Get On The Damn Unicorn!

Today I had one of those wonderful moments in mathematics. I was teaching the quotient rule, feeling bored, and suddenly I found myself at the perfect intersection of calculus, geometry, history, and humor!

Imagine me, proving the quotient rule, 66 half-lidded eyeballs staring my way, and just when I thought all was lost, I recalled the most wonderful function:

At first you think, sure, I can differentiate this using the quotient rule, but what does it all mean?

To construct the graph of this function, imagine a circle with diameter 1, squeezed between two parallel lines. The circle passes through the origin, O. Now for every point A on the circle, draw the secant line through A and O. Once you've done that turn the part between A and the top line into the hypotenuse for a right triangle, and then draw a point at that right angle. Better yet, check out the diagram below (courtesy of Wikimedia Commons):

Do that for every point on the circle, and eventually you will trace out this entire beautiful curve which gently slopes up and hugs the top of this circle. Check out the plot below, courtesy of Sage:

This is called the Witch of Agnesi, and was made popular by Italian mathematician Maria Agnesi in the 1700's. Yeah, that's right, I said Maria -- she was the first woman ever to be appointed a university professor. What Maria did not know, is that aside from having this really beautiful geometric construction, this function would also have some important implications in probability theory, particularly Cauchy densities. First came the beautiful geometric construction, then came the application.

So this long digression is all to say that today, as I was teaching the quotient rule, and feeling really bored by the whole thing, this wonderful function came marching along. And as I recounted the tale of the Witch of Agnesi, I remembered my favorite comic from Abstruse Goose, and I actually caught myself grinning at my class and saying "Yeah, you know, eventually all math becomes applied math."

I don't think my audience gripped the hilarity of it all, but nothing could get me off that unicorn right then. I was riding high.

<3