by Rudy Rucker
looking or, in particular, of an endlessly detailed fractal nature, they are called strange attractors.
To return to the surf example, you might notice that the waves near a rock tend every so often to fall into a certain kind of surge pattern. This recurrent surge pattern would be an attractor. In the same way, chaotic computer simulations will occasionally tighten in on characteristic rhythms and clusters that act as attractors.
But if there is a storm, the waves may be just completely out of control and choppy and patternless. This is full-blown chaos. As disorderliness is increased, a chaotic system can range from being nearly periodic, up through the fractal region of the strange attractors, on up into impenetrable messiness.
Quite recently, some scientists have started using the new word complexity for a certain type of chaos. A system is complex if it is a chaotic system that is not too disorderly.
The notions of chaos and complexity come from looking at a wide range of systems - mathematical, physical, chemical, biologi-
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cal, sociological, and economic. In each domain, the systems that arise can be classified into a spectrum of disorderliness.
At the ordered end we have constancy and a complete lack of surprise. One step up from that is periodic behavior in which the same sequence repeats itself over and over again - as in the structure of a crystal. At the disordered end of the spectrum is full randomness. One notch down from full randomness is the zone of the gnarl.
NO DISORDER LOW DISORDER GNARLY HIGH DISORDER
MATH Constant Periodic Chaotic Random
MATTER Vacuum Crystal Liquid Gas
PATTERN Blank Checkers Fractal Dither
FLOW Still Smooth Turbulent Seething
Table 1: Spectrums of Disorderliness for Various Fields.
As an example of the disorderliness spectrum in mathematics, let's look at some different kinds of mathematical functions, where a function is a rule or a method that takes input numbers and gives back other numbers as output. If f is a function, then for each input number x, the function fassigns an output number f(x). A function fis often drawn as a graph of the equation y = f(x), with the graph appearing as a line or curve on a pair of x and y axes.
The most orderly kind of mathematical function is a constant function, such as an ffor which f(x) is always two. The graph of such a function is nothing but a horizontal line.
At the next level of disorder, we might look at a function ffor which f(x) varies periodically with the value of x. The sine function sin(x) is an example of such a function; it fluctuates up and down like a wave.
The gnarly zone of mathematics is chaos. Chaotic functions have finitely complicated definitions, but somewhat unpredictable patterns. A chaotic function may be an extremely irregular curve, unpredictably swooping up and back down.
A truly random mathematical function is a smeared-out mess that has no underlying rhyme or reason to it. A typical random function has a graph that breaks into a cloud of dots, with the curve continually jumping to new points.
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Formally, something is truly random if it admits to no finite definition at all. It is an old question in the philosophy of science whether anything in the universe truly is random in this sense of being infinitely complicated. It may be the whole universe itself is simply a chaotic system whose finite underlying explanation happens to lie beyond our ability to understand.
Before going on to talk about the disorder spectrums of the Matter, Pattern, and Flow rows in Table 1, let's pause to zoom in on the appearance of the math row's disorderliness spectrum within the gnarly zone of chaos. This zoom is shown in Table 2.
LESS DISORDER MORE DISORDER CRITICAL HIGH DISORDER
CHAOS Quasiperiodic Attractor Complex Pseudorandom
Table 2: Spectrum of Disorderliness for the Chaos Field.
The most orderly kind of chaos is "quasiperiodic," or nearly periodic. Something like this might be a periodic function that has a slight, unpredictable drift. Next comes the "attractor" zone in which chaotic systems generate easily visible structures. Next comes a "critical" zone of transition that is the domain of complexity, and which is the true home of the gnarl. And at the high end of disorder is ''pseudorandom" chaotic systems, whose output is empirically indistinguishable from true randomness - unless you happen to be told the algorithm which is generating the chaos.
Now let's get back to the other three rows from Table 1, back to Matter, Pattern, and Flow.
In classical (pre-quantum) physics, a vacuum is the simplest, most orderly kind of matter: nothing is going on. A crystalline solid is orderly in a predictable, periodic way. In a liquid the particles are still loosely linked together, but in a gas, the particles break free and bounce around in a seemingly random way. I should point out that in classical physics, the trajectories of a gas's particles can in principle be predicted from their starting positions - much like the bouncing balls of an idealized billiard table - so a classical gas is really a pseudorandom chaotic system rather than a truly random system. Here, again, chaotic means "very complicated but having a finite underlying algorithm."
In any case, the gnarly, complex zone of matter would be identi-
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fied with the liquid phase, rather than the pseudorandom or perhaps truly random gas phase. The critical point where a heated liquid turns into steam would be a zone of particular gnarliness and interest.
In terms of patterns, the most orderly kind of pattern is a blank one, with the next step up being something like a checkerboard. Fractals are famous for being patterns that are regular yet irregular. The most simply defined fractals are complex and chaotic patterns that are obtained by carrying out many iterations of some simple formula. The most disorderly kind of pattern is a random dusting of pixels, such as is sometimes used in the random dither effects that are used to create color shadings and gray-scale textures. Fractals exemplify gnarl in a very clear form.
The flow of water is a rich source of examples of degrees of disorder. The most orderly state of water is, of course, for it to be standing still. If one lets water run rather slowly down a channel, the water moves smoothly, with perhaps a regular pattern of ripples in it. As more water is put into a channel, eddies and whirlpools appear - this is what is known as turbulence. If a massive amount of water is poured down a steep channel, smaller and smaller eddies cascade off the larger ones, ultimately leading to an essentially random state in which the water is seething. Here the gnarly region is where the flow has begun to break up into eddies with a few smaller eddies, without yet having turned into random churning.
In every case, the gnarly zone is to be found somewhere at the transition between order and disorder. Simply looking around at the world makes it seem reasonable to believe that this is the level of orderliness to be expected from living things. Living things are orderly but not too orderly; chaotic but not too chaotic. Life is gnarly, and A-life should be gnarly too.
Sex
When I say that life includes gnarl, sex, and death, I am using the flashy word "sex" to stand for four distinct things:
Having a body that is grown from genes
Reproduction
Mating
Random genetic changes.
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Let's discuss these four sex topics one at a time.
The first sex topic is genes as seeds for growing the body.
All known life forms have a genetic basis. That is, all living things can be grown from eggs or seeds. In living things, the genes are squiggles of DNA molecules that somehow contain a kind of program for constructing the living organism's entire body. In addition, the genes also contain instructions that determine much of the organism's repertoire of behavior.
A single complete set of genes is known as a genome, and an organism's body with its behavior is known as the organism's phenome. What a creature looks like and acts like is its phenome; it's the part of the creatures that shows. (The word "phenome" comes fr
om the Greek word for "to show"; think of the word ''phenomenon.")
Modern researches into the genetic basis of life have established that each living creature starts with a genome. The genome acts as a set of instructions that are used to grow the creature's phenome.
It is conceivable that somewhere in the universe there may be things with phenomes that we would call living, but which are not grown from genomes. These geneless aliens might be like clouds, say, or like tornadoes. But all the kinds of things that we ordinarily think of as being alive are in fact based on genomes, so it is reasonable to base our investigations of A-life on systems which have a genetic basis.
If we're interested in computer-based A-life, it is particularly appropriate to work with A-life forms whose phenomes grow out of their genomes. In terms of a computer, you can think of the genome as the program and the phenome as the output. A computer A-life creature has a genome which is a string of bits (a bit being the minimal piece of binary information, a zero or a one), and its phenome includes the creature's graphic appearance on the computer's screen. Keep in mind that the phenome also includes behavior, so the way in which the creature's appearance changes and reacts to other creatures is part of its phenome as well.
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The second sex topic is reproduction.
The big win in growing your phenome from a small genome is that this makes it easy for you to grow copies of yourself. Instead of having to copy your large and complicated phenome as a whole, you need only make a copy of your relatively small genome, and then let the copied genome grow its own phenome. Eventually the newly grown phenome should look just like you. Although this kind of reproduction is a solitary activity, it is still a kind of sex, and is practiced by such lowly creatures as the amoeba.
As it happens, the genome-copying ability is something that is built right into DNA because of the celebrated fact that DNA has the form of a double helix which is made of two complementary strands of protein. Each strand encodes the entire information of the genome. In order to reproduce itself, a DNA double helix first unzips itself to produce two separate strands of half-DNA, each of which is a long, linked protein chain of molecules called bases. The bases are readily available in the fluid of any living cell, and now each half-DNA strand gathers unto itself enough bases to make a copy of its complementary half-DNA strand. The new half-DNA strands are assembled in position, already twined right around the old strands, so the net result is that the original DNA genome has turned itself into two. It has successfully reproduced; it has made a copy of itself.
In most A-life worlds, reproduction is something that is done in a simple mechanical way. The bitstring or sequence of bits that encodes a creature's program is copied into a new memory location by the "world" program, and then the two creature programs are run and the two phenotypes appear.
The third sex topic is mating.
Most living creatures reproduce in pairs, with the offspring's genome containing a combination of the parents' genomes. Rather than being a random shuffling of the bases in the parents' DNA, genomes are normally mated by a process known as crossover.
To simplify the idea, we leave out any DNA-like details of genome reproduction, and simply think of the two parent genomes as a chain of circles and a chain of squares, both chains of the same
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length. In the crossover process, a crossover point is chosen and the two genomes are broken at the crossover point. The broken genomes can now be joined together and mated in two possible ways. You can have squares followed by circles, or circles followed by squares. In real life, only one of the possible matings is chosen as the genome seed of the new organism.
In computer A-life, we often allow both of the newly mated genomes to survive. In fact, the most common form of computer A-life reproduction is to replace the two original parent programs by the two new crossed-over programs. That is to say, two A-life parents often "breed in place."
In a world where several species exist, it can even sometimes happen that one species genome can incorporate some information from the genome of a creature from another species! This phenomenon is called "exogamy." Although rare, exogamy does seem to occur in the real world. It is said that snippets of our DNA are identical to bits of modern cat DNA. Gag me with a hairball!
The fourth sex topic involves random changes to the genome.
Mating is a major source of genetic diversity in living things, but genomes can also have their information changed by such randomizing methods as mutation, transposition, and zapping. While mating acts on pairs of genomes, randomization methods act on one genome at a time.
For familiar wetware life forms like ourselves, mutations are caused by things like poisons and cosmic rays. Some mutations are lethal, but many of them make no visible difference at all. Now and then a particular mutation or accumulation of mutations will cause the phenome to suddenly show a drastically new kind of appearance and behavior. Perhaps genius, perhaps a harelip, perhaps beauty, perhaps idiocy.
In the A-life context, where we typically think of the genome as a sequence of zeroes and ones, a mutation amounts to picking a site and flipping the bit: from zero to one, or from one to zero.
Besides mutation, there are several other forms of genome randomization, some of which are still being discovered in the real world and are as yet poorly understood.
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One interesting genome changer is known as transposition. In transposition, two swatches of some genomes are swapped.39
Another genome randomizer that we sometimes use in A-life programs is zapping, whereby every now and then all of some single creature's genome bits are randomized. In the real world, zapping is not a viable method of genetic variation, as it will almost certainly produce a creature that dies instantly. But in the more forgiving arena of A-life, zapping can be useful.
In the natural world, species typically have very large populations and big genomes. Here the effects of mating - sexual reproduction - are the primary main source of genetic diversity. But in the small populations and short genomes of A-life experiments, it is dangerously easy for all the creatures to end up with the same genome. And if you cross over two identical genomes, the offspring are identical to the parents, and no diversity arises! As a practical matter, random genome variation is quite important for artificial life simulations.
Death
What would life be like if there were no death? Very crowded or very stagnant. In imagining a situation like no death, it's always a challenge to keep a consistent mental scenario. But I'm a science fiction writer, so I'm glad to try. Let's suppose that Death forgot about Earth starting in the Age of the Dinosaurs. What would today's Earth be like?
There would still be lots of dinosaurs around, which is nice. But if they had been reproducing for all of this time, the dinosaurs and their contemporaries would be piled many hundreds of meters deep all over Earth's surface. Twisted and deformed dinosaur mutations would be plentiful as well. One might expect that they would have eaten all the plants up, but of course there would be no death for plants either, so there would be a huge jungle of plants under the mounds of dinosaurs, all of the dinos taking turns squirming down
39. John Ronnie, "DNA's New Twists," in Scientific American, March 1993, pp. 122132 contains a discussion of transposition and some of the other methods of genome variation being currently investigated.
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to get a bite. The oceans would be gill to gill with sea life, and then some. I think of the Earth before Noah's flood.40
Would mammals and humans have evolved in such a world? Probably not. Although there would be many of the oddball creatures around that were our precursors, in the vast welter of life there would be no way for them to select themselves out, get together, and tighten up their genomes.
An alternative vision of a death-free Earth is a world in which birth stops as well. What kind of world would that lead to? Totally boring. It would be nothing but the same old creatures
stomping the same old environment forever. Like how the job market looks to a young person starting out!