- You have your second exam this week. Think of it as three quizzes together....
It will be over fractals, wallpapers, Platonic solids, and symmetry.
- Speaking of symmetry, I just saw this article in Science News:
Messed-up metabolism during development may lead guts to coil the wrong way: Experiments in tadpoles may provide clues to a condition in people called intestinal malrotation
"In frogs - and in people - the intestine typically winds counterclockwise as it develops. This happens early in a tadpole's development, roughly between the third and fourth day. The researchers exposed Xenopus laevis embryos to atrazine, a common herbicide and known hormone disruptor in humans, both before and during this stage of development (SN: 1/21/15). They found that many of the frogs' intestines coiled clockwise - the wrong direction."
(Long's take: "atrazine" is RoundUp -- please avoid it, for your own sake -- but especially if you're pregnant!)
- 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
- We discussed the quiz, and hopefully know how to do fractals a
little better
- We took a special look at the Borromean rings, seeing how to think
of them as popping
out of a trefoil knot -- just like those twist knots.
- I quickly showed that there are some other nice, somewhat mechanical methods for drawing and studying knots.
Are we ready for this exam?
- Let's begin with some fractal practice. Remember your licorice and
your cheese nibbling!
No joke, I was reading another article in Science News today, and came across the "nibblerids" (which could really do a number on our licorice!).
- Let's construct a fractal on squares with order 2 rotational symmetry.
- Let's construct a fractal on triangles with order 3 rotational symmetry.
- Let's contruct a stick fractal with rotational symmetry of order 2.
- Let's contruct a stick fractal with rotational symmetry of order 1, and a single line of reflection symmetry.
- Do you know your features of each of the Platonic Solids?
- Which are dual pairs?
- Can you draw the Platonic solids, e.g. the Tetrahedron,
cube, or octahedron? (I wouldn't make you draw the
others!).
It's related to another issue: can you draw the duals?
These two dimensional projections might help
- How many faces, edges and vertices does each one have?
- What are their defining face shapes?
- Which one is the same as \(K_4\)?
- How do you go about classifying wallpaper patterns?
- Polya's classification scheme
- Here's a lovely intro to the wallpapers, focusing on the work of M. C. Escher, a Dutch artist famous for many wonderful works of art (especially for his optical illusions, but my favorite is his Drawing Hands....).
- It includes this diagram to the 17 groups.
- Can you construct objects with rotational and reflection symmetries? Can you recognize the order of rotational symmetry in an object, or the number of lines of reflection?
- 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...:(