Day 21, MAT115

• You have your second exam this week. Think of it as three quizzes together....

  It will be over fractals, wallpapers, Platonic solids, and symmetry.

  We'll review tomorrow.

• You have a homework that will be a small part of the exam: to make some pretty knots, and take a picture, and send to me.

  Then we'll have another art contest.

• Reminder: You're invited! 2024 Math/Art Pop-Up Conference We'd love to have you:
  • Fun activities
  • Free lunch! (Yes, there is!)
  • Here's the link to sign up!
  • Schedule:
    • 9:00: Hyperbolic crochet
    • 10:30: Origami
    • 11:30: Lunch
    • noon: Korean drumming

  This could be an opportunity to create something that could serve as the focal point of your project.
• Let's have a look at the winners (and runner-ups) from the Fractal Art Gallery!

• We began by coming up with a slightly different way of drawing all the knots or links that can be obtained with a twisted band (all those bands with from 0 to five twists). We call these "toroidal knots", because we can think of them as being drawn along the surface of a torus (a donut):
  • The unknot
  • The unlink
  • The Hopf link
  • The trefoil knot
  • Solomon's "Knot" (which is actually a link)
  • The cinquefoil (five leaved) knot

• Then we talked about twist knots, which can't be obtained from a twisted band (although the first, the trefoil is both a toroidal knot and a twist knot); but come about by modifying a trefoil knot:
  • The trefoil knot
  • The figure-eight knot
    
  • The other five knot: $5_2$

• I showed a table of the first bunch of knots (up to seven crossings):
  

  You only need to know the first row!:) But can you see some of the other toroidal and twist knots down the road?

How can we turn a trefoil knot into the Borromean rings?

• But first let's talk about that quiz....
  • Median: 6 (that's rough)
  • The Key

• The trefoil knot is our simplest interesting knot. Can we make a simpler knot -- one with either 1 or 2 crossings? Why not?

• There are two different ways to think about the trefoil knot: as a
  1. torus knot, or as a
  2. "twist knot".

  The twist formulation (which is more like what we drew last time) is a good place to start in order to draw the Borromean rings:

  

• Now let's migrate our discussion of knots over to "Knot Mosaics" (and other "mechanical" ways to draw knots):

  • Celtic Knot Mosaic Examples: (not quite the same idea, but pretty! And again, you'll see that there are links contained therein -- including Borromean rings)
    • Triangular paper (in case you want to try making some of those).

  • Knot Mosaics
    • help us to draw Borromean Rings, for example: use your Mosaic Graph Paper to create your own version of the Borromean Rings.
    • Rectangular Mosaic Tiles and symmetry
    • We can use them to make any knots: try your hand!

• Something fun to do with a few friends: can we make a human knot in the shape of a trefoil knot? If so, then we won't be able to undo it -- you and your friends will be knotted together forever! I hope that's a good thing....

• Triangle and rectangle paper
• Shodor:
  1. Snowflake on-line
  2. Gasket on-line
  3. The Fractal Microscope (click on the image)

• L-Systems in PostScript, from which I generated this image:
  

• Vi Hart
  • Vi Hart draws fractals
• L-System Fractals
• Nice website with examples of Cantor, Koch, and other fractals
• Here are some more lovely examples
• l-system generator: world's simplest math tool
• I first encountered the L-Systems fractals at britton.disted.camosun.bc.ca/fractals_arcytech/lsystems.html, but that's a dead link. However, I copied the files, and have a local copy here. However, web browsers these days don't seem to like the applet tag in javascript, so this doesn't work anymore in a standard web browser...:(