calculus at the beach

logarithmic spirals everywhere... 

logarithmic spirals everywhere... 

I'm on vacation at Edisto Beach, SC, this week, so I'll have a MOOC update when I return (vacations are good and bad for MOOCs, obviously).  Still, being the good math geek that I am, I can't help noticing mathematics all over the place, even when I'm supposed to be taking a break.  I found the shells in the picture above in a fairly short time just by looking carefully.  I have jars full of these at home; it's a bit of a problem, really.  I love the different color combinations that occur, and of course I also love logarithms, so win-win. 

Tides are interesting to think about.   Newton spent a lot of his spare time, you know, between alchemy and universal gravitation, trying to figure out a formula that would predict the tides.  It's extremely complicated, of course, because it depends on the Earth, Sun, and Moon, but sitting on the beach all day allows you to notice certain things.  For example, when is the water level rising or falling the fastest?  There is an analogy here with the length of the day, something which can be predicted nicely using a sine function.  Day length increases at the fastest rate at the spring solstice because that is when the derivative of sine (i.e., cosine) is greatest.  If you've ever paid attention, you've probably noticed that.  Well, when should the sea level be rising fastest?  Halfway between low and high tide, right?  That's what I noticed anyway, but of course I couldn't make careful measurements while hiding from the sun under an umbrella. 

And then there is the 12-variable optimization problem arising from trying to maximize the happiness of a whole house full of people.  Luckily, my mother-in-law is pretty good at solving that one, although it's really only possible in practice to find saddle points.  Local maxima are hard to come by with small children involved.

Back to work on Monday for a short week.  And back to the MOOC, too.