Pinscreen Approximations

I’ve always enjoyed playing around with pinscreens, but only recently did I realize what cool mathematical concepts they display!

For example, the image above shows an approximation of the volume of a hemisphere using right cylinders.  A pinscreen Riemann sum!

And the images below suggest how we might approximate the areas of a circle and a square using pinscreens.

Compute the ratio of raised pins to total pins, and multiply by the total area of the pinscreen.  A pinscreen Monte Carlo method!

Any other cool math hiding in there?