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 shortly, after we wrap up something with a little knot...

• 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 tried our hands at 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

• Links:
  • The most famous links are the Olympics rings
  • We also examined an historically important link called the Borromean rings
    

    which I always think of in the context of this quote:

    "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.

    We saw how many different groups have used it in their logos, whenever three things are intimately intertwined.

  • Then we had a quiz.

How can we turn a trefoil knot into other knots (and the Borromean rings)?

• But first let's talk about that quiz....
  • Median: 7.25
  • The Key

• Now it's back to knots. Let's see how to create our last two knots, which are called "twist knots". The ones that come from a twisted bands we actually call toroidal knots -- knots on a torus. But these twist knots can't be obtained that way.

  We begin with the Trefoil Knot -- the simplest interesting knot. It has three crossings, and there's no way to make it any simpler! We will also call it the 3-knot (since it's the only knot with three crossings):

  

• There are two different ways to think about making the trefoil knot: as a
  1. torus knot, and as a
  2. twist knot.
  Here it is drawn to emphasize the "twist":
  

  The twist formulation is a good place to start in order to draw the Borromean rings:

  

• The figure-eight knot is a twist knot that is not toroidal:
  

• The three-twist five knot $(5_2)$ is a simple continuation of that. I want us to know that one because there are two (and only two) different five-crossing knots: this one, and the cinquefoil $(5_1)$.

• Here's a table of the first few knots:
  

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

• A fractal (or two)
• Some symmetry problems, like our original homework: reflective and rotational symmetry
• An example wallpaper, and its classification.
• Platonic problem, perhaps having to do with duality, numbers of faces, edges, and vertices; symmetry, etc.
• Regular polygons

• 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...:(