Saturday, January 3, 2009

Eccentricity

Being a math teacher, I know my conic sections. Perhaps of all the cross sections of a double-napped cone (has better hair than it actually sounds), degenerates excluded. My favorite has to be the ellipse, commonly known in less mathematical circles among the non-nerdy as "ovals." Not that the circle, parabola, and hyperbola aren't fascinating and amazingly useful conics, the ellipse has always held a special place in my heart ever since I was called weird by my 1st grade teacher, and before that by my mom and dad, but what's a newborn with asthma, pneumonia, and a giant smile to do?

Yes, the ellipse is such a remarkable section of a cone. Obtained by slicing a right circular cone at such an angle is beyond parallel to the base, but not parallel to the side of the cone, so that the slice exits the other side of the cone (practice makes perfect . . . . practice!) they certainly come in all shapes and sizes and have such fascinating reflective properties. From the smallest elliptical cam gear in an engine, to the vat of water in the lithotripsy procedure, to the semi-ellipsoidal whispering rooms ("pssssst, I like math . . . . shhhhh, don't tell anyone . . ." ) in capital rotundas, to the giant orbital paths of celestial bodies, ellipses certainly are hard to pigeonhole.

A lithotripter and a lithotriptor

It's the measure of the "oblongness" of an ellipse that has stolen my heart. Some "ovals" are almost as perfectly circular as a poorly balanced tire, while some are nearly as flat as a math student's enthusiasm in the classroom after Christmas break. The measure of this "roundness," "flatness," "oblongness," or "who cares-ness" is called the eccentricity for the ellipse, where "eccentricity" literally translates to "out of round" in some obscure, non-English language.

You see, every ellipse has two points along the longer axis set in from the edges called the "foci," pronounced "Fo-ci." The numerical signature of every ellipse is simply the ratio of the distance from the center to a single focus (singular for "fo-ci"), called the focal length to the distance from the center to one edge along the longer side, called the "semi-major axis length." The closer these foci are to the center, the rounder the ellipse and the closer the eccentricity is to zero, a number whose symbol looks startlingly like a round circle. The further these foci are from the center (and the closer they are to the edge), the flatter the ellipse and the closer the eccentricity is to one, a number whose Arabic symbol looks startlingly like a flat line standing on edge.

How "out of round" are you??

So I guess my penchant for ellipses comes from our affinity for being "eccentric," a term we prefer over "weird," or "oval," and much like the (former) planet of Pluto, whose orbit had the most eccentric orbit in our Solar System, we have grown up ostracized, criticized, misunderstood, and even castigated, abandoned, and excommunicated from our peers.

Oval Outcasts

Anyway, to make a long, boring story short and boring, I've been thinking about ellipses again recently not only because of my non-sequitur, high-brow humor among my friends and family, but mainly because of my new, low-impact ELLIPTICAL running machine.

With my bad knee and inability to pound the pavement like I'm akin to do, I've purchase an artificial running device that allows me to torture myself without leaving the relative comfort of my garage. Supposedly "better" on the knees because of its smooth, eccentric motion, the experience feels very unnatural, synthetic, and goofy. Having not exercised since July through no lack of trying, my first 30 minutes on it this morning sent me into what I called my "post elliptical cardiac infarction." I really felt like I was going to die, not just because a few of my neighbors saw me bouncing up and down in front of my table saw, but because a felt like my heart was going to explode.

But whether it's enduring criticism or exercising, if you do it enough, you get increasingly better at it.


In the case of the elliptical, I'm hoping that the more time I spend on it, ironically, the LESS out of round my body will be.

I'm not sure there's much hope for my personality (I'm an INTJ).

2 comments:

bob s said...

You almost make me miss classes on conic sections. What a great piece of mathematics. Good luck tomorrow and I hope your students return refreshed and ready for new mathematical challenges.

LaurenC said...

I agree with Bob, a very good blog today. It almost makes me want to teach conics.

Happy Anniversary to you and Shealynn (saw it in today's paper).

My sister watches movies while she exercises on her machine --- she loves it. She usually watches TV series on DVD (like Ghost Hunter, BattleStar G, etc.) She watches it on her computer, but a TV would be better. Just an idea.

Let's hope tomorrow is a great day.

Happy New Year.