- Your quiz today is over Fractals.
- We have an exam next week, Thursday. It will be over fractals,
wallpapers, Platonic solids, and symmetry.
We'll review on Tuesday, but do some more knots and links on Monday.
- 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!
- Let's take a vote on our favorites in the Fractal Art Gallery
Vote for three. Put your name on your ballot (there will be no ballot stuffing!:).
- We began by cutting a Mobius band into thirds, to discover a Mobius band inside a Mobius band.... We're on our way to a fractal! The longer band that popped out was twice-twisted. Each was a third the width of the original band.
- I mentioned that a belt is a great tool for creating mobius
bands, and trefoil knots, and the like.
- We created some of our knots and links in a methodical fashion, by
drawing them. We'll reprise that today, because I think that I have a
little better way of doing it that I want to share.
- Links: we talked about several common ones:
- Perhaps the most famous links in
the world are the Olympic rings:
Wikipedia Here are the originals - Borromean
Rings: "Is it possible to create a link of three loops (or
rings) in such a manner that they are indeed linked, yet if we
remove any one of the rings the other two become unlinked?"
Yes, indeed!
Standard diagram of the Borromean rings
A realization of the Borromean rings as ellipses
Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration
- Perhaps the most famous links in
the world are the Olympic rings:
Before we get to that, however, I want to thank Dominik for sharing this link to the tree of life: it's a fractal-like way to investigate any creature, and its derivation from its roots.
- Now it's on to twist knots. We've seen most of the knots and links
you'll need to know, so let's go back through the list by drawing them
(reprise):
- Draw what I call "the wedding band" -- an untwisted band.
- Draw the edge of a Mobius band (once-twisted).
- Draw the edge of a twice-twisted Mobius band.
- Draw the edge of a thrice-twisted Mobius band.
- Draw the edge of a fourice-twisted Mobius band.
- Draw the edge of a fiveice-twisted Mobius band.
- Knot Theory
- We begin with the "unknot", of course: a circle. This is the simplest knot, but not very interesting.
- 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:
- 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
- torus knot, or as a
- "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:
- 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?
- We begin with the "unknot", of course: a circle. This is the simplest knot, but not very interesting.
- Now let's migrate our discussion of knots over to "Knot Mosaics":
(an example of Solomon's Knot -- which you might notice is actually a link!)
- 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!
- 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)
- 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:
- Snowflake on-line
- Gasket on-line
- The Fractal Microscope (click on the image)
- L-Systems
in PostScript, from which I generated this image:
- Vi Hart
- 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...:(