The Mathematics of ... Juggling
Article by Bill Donahue from Discover Magazine
Synopsis and resource annotations by Max Grinnell
Can reciting a number sequence such as "6-6-1-5-1-5" help a person juggle?
According to Allen Knutson, a mathematics professor at the University of
California at Berkeley, these sequences might be the best technique to use
while attempting to learn and perform this rather challenging activity. Knutson
makes juggling look easy, and for him, it's equal parts dexterity and algebraic
In this profile from Discover Magazine,
Bill Donohue describes how Knutson deploys his nuanced knowledge of algebraic
combinatorics to further his juggling prowess and mastery. Knutson also uses
his juggling skills to demonstrate the basic premise of discrete mathematics in
his classroom. Briefly stated, the premise of discrete mathematics is that one
input (or throw) will yield one output (or falling ball).
The aforementioned juggling sequences are part of siteswap, which is a
mathematical language that describes juggling routines. Essentially, siteswap
assigns a number to each juggling motion, so a 3 is a throw that goes about
chin high and stays in the air for approximately three beats of time.
Odd-number throws are passed from one hand to another, and so on. The rather
amazing thing about all of this is that siteswap allows jugglers to codify
routines and share information with other aficionados of this craft. Knutson
comments that juggling is much like a solid mathematical theorem: "It holds
together. It makes sense, and it also delivers pleasant surprises."
Found below is a list of useful resources that will illuminate and enhance
understanding of several of the topics explored within this piece. Broadly, the
first three links lead to resources for college level mathematics instructors
and students with a curious streak. The next two offer
interactive mathematics resources, and the last link provides information on
Knutson's own academic specialty, combinatorics.
The first entry
leads to a set of tutorials on college algebra created by Kim Steward at West
Texas A&M University. The second entry will take visitors to the homepage of the
Macalester College Problem of the Week, which features math questions with
titles that include "A Tale of Two Sons" and "Charge Your Batteries". The third entry leads to a
series of fine online tutorials, from Harvey Mudd College, on topics such as
derivatives, linear algebra, and differential equations. The fourth entry will take
visitors to the Maths Online Gallery, which includes some interactive
multimedia units on variables, equations, and sets. The fifth entry leads to the
Fun Mathematics Lessons website. Here visitors can take in well-illustrated
lessons on the mathematics of cartography and fractal geometry. The final entry leads to a
page that provides some informal insights into various aspects of combinatorics
including Dedekind's problem, solving magic squares, and much more. Overall,
these resources should provide greater scope and help contextualize the ideas
and concepts found within in the featured work. The list provides links to
resource records in the Applied
Math and Science Education Repository (http://amser.org).