- Today we are going to continue with one of my favorite topics --
Fractals! -- but transition to our final topic: infinity!
- I hope that you're thinking about your mathematical logo. Those
will be due soon....
- Sorry I haven't gotten to the grading from last week -- tests in
other courses have got me behind. I'll have things graded by Thursday.
- Materials for today:
- A video covering today's material: Fractals to Infinity
- My knot fractal and infinity fractal graphics....
- The Harriss fractal (and review of the Golden Rectangle fractal)
- "L-system" fractals:
This one you can do on-line, without downloading anything.
The Koch fractal is given by (you can copy and paste):
- Axiom: f
- Rule 1: f=f-f++f-f
- Angle: 60
Play around, have fun!
- Vi Hart, and Sierpinski's triangle from Pascal's Triangle....
- Finally I asked you to create a fractal image, like my image of
"Flirting with Death Spiral" I shared with you last time:
For that you will go to my special pages
- making fractals out of almost anything!
- Fractals leading to infinity....
- For the first part, I'm going to illustrate how to make fractals out of
almost anything (well....), and I'm going to use a knot, and then the
infinity symbol. The key, as you should already know, is to be able to
"do it again, do it again,...." forever....
- A knot fractal
- An infinity fractal
- For the second part, I want to focus on the Koch fractal one last
time.
- The Koch fractal grows in length each time: the generator is 4/3 times as long as the initiator.
- This means that the fractal grows by a factor of 4/3 with each step.
- If the initial stick has length 1, then the first iteration of the fractal has length 4/3. The second iteration has length $(4/3)^2$, the third $(4/3)^3$, and so on.
- This is an example of exponential growth. If we compute the length of the Koch fractal after 10 iterations, it will have length \[ L = \left(\frac{4}{3}\right)^{10} \approx 18 \]
- If we do it forever, the Koch fractal has length $\infty$!
It's infinitely long, yet lives in a finite space.
This is only our very first example of something really mysterious about infinity....
- Nothing to report.