My challenge was to find a rule with fixed point behavior, one with oscillating behavior, with period 2, and one with complex behavior, leaving you to judge what complex means. There are many answers to these questions. There's many rules that have these properties. I picked some examples. My fixed point rule was Rule 232. So let me go to that and I'll show you what it looks like. This is a majority voting rule. That is, if you took the majority state in each neighborhood, and used that as the center cell update, this is what you would get. The neighborhoods with majority white go to white, and so on. OK let's see what this looks like. Well, after an initial random period, we get this fixed point pattern, with bars of black and white. That was my fixed point rule. My oscillating rule was Rule 127. You can see that it oscillates with a period of 2, between two different patterns. If I picked another random initial configuration, it would look slightly different, but pretty much the same pattern. My complex rule was a famous complex rule among the elementary cellular automata, one that has been very well studied, and that is Rule 30. So if I do a "Setup " and "Go", we get another hard-to-describe behavior. Let me do this starting from a setup of one cell on. You can see the interesting looking behavior that you get from that. Well Rule 30 along with Rule 110 were two very well-known elementary cellular automata that give complex behavior and were well studied by Stephen Wolfram. In fact it was observing the behavior of these rules in particular, that got him particularly interested in cellular automata. Later, after studying it for many, many years, Wolfram said in Forbes magazine, "The Rule 30 automaton is the most surprising thing I've ever seen in science "It took me several years to absorb how important this was." "But in the end, I realized that this one picture contains " "the clue to what's perhaps the most long-standing mystery in all of science, " "where, in the end, the complexity of the natural world comes from." So if we look back at what the rule looks like, what Wolfram's saying I think, is that the rule which produces very complex patterns, does so as an emergent behavior out of extremely simple rules. Even just with, as you recall, one cell starting at black, and the rule which is specified by eight bits, you get these enormously complex patterns. Wolfram felt that that was a clue for thinking about how complexity could arise from simple rules in the real world. I'm not sure I'd go as far as Wolfram, to say that this is the most surprising thing I've ever seen in science, but it is quite intriguing. More practically, Wolfram actually patented this rule's use as a pseudo-random number generator.