the pictures (some pictures won't be there yet, since you may have embedded them in a word doc, or just recently sent them).
If you haven't given me
County,
Precinct info, and
exact vote totals,
I don't have enough from you! Check what you sent me, and make sure that you gave me all of that....
What do I do with this?
"Out of the twenty members we had participated
only one voted for Conway."
Please be absolutely specific about the vote! How
many for Conway, how many for Paul.
Today: More Chaos
Laplace's notion: that, given the positions and velocities of all
the particles in the universe, we should be able to predict the future
of "the system" (the universe) exactly for all time. This is the
"determinism" in "free will versus determinism". Fortunately for free
will, determinism is shown to fail.
Chaos is roughly defined as "sensitive dependence on initial
conditions", but what this means is that if you start at two different
points in a system (for example, in weather you might be thinking of a
state of three variables, temperature, barometric pressure, and
wind speed), very close to each other but distinct, then eventually
the system will produce wildly different behaviors (even those the
starting conditions were extremely close).
Some classic examples of chaos
Hexstat Probability Generator -- out of chaos comes some predictability
"The Swinger"
the
fractal diagram, showing which magnet will result if you
start at a given point (each of the five magnets on the
pentagon is associated with a color)
The "bifurcation diagram". This tells us that, as we vary r, the behavior of the system is different. And at particular values of r, fairly dramatic differences occur.
A related topic (complicated dynamics from simple rules): introduction to the Game of Life:
The game is generally played on an infinite checkerboard, or on a
world that wraps back around on itself.
The rules are:
A living square will remain alive in the next generation
if exactly two or three of the adjoining eight squares
are alive in this generation; otherwise, it will die.
A dead square will come to life if exactly three of its
adjoining eight squares are alive; otherwise, it will
remain dead.
Create your own creatures!
Can you think of one that will lead to extinction?
Can you think of one that will lead to a stable
pattern?