### Site Swap Notation

Daniel Kahneman in Thinking, Fast and Slow
noted that the remembering mind rates experiences using a peak-end rule. Although I had not known this particular fact when I designed laboratory 15 five years ago, I had always shared George M. Cohan's belief that one should "always leave them laughing when you say goodbye."

A class that ends on a pleasant activity makes for better memories looking back on the course. SC 130 Physical Science Laboratory 15 is both fun and yet is also a chance to introduce a mathematical model that is very different.

Laboratory fifteen in physical science sought to push the boundaries on the mathematical box for the students. In laboratory one a quote from Freeman Dyson was used to start a journey through the mathematical models 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.

There are other mathematical relationships that govern physical systems. There are systems that are modeled by exponential and logarithmic functions. The path of a RipStik formed a sine wave on a sheet of paper. 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 described 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. This laboratory seeks to broaden the students mathematical horizons by introducing a mathematical model and notation that is not algebraic. In laboratory fifteen the students were introduced to the mathematics of site swaps.

In an attempt to connect site swap theory back to the language of algebraic equations, after introducing site swap notation I referred to sequences such as 33342333 as site swap equations. The sequence is a mathematical statement that can be true (can be juggled) or false (cannot be juggled).

This laboratory continues to provide a fun way to wrap up a term of exploring the mathematics at the core of physical science while expanding the students thinking with a mathematics system like nothing they have ever seen before.

The laboratory is also an end of term enjoyable experience. As the course is not required by any major, the students are primarily from majors other than those in the natural sciences. For many of these students science is a requirement, possibly even a dreaded requirement.

I want the students to have the chance to do science, engage in exploring systems, grapple with the mathematical language underneath physical science. I can only hope that the students catch a glimpse of the beauty of science - of even pure science for science's sake.

Kanisia |

Brenda |

Terrence |

Josephina |

Joseph |

Gister |

Yvonne, site swap space-time diagrams on the board behind her |

Afilina, a 51 diagram on the board to her left |

Lilly Nora, a newly minted juggler |

Amyleen amply demonstrating the joy possible in an unconventional |

Amyleen demonstrates a two-ball juggle commonly done out in the neighborhoods using mangoes |

Amyleen with a tightly controlled three ball cascade |

Shra Ringlen |