- Your recent assignment is returned.
- If you only see three crossings, you can't have a four (or five, or six) knot!
- A transition from one four knot to the other.
- Only trefoil and 6_1 are tricolorable in the first knots with six or fewer crossings:
Tricolorable 6 knotI wish that the trefoil in black were red -- that would make the tricolorability more apparent. You do see that it doesn't matter, right? If we colored it red, it would clearly be tricolorable; and once tricolorable, always tricolorable....
Notice that it's just a trefoil with another trefoil glued onto it! That's a way to find more complicated knots that are tricolorable -- just splice any old knot, colored one of the three colors, onto the same-colored strand of a trefoil knot.
- I've posted your assignment about Mobius bands.
- Reminder: our next exam is Monday, 11/21 -- one week from today. It will cover through Mobius bands.
- Question of the day: What is a fractal?
- Fractals
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- 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! (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. 102):
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