, along with a mechanism for converting between axis ratios when a change in direction is required.

Another problem is that this model has trouble compensating for the kind of spatial distortion witnessed in Relativistic systems. Specifically: there’s no room in the model for spatial expansion or contraction. That means no Big Bang, at least, not one that’s compatible with cosmological observations. Ideally, we’d like to choose a model that rules out none of the kinds of behavior we’d like to later produce.

Another approach that’s been explored is to once again use a grid, but to have the glider change axis of motion with each step based on some probability function. Thus when headed NNE, this glider has a 75% likelihood of going North, and a 25% likelihood of going East, but we don’t know which way it’ll turn for each step. While this approach gets around the problem of the awkward issue of ratios on different axes, it replaces it with dependency on a continuously varying probability value. Such variables are exactly the kind of tools we’d like to avoid using. Furthermore, the use of a grid once again rules out large chunks of Relativity.

What I do instead is use a densely-connected, irregular graph as my discrete medium, and define my glider as a function operating over sets of nodes on that graph. I define two sets, front nodes and back nodes, if you like, and then employ a simple algorithm I call ‘Jellyfish’ to find a new set of front nodes with each iteration.

The formula for Jellyifish is outlined in my NKS Midwest 2008 presentation slides, which you can find here:
http://www.cs.indiana.edu/%7Edgerman/2008midwestNKSconference/Lamb_Slides.pdf
The slides outline the formula, so I won’t duplicate it here.

If you want a more in depth explanation, my paper on this system will be published in the journal Complex Systems shortly, but if you don’t want to wait, send me your email address and I’ll send you a preprint. Alternatively, if you’d just like to see the results, you can alway go to YouTube and watch the glider, or ‘pseudo-particle’ moving for yourself.
You can find it here:
http://www.youtube.com/user/alexlamb#p/u/8/Y_yCxcjYPmo

Using an irregular graph means that the bulk properties of the medium are the same in every direction. There are no preferred directions of motion, so most of the anisotropy problems associated with Cellular Automata disappear immediately. Furthermore, the medium can be distorted in any way we like. Its geometry is not fixed. This means that nothing is stopping us from exploring the implications of Relativity later. Defining a pseudo-particle in terms of operations over sets of nodes also allows us to define orientation of motion as a group effect, and thus to describe motion over an arbitrarily large number of directions with ease.

To some, the Irregular Graph/Jellyfish approach feels rather more random than Cellular Automata, more fundamentally complex, and certainly less likely to produce pretty patterns. However, though we lose a little in terms of algorithmic succinctness, we seem to gain at least as much in terms of descriptive power, and, as you’ll hopefully see in later posts, what we gain often looks eerily like physics.

The Jellyfish algorithm works equally well in 2, 3 or any dimensionality, as well as on curved surfaces. It shows potential compatibility with Lorentz invariance, as I illustrate in the slides, and even some properties similar to those of Quantum Mechanical systems. What Jellyfish doesn’t have is wavelength, polarization, or the habit of following all paths at once. It’s a long way from being a physical particle, and that’s okay because it’s not supposed to be one. However, what it does demonstrate is that getting something like basic particle behavior out of a discrete system is extremely easy. Natural, even.

I believe that this work work leaves us with a new class of automata to explore, and an important question to answer: What is the simplest algorithmic model that fulfills the constraints that physical law imposes on a system, without resorting to the classical formalism of that system. In other words, if we tie one hand behind our back, and forgo the use of differential equations, axes, and smooth numbers, can we still wield the rapier of science? I bet we can. In solving this and similar puzzles, we may be opening the doors to a new era of science. The answers are just a few simple experiments away and anyone with a computer and a little curiosity can start looking.

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