- Return of your graph homework (from this worksheet
on the web).
- Some of you forgot problem 0....
- In this problem, a tree is useful -- it helps you to realize that there are repeated vertices in the nine solutions! In other words, the problem is actually asking that you repeat no road twice -- a type of Eulerian path.
- Some forgot connectedness....
- The answer to 3 was at the bottom of 4!
- The "Knotty Tale" homework is due....
- You have a new assignment (due next Tuesday)
- Reminder: on 4/26, we'll have "Logo Day". It's about time that you
began wrapping up your logo. What -- you haven't started yet?! Time to
get started....
From the syllabus:
LOGO: Companies have logos; you should have one, too! Your math logo will be something (like a family crest) that represents you. It will be created using elements from this course (or other mathematical elements of your own choosing). You will write a one page sheet, illustrating and explaining your choice. These will be presented the last day of the course.
- Let's get back to that question about what happens when you cut a Mobius band into thirds: it turns out that you can think of it using a diagram, that hopefully will explain the strange result: The last "Question of the day" was:
What if you cut a Mobius band in thirds?
- Our intuitive definition of a Fractal: "A world within a world!"
This key notion is more formally called "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)."
- Some examples you have seen before:
- Our Fibonacci spirals
- The tree diagram of the rabbit problem that gave rise to the Fibonacci numbers.
- The "Russian Dolls" of the Platonic solids (e.g. within a tetrahedron is another, smaller tetrahedron, and so on forever....).
- Often these arose out of the idea that we carry out a process, then "do it again, do it again"....
- Some examples you might have seen before:
- Here's a suitable mascot for this unit: Bessie the fractal cow:
- Have you ever lined up a couple of mirrors and seen an infinite number of
copies of yourself disappearing off into the distance?
Well, try my funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them on our "website gallery".
- Romanesco broccoli -- a fractal in nature
- World's Most Famous Fractal (Benoit Mandelbrot, Novel Mathematician, Dies at 85: 10/14/2010 -- a good image of him)
- Here's a suitable mascot for this unit: Bessie the fractal cow:
- So what is a fractal? A
couple of things we'll generally see:
- It's a process or graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative)
definition.
- Let's have a glimpse at a beautiful gallery of natural fractals.
- Deterministic
fractals: delve into an infinite, dynamic world, created by
a simple mathematical process:
- a simple rule
- repeated over and over and over.... but
- simple rules, intense content!
- Building deterministic fractals with sticks
- Let's see the process, but a little slower (check out the beautiful bushes...).
- Now let's introduce a little bit of randomness:
- Sierpinski triangle (p. 102):
- First deterministically: fractals by subtraction
- 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!
- Here's
a better version of that, that has more general options.
From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose....
- Sierpinski triangle (p. 102):
