Saturday, December 27, 2003

Elliptic Curves, Part I

The Directed Reading Program I took this quarter comes to an end with a presentation, Thursday of the first week of classes. So, over break, I'm preparing notes and slides for my presentation on elliptic curves. Unfortunately, to do so, I needed to find a graphing utility which can handle implicit functions, hopefully a free utility at that. Well, I can't recommend GrafEQ highly enough now. Not only does it handle elliptic curves like y^2 = x^3 - 5x +11, it can take on such bugaboos as y = x^x, as well as number-theoretical experiments like gcd(|10x|,|10y|) = 1.

As far as I can tell, GrafEq takes your relation between x and y and a 2-variable function in z, then analytically finds the zero set. It can take a while to work, if you give it a nasty function: for example, take a good implicit function 0 = f(x,y) and try 0 = cos(f(x,y)). But work it does, and I highly recommend it.

In a few days, I'll post a little explanation of elliptic curves and the group law (if I'm doing this right, it should be understandable to anyone who still remembers a good calculus course), along with a few pictures (if I can figure out how to upload to the U of C website).

And Christmas Day

Each year, the Christmas morning video shows the further development of a few cheesy traditions. We give our dog Nacci a new chew toy, and watch her rend it asunder in the space of a few minutes. Dad takes the bow from one of his gifts and affixes it to his head, acting as if nothing is out of the ordinary. Mom cheers over the bubble wrap in the box, ignoring the actual gift. I search my stocking for the button that plays Christmas music (a part of the stocking itself), and claim it as another present for me. My sister Katie gets increasingly embarrassed and begs us to stop.

This year added a few things. My grandma jitterbugged to her new Glenn Miller CD. Katie turned her herbal mask into a superhero's costume for the camera. Most dramatically, there was a loud crash in the living room as we sat in the family room; Dad had balanced our glass chess set atop the scanner, on the highest shelf, and eventually it slid off and shattered as we all jumped.

Later, we all headed to my aunt Jill's house for Christmas dinner (including multiple pies). For once, all of us sat at the adult table (only to discuss Hey Ya along with the Gnostic Gospels and college life). It was a nice ending to a most merry Christmas.

No comments: