Growing new eyes

Introduction

by Meghan Kofod and Maria Droujkova

hapuu6.jpg

Maria,

I thought of you when looking up this plant:

http://www.instanthawaii.com/ cgi-bin/hawaii?Plants.hapuu

The description made reference to its unfurling in a fractal-like pattern!


Thank you for the link, Meghan! What a neat, weird plant!

Once you become aware of a math topic, it SUDDENLY starts to come up everywhere you look. We recently talked about it with Hope from our math club - for her, the recent math topics were patterns and stars. It feels quite trippy when the world around you is suddenly full of math objects of a particular kind. My daughter calls it "growing new eyes" - she would ask: "Was I blind before or what? How could I not notice this everywhere?!!"

My latest (it came up yesterday) notice-math topic is "chirality" - objects that have almost the same structure, but are mirror images or "inside-out" images of one another, like left and right glove.
http://en.wikipedia.org/wiki/ Chirality
chirality_nodal_spiral.jpeg
Image by http://scienceblogs.com/pharyngula/2009/04/snails_have_nodal.php


Everybody - what topics helped you to grow new eyes and notice math around you?

Abstract algebra of juggling

by Mike South
Abstract algebra was a complete life changer for me when someone explained it to me for the first time. I think it was in context of juggling and how to look at juggling throws as examples of operations on the symmetries of a triangle. I was hooked (lined and sinkered, too). I still think of that as the first time I really "saw" math, like saw math for what it was (consider that I was halfway through a bachelor's in physics, so I had been exposed to a lot of math up to then, but pretty much only the very narrow selection used in the standard science toolkit.

It was like having only heard military marches all your life and then someone playing you rock, and hinting to you about classical, blues, jazz, etc. You suddenly realize that there's another universe out there.

Anyway, back to the "eyes" thing (or as Devlin put it, "Making the invisible visible"), the pull chain on a ceiling fan is an example I have used before of something that became visible to me via abstract algebra. It's an example of a simple algebraic group, where the only element you get to work with directly is "+1". In a fan with "off, high, medium, low" settings, that means you pull once to go to high, pull again to go to medium, pull again to go to low, and pull again to go to off. But what if you weren't sure about whether you were at medium or low, and you pull, thinking...."is that off, or did it just switch to low?" so you pull it again...and it was off, just still turning, so you're back on high.

What do you need now? -1. You want to go back to off. There's no way to go backwards, directly, but if you know that the inverse of 1 in this group is 3, you can just pull the chain three times and walk away.

Another example would be juggling with two people and six clubs. You have a couple standard patterns (like "everies" or "every others" where you throw either everything (in everies) or every other thing that lands in your right hand to the other juggler. Because of the way the timing works out, you can, instead, throw a left double early (throw it to the other juggler) and it will end up in his hand at the same time the later right hand throw would have got there.

So I worked out the elements of the permutation group that were being exercised by these throws and found out lots of interesting things about how the clubs moved between the two players that I had never known even though I had been club passing for years.

Even with a standard three ball cascade the math can tell you interesting stuff. I would bet that less than one percent of jugglers know that all six possible arrangements ("ball one in the air, ball two in left hand, ball three in right hand" being one, there are five others) of the three balls occur every six throws. Not that they need to. But it's cool. I mean, they don't _need_ to juggle, right? :). Although it can definitely seem like it :).
juggling-17.jpg
by http://entertainment.howstuffworks.com/juggling.htm/printable


Fractals and stars

by Erin Henry
For me it was fractals, last year in Math club. It was winter, and started seeing all the bare trees as (imperfect) fractal structures, and wondering if there were formulae to describe the branching angles, radial and length distances that could be used to describe each type of tree. I wondered if that is what ultimately translates into the distinctive shapes that trees take on (bulbous, or cone shaped, tall and thin, or wide and spreading). For a few days, I just drove around marveling at trees, seeing how I could tell certain types of trees from their shape, and looking at branches to see if I could see repetitive patterns in and among the branching points. I even looked up a few studies about it, but didn't find the answer to my question.

The stars topic that hooked Hope got me too. I saw a coffee mug at someone's house and was able to describe the pattern of the star painted on the outside. Then I noticed that I had a cut glass cake plate with two 12 point stars superimposed on it, and was able to figure out patterns for them, too. It added an interesting level of depth and "ownership" to be able to look at the stars and say, "I know how to create that pattern using a simple formula."



Fibonutsy

by Carol Cross
In terms of a math concept that suddenly I see everywhere--

For the past couple of months, we've been "Fibonutsy" about Fibonacci. Of course, we have covered this before. But in September, I happened to see a wonderful picture book by an author we really like about Fibonacci (for more information, see http:// teachingyourmiddleschooler. blogspot.com/2010/10/book- review-blockhead-life-of- fibonacci.html ). Then we started to explore it in more traditional math ways, particularly by looking for Fibonacci sequences in nature. Then we applied it to cooking (http:// teachingyourmiddleschooler. blogspot.com/2010/10/math-for- dinner.html ). Then I discovered it in POETRY! (http:// teachingyourmiddleschooler. blogspot.com/2010/10/lesson- plan-writing-fibs-fibonacci. html ). Most recently, I've been seeing it in art a lot--especially that graphic of the Fibonacci sequence as a curve. That drawing of the curve seems to be popping up in lots of visuals I see that have nothing to do with math or Fibonacci or anything.

So my "new eyes" have been more focused on how many disciplines use this concept, rather than seeing this concept in math per se.
Blockhead1.jpg

Dimensions and grids

by Maria Droujkova

We were making sets for "Odd one out" by Tom O'Brien. The topic of dimensions came up in both math clubs doing the activity. Erin also brought up grids, which we recalled using sidewalk chalk. Both ideas are powerful for "growing new eyes.