Today's Google doodle on google.co.uk, google.de, google.it, google.fr, google.ch, google.at and probably several others but not on google.com celebrates Maria Gaetana Agnesi, whose birthday is today. She is the eponym of the "witch of Agnesi":
This is the curve $f(x)=\dfrac{1}{1+x^2}$.
This has a somewhat surprising number of interesting properties, starting with things like $f\left(\dfrac 1 x\right) = 1-f(x)$ and including things like the fact that if (capital) $X$ is a random variable whose probability density function is proportional to this function then $1/X$ has the same distribution as $X$, and if $X_1,\ldots,X_n$ are probabilistically independent copies of $X$, then their average also has that same distribution.
Has any published expository account attempted to enumerate the principal interesting properties of this curve in geometry, trigonometry, algebra, probability, combinatorics, etc.?