Erik Tou

DO THE MATH!: Juggling with Numbers

I f you love puzzles, as I do, you might already know the “word morphs” game. The rules are simple: You are given a starting word and an ending word, and must gradually change (or, morph) the starting word into the ending word in stages. At each stage, you are allowed to change a single letter from the word, and the result must be a valid word. For example, suppose you want to morph the word “bird” into “park.” Here is one way to do it: BIRD BARD BARK PARK Simple, right? Well, it turns out that one of the foundational results in the mathematics of juggling relies on a numerical variant of the word morph game. But to understand that fact, we need to explore how to make juggling numerical. Juggling has a long history, with the oldest known depictions appearing in one of the ancient Egyptian temples at Beni Hasan (c. 1994–1781 BCE). Georg Forster, a Prussian scientist who accompanied Captain James Cook on his second voyage to the Pacific Ocean (1772–1775), writes of Tongan women who could juggle up to five gourds at a time. During most of this long history, juggling was the province of entertainers and artists, though there was some overlap with other intellectual pursuits, including mathematics. Only in the 1980s did jugglers develop a way to keep track of different juggling patterns using a numerical code, now known as a siteswap. To understand the siteswap, I invite you to try a thought experiment. Imagine a juggler is standing in front of you and is juggling three balls in a uniform way, free of tricks, gimmicks, and flaming torches. Now close your eyes (in your imagination, that is). You will hear a regular sequence of “thuds” (or, beats) as the balls hit the juggler’s hands—left and right in alternation. The pattern you may be visualizing can be seen, beat by beat, in figure 1. Imagine now that you open your eyes and follow the motion of a single ball. Jugglers define the height of a throw as the number of beats that occur between the time the ball is tossed and the time it lands (including the landing beat itself). Be careful here—this version of height is not directly related to the physical altitude of the ball. Rather, height has more to do with the tempo of the pattern being juggled. Also, a throw of height 0 corresponds to a skipped beat; no ball is caught or thrown on that beat. Since most juggling patterns repeat themselves at some point, it is enough to describe a pattern by listing the heights of the throws up to the point at which they repeat. For example, if a juggler throws each ball to a height of 3, the pattern 3, 3, 3, 3, . . . would be denoted by a siteswap of (3). This pattern is shown in figure 1. The siteswap (b) is called a b-ball cascade pattern and is one of the most common juggling patterns. Two other common three-ball patterns are (531) and (441). Once we have a list of throws, it is possible to construct an arc diagram in which each beat corresponds to a dot and the throws are drawn as arcs. Think of the arc diagram as a musical score: The arcs and the dots tell us what is happening at each moment of time. Figures 2, 3, and 4 show the arc diagrams for the do the math!

A Typology for Finite Groups

Summary This project classifies groups of small order using a groups center as the key feature. Groups of a given order n are typed based on the order of each groups center. Students are led through a sequence of exercises that combine proof-writing, independent research, and an analysis of specific classes of finite groups (including the dihedral, symmetric, and alternating groups) to produce a list of groups of each possible type. The project can serve as a capstone experience that unifies group theory with expository mathematical writing.