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 |    Not for the faint of art. |
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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| This article is a few years old now, but I haven't seen anything to contradict it in the last four years, so it's probably still current science. New Model of Evolution Finally Reveals How Cooperation Evolves  By treating evolution as a thermodynamic process, theorists have solved one the great problems in biology. It also addresses a few of the misconceptions about evolution, but there are a few points I'd quibble with. The first misconception is that "survival of the fittest" necessarily equates with "survival of the strongest." Bunnies, for example, didn't get to where they are by being strong; they exist because they're famously fast and, even more famously, prolific reproducers. And they're very cute. One of the great unanswered question in biology is why organisms have evolved to cooperate. The long-term benefits of cooperation are clear—look at the extraordinary structures that termites build, for example, or the complex society humans have created. In other words, we know that intraspecies cooperation (and sometimes interspecies cooperation) exists, but how such cooperation evolved has been an ongoing subject of debate. But evolution is a random process... Uh... not really. There's a random element, sure, but chance isn't the biggest factor. I mention this because some people have used "random chance" as a specious argument against evolution. No, we humans didn't get to where we were able to invent computers merely "by chance," and bunnies didn't suddenly and randomly develop the ability to hop. ...based on the short-term advantages that emerge in each generation. That whole sentence is problematic, but I understand that its purpose is to provide background for later arguments. It's close enough for that purpose, I think. Of course, individuals can cooperate or act selfishly, and this allows them to accrue benefits or suffer costs, depending on the circumstances. But how this behavior can spread and lead to the long-term emergence of cooperation as the dominant behavior is a conundrum that has stumped evolutionary biologists for decades. I'll take that as given. All I'll add is that neither cooperation nor competition is the only way for a species to evolve. I'm sure you can think of plenty of examples of both, but in many cases it's some mixture of the two. Today, that could change thanks to the work of Christoph Adami and Arend Hintze at Michigan State University in East Lansing. They have created a simple mathematical model using well understood physical principles to show how cooperation emerges during evolution. So, basically, while it's a model with solid grounding, it's not like an experimentally verified result. Still, it's worth noting, because the first thing you have to do to draw a conclusion is ensure that your hypothesis is at least plausible. Their model suggests that the balance between cooperation and selfish behavior, called defection, can undergo rapid phase transitions, in which individuals match their behavior to their neighbors. What’s more, a crucial factor turns out to be the process of punishment. Well, good news for BDSM fans. The article goes on to explain, in what I think are very accessible terms (so don't be scared off by the "mathematical model" phrase above), the way they looked at the problem. That’s an interesting result. It implies that behavior can be manipulated on a large scale by the introduction of certain costs. It also implies that the result can be modeled using relatively simple physics. Well, they could have just taken the route of "ask an economist." The only danger there is that if you ask n economists a question, you'll get n+2 different answers. Anyway, it seems to my completely unschooled opinion to boil down to game theory. But it struck me as an interesting result because I'd been thinking about how competition and cooperation interact. Like in a game. Two players sit down to play chess. At first glance, it's a competition: either one will win and the other will lose, or it will end in an unsatisfying stalemate. But the thing is, both players have agreed, formally or not, on a set of rules governing that competition. Someone who breaks the rules is called a "cheater" and no one likes a cheater. The rules are a form of cooperation, and the social disapprobation heaped upon cheaters serves to discourage such behavior. Doesn't matter if the game is chess, football, Monopoly, or whatever. You have to cooperate before you can compete. And even when there's not an actual game involved ("game theory" isn't actually about games), there are rules and consequences for breaking them -- even in the ultimate human competition called war. I know the saying is "all's fair in love and war," but that's simply not the case. Like I said, the title is a bit misleading, because nothing's been settled (or "solved") here -- but it does provide a fresh way of looking at things, and sometimes that's all it takes to eventually make a breakthrough. Another advantage of cooperation. |
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