- Our next exam is a week from today, Thursday, 11/18. It will cover
- Platonic Solids
- Rings and Knots
- Mobius bands
- Graphs
- Fractals
- Voting
- Chaos
- Our next topic is logic (as "played" by Lewis Carroll). Following
that we'll study infinity, and that will be it! The semester
draws to a close....
- Election results (so far):
- the
pictures and descriptions (everything should be there now!
Check if you don't see your info....).
- The grading (which I've not finished yet) is based on these things:
- Did you give me what I asked for?
- Clear vote results, with precinct info.
- A photo
- The writing
- Style
- Quality (spelling, grammar, etc.)
- Did you give me what I asked for?
- the
pictures and descriptions (everything should be there now!
Check if you don't see your info....).
- You should have received a reply if you sent me a project proposal
revision. If not, please let me know (send a follow-up email, using
your project email to reply, as always), and I'll get back to you.
What questions do you have about the projects?
Projects will be presented in class the last week of classes, and I will put them on the web (we'll be creating a new book of great ideas in mathematics). You'll have at most 7.5 minutes to present!
- Iterated maps: last time we had a look at one particular iterated map, called
the logistic map:
- The recurrence relation:
- The graph of
is a parabola, so we shouldn't be surprised that a parabola appears in our diagrams. There are several graphical means we have of understanding the "dynamics" of this map (how it changes in time, or as we iterate).
- One is the idea of "Cobwebbing". Let's
try an example: use r=3.5, and
- Here's a nice animation of the process, including the cobwebbing.
- Excel makes this easy!
- 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.
- The recurrence relation:
- A related topic (complicated dynamics from simple rules): John Conway's Game of Life:
- The game is generally played on either
- a finite checkerboard (with edges),
- an infinite checkerboard, or
- 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.
- 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.
- Now: let's create our own universe, using this version of
the Game of Life:
www.bitstorm.org/gameoflife/
(with a finite checkerboard)
- Can you think of a pattern of cells that will lead to "extinction"?
- Can you think of one that will lead to a stable (unchanging) universe?
- Now: on what basis could we argue that this is a chaotic "dynamical system"?
- The game is generally played on either
