Day 28, MAT115

1. All the knots you need to know....
2. Solution to the Knotty Tale
3. You need to distinguish between the two five-knots
4. What's the difference between the trefoil knot and 3-knot? (Trick question!)
5. Almost everyone missed the complicated unknot....
6. Tricoloring....

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.

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)."

• 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"....

• 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)

1. It's a process or graph that possesses "self-similarity".
2. It often possesses a simple and recursive (or iterative) definition.

• a simple rule
• repeated over and over and over.... but
• simple rules, intense content!

• Let's see the process, but a little slower (check out the beautiful bushes...).

• 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!
    • We've discovered that simple rules may lead to complex objects. This time, however, we don't proceed methodically -- deterministically -- but rather haphazardly - and yet we still produce the complex images!
      1. Draw an equilateral triangle.
      2. Number the vertices 1, 2, 3 (clockwise, from the top).
      3. Start anywhere on the page, by putting a dot.
      4. Now the simple rule, to be done again and again, is to roll a die. If it's
         • 1 or 4, go halfway to 1
         • 2 or 5, go halfway to 2
         • 3 or 6, go halfway to 3
         You'll generate a beautiful Siepinski Gasket!

    • 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....

1. Construct a rather large Pascal's triangle (say up to the eighth row).
2. Color all even numbers one color, all odds another. What do you find?