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.