The Sands of Calculus

LargeCircleFilling 2

Physics sandbox programs such as PowderToy create an entertaining environment for playing with mechanics. Sometimes, the physics is a bit surreal, as with MineCraft, but that is fine as long as the rules are uniform, repeatable, and easy to understand. I worked on a design with Scott Easum here in our lab, and he produced a nice sand integrator inside of Powdertoy using a digital counter thats someone else had developed within the Powdertoy community. The learning theory is simple: if a student likes PowderToy, then deliver content such as calculus to that student through PowderToy. The goal of this machine is to measure the area of the circle. Note the digital counter with some very small multi-colored pixels beneath it. These pixels form a structure that represents a digital circuit required to make the counter work. The count begins once the sand starts pouring into the circle. The circle's area can then be measured mechanically using a feedback mechanism so that when the container is full, the overflow sand triggers the digital display to stop counting. The final count is read off to yield the area (an adjustment coefficient is required to obtain the area in common metric units). The operating principle is similar to the hourglass. Unlike the hourglass, though, we can quickly create any geometry we like in PowderToy and use our sand calculus machine to determine the area of an arbitrary shape.