- Homework set returned:
- Platonic Euler
- Can't find a soccer ball?
- Easier edges.
- Harder edges!
- Applications of Platonics
- There's a homework due today.
- It's time to start thinking of your projects.... Recall:
"You will select a mathematical topic outside of (or beyond) those covered in our class and textbook, read and teach yourselves any necessary background to understand it, and then write up a four-page description just like those Tony Crilly presents in our textbook. You are strongly encouraged to work together in pairs on this project: by working together, you can learn from each other and share the burdens and joys of the experience."
- Moebius bands are a part of geometry, obviously, but
relate to a special area called "topology".
Riddle: What's the difference between a donut and a coffee cup?
Answer: There isn't any! (to a topologist....)
- What is a Moebius band?
- Let's make some!
- How do we know that the moebius band is not "topologically equivalent" to a regular band?
- What happens when you cut a moebius band "in
half"? Let's check out a bug on a band....
- What if you cut it into "thirds"?
- By the way, did you know that there are two different
Moebius bands? They're mirror images of each other! (Symmetry, again....)
- Check yours against someone else's, and convince yourself that they're really different.
- Make one of each!
- My dad loved moebius bands so much....
- Applications of Moebius bands:
- Conveyor belts
- Kazakhstan's National Library in Astana
- Recycling Topology
- a history of the recycling symbol
- Recycling -- Moebius, or only Seemingly Moebius?
- A joke: "Why did the chicken cross the Mobius band?"
- 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. 436):
- 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! (p. 447)
- 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. 436):
