Site Swap Laboratory

The final laboratory of the year is both for fun and to expose the students to a whole new form of mathematical model called site swap notation.

Jeanine working on a three ball cascade

Early in the course a quote from Freeman Dyson was used to start a journey through the mathematical models (relationships) that explain physical systems. Dyson calculated how an electron ought to behave. Later someone went into a laboratory and the electron behaved as predicted by the mathematical model. In laboratory two a linear model predicted the location of a rolling ball. In laboratory three a falling ball obeyed a quadratic mathematical relationship. The behavior of a marble rolling off of a banana leaf obeyed a square root relationship. And in laboratory four the marbles knew what to do in order to mathematically conserve momentum. Sound, the relative depth of an image, and Ohm's law all exhibited linear relationships.

Leslie juggling three karer (key limes)

There are other mathematical relationships that govern physical systems. There are systems that are modeled by exponential, logarithmic, and circular functions. There are exotic functions such as the hyperbolic sine and hyperbolic cosine. Some systems are best described by complex variables that include a real and an imaginary component. Many of these systems are beyond the mathematical scope of this course.

The relationships above are algebraic mathematical models. Much of the mathematics curriculum is centered on algebra in part because algebra is important to describing the physical world. There are, however, other mathematical models, non-algebraic models. Laboratory fifteen seeks to broaden the mathematical horizons of the students by introducing a mathematical model and notation that is not algebraic.

I first demonstrate site swap notation and then demonstrate how to learn to cascade by starting with one ball, adding a second, and then doing a three ball cascade using the 3 pattern.

Juanita with three Space Balls under control

Occasionally I capture the balls floating in mid-air. I have three "Space Balls" that usually produce visually more interesting images. In a photo, the balls seem to float almost magically in mid-air.

The orange Space Ball floats as if a bubble in the air in front of Kyoko