- I submitted your midterm grades yesterday, under somewhat rosey
assumptions about your performance in the future: in
particular, I dropped your two lowest quizzes (which means that
most of you have only a single drop left); and I assumed that
you would amaze and astound with your projects and logos.
- Speaking of projects and logos, next time we'll take a trip up
to visit with Mathemalchemy, to see where we are, and to talk
about your projects.
- Your quiz this week will concern the wallpaper groups, as well as include a novel element: you will produce an image to send to me, to create a gallery of fractals.
I expect that for your quiz you're going to be asked to match a pattern to one of those ....
- Fractals were introduced:
- Informal definition:
A world within a world! there's a perfect copy of the fractal contained within itself -- and perhaps infinitely many!
- We saw examples of fractals:
- A0 paper, which we discovered had side relative ratio of $\sqrt{2} \approx 1.4142$. It, too, can spiral! Not all spirals lead to gold, but those using squares do.....
- "Russian Dolls"
- Fractals in nature (e.g. broccoli, tea oil in lakes,...)
- Fractals in graphs (e.g. Fibonacci's immortal, incestuous rabbits)
- Fractals in the duality of Platonic solids
- World's Most Famous Fractal discovered/created by the late Benoit Mandelbrot
- Koch stick fractal, and other stick and area fractals, most easily produced as "L-System Fractals". We made a stick fractal last time, a variation on the theme of the Koch fractal.
- Spiral homework fractals, e.g.
You'll be making one of very own Spiral Fractal as half of this week's quiz grade.
- Informal definition:
- The key notion behind fractals is formally known as "self-similarity":
"a self-similar object is exactly or approximately similar to a part of
itself (i.e. the whole has the same shape as one or more of the
parts)."
Often we start with an idea or object ("initiator", and we carry out a process ("generator") that results in more of the same; then we "do it again, do it again"....
As we have seen, a fractal is often characterized by these things:
- It's a process or graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative)
definition -- that is, we do something, and then we "do
it again!"
- Start with a straight stick.
- Replace the middle third of the stick with a little "tent" -- two third-length sticks.
- Do it again and again!
Let's do two calculations: of number of sticks, and length of the fractal.
This fractal becomes infinitely long, but in a confined space! Very strange.... but this type of strange behavior is typical of fractals.
There are changes you can make to produce new and interesting fractals.
Let's create one by subtraction.
- I've created a template that's "pre-made" for some triangular fractal creation...
- After we're finished removing all that stuff, how much area is left in the triangle?
The Chaos game - generating fractals using random movement!
- Once again, simple rules lead to complex
objects. This time, however, we don't proceed
methodically, but rather haphazardly - and yet we still
produce the complex images.
One of the most interesting fractals arises from what Michael Barnsley has dubbed ``The Chaos Game'' [Barnsley]. The chaos game is played as follows. First pick three points at the vertices of a triangle (any triangle works---right, equilateral, isosceles, whatever). Color one of the vertices red, the second blue, and the third green.
Next, take a die and color two of the faces red, two blue, and two green. Now start with any point in the triangle. This point is the seed for the game. (Actually, the seed can be anywhere in the plane, even miles away from the triangle.) Then roll the die. Depending on what color comes up, move the seed half the distance to the appropriately colored vertex. That is, if red comes up, move the point half the distance to the red vertex. Now erase the original point and begin again, using the result of the previous roll as the seed for the next. That is, roll the die again and move the new point half the distance to the appropriately colored vertex, and then erase the starting point.
From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose.
