Appreciation

While in Maine, I took some nice photos of the boundary between the beach and the sea:

It made me think of something I saw a long time ago (maybe on 60 Minutes?) about a scientist who thought deeply about coffee spills on his countertop. The power of the internet helped me locate Sidney Nagel, a physicist who studies the physics of drops, why things get “jammed”, and why a coffee spill leaves a dark ring after it dries.

Is there any way to predict the kind of edge this water will make as it crawls up the beach? Is there any order in this chaos? If this inspires you to great scientific accomplishment, please remember where you got your start.

I was trying to construct a simple, introductory intersection problem for the first day of Calculus class, so I started with a well-behaved circle:

This is a circle centered at (3,2) with radius 5.  I picked two points on the circle, (0,6) and (7,-1), found the equation of the line between them, and put together my system of equations:

So I had successfully reverse-engineered my circle-and-line intersection problem with two nice solutions:  (0,6) and (7,-1).

Unfortunately, I made a typo on the handout.  At the end of the left side of the circle equation I wrote ” + 12″ instead of ” – 12″.

So all my work was for naught.  Or so I thought.  Turns out, at least two amazing things happen:

First, the new circle still ends up having a nice radius, namely 1.  What’s even more amazing is that the new circle also ends up having two nice intersections with the given line, (3,3) and (4,2)!

I wish my intentional work always turned out as well as this mistake!