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A Zeta Function for Juggling Patterns
Erik R. Tou Carthage College
[email protected]
MAA MathFest
Madison, Wisconsin
4 August 2012
Joint work with Dominic Klyve and Carsten Elsner
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How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
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How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
X 1
-
How To Make Juggling Mathematical Measure height of throw
according to number of “beats” until it comes back down
(usually, “beats” = “thuds”)
Can track this with an arc diagram
Repeated throws of height 3 Siteswap notation is (3)
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More Examples — (441) and (531)
(441) (531)
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Multiplication for Juggling Patterns
Definition: Multiplication of juggling patterns Patterns can
be combined by concatenation (usually).
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Multiplication for Juggling Patterns
Definition: Multiplication of juggling patterns Patterns can
be combined by concatenation (usually).
Definition: Primitive juggling patterns Most juggling
patterns can be broken up into smaller ones If not, it’s a
primitive juggling pattern
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Multiplication for Juggling Patterns
Definition: Multiplication of juggling patterns Patterns can
be combined by concatenation (usually).
Definition: Primitive juggling patterns Most juggling
patterns can be broken up into smaller ones If not, it’s a
primitive juggling pattern
Example: (5304252612) = (5304)(252)(612)
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Multiplication for Juggling Patterns
Definition: Multiplication of juggling patterns Patterns can
be combined by concatenation (usually).
Definition: Primitive juggling patterns Most juggling
patterns can be broken up into smaller ones If not, it’s a
primitive juggling pattern
Example: (5304252612) = (5304)(252)(612) So: we have a set
in which we can multiply and factor,
and we have an analogue of the primes.
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
Our decision: difficulty is the appropriate measure
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
Our decision: difficulty is the appropriate measure One
aspect of difficulty: number of balls (b)
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
Our decision: difficulty is the appropriate measure One
aspect of difficulty: number of balls (b) Another aspect: the
length of the siteswap (n)
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
Our decision: difficulty is the appropriate measure One
aspect of difficulty: number of balls (b) Another aspect: the
length of the siteswap (n)
Our definition: For a given juggling sequence j, the norm N(j)
is given by bn.
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Measuring the Size of a Pattern
What aspects of a juggling pattern measure “size” in a
meaningful way?
Our decision: difficulty is the appropriate measure One
aspect of difficulty: number of balls (b) Another aspect: the
length of the siteswap (n)
Our definition: For a given juggling sequence j, the norm N(j)
is given by bn.
Some examples… N(3) = 31 = 3 N(531) = 33 = 27
N(5304252612) = 310 = 34 33 33 = N(5304)N(252)N(612)
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The Zeta Function for 3-Ball Patterns
Let’s recall the Riemann zeta function:
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The Zeta Function for 3-Ball Patterns
Let’s recall the Riemann zeta function:
Using the norm N(j), we can define a similar series for 3-ball
patterns:
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The Zeta Function for 3-Ball Patterns
Let’s recall the Riemann zeta function:
Using the norm N(j), we can define a similar series for 3-ball
patterns:
Big Question: how many b-ball patterns will have the same norm
(i.e., the same length)?
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The Zeta Function for 3-Ball Patterns
Fortunately, this is known [Chung & Graham]:
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The Zeta Function for 3-Ball Patterns
Fortunately, this is known [Chung & Graham]:
For the 3-ball situation, this means:
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Analytic Continuation We can use these multiplicities to
condense the series:
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Analytic Continuation We can use these multiplicities to
condense the series:
A little manipulation…
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Analytic Continuation We can use these multiplicities to
condense the series:
A little manipulation…
…and we can see it’s geometric! So:
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Finding Zeroes and Singularities
We know the singularities occur whenever 3s = 4.
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Finding Zeroes and Singularities
We know the singularities occur whenever 3s = 4. Finding the
zeroes is simple algebra: set z = 3s, and set ζ3(z) = 0 … (algebra
ensues) …
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Finding Zeroes and Singularities
We know the singularities occur whenever 3s = 4. Finding the
zeroes is simple algebra: set z = 3s, and set ζ3(z) = 0 … (algebra
ensues) …
So there are two classes of zeroes, occurring along two
vertical lines in the complex plane:
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Visualization
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Generalization It’s easy to generalize this to the set of
b-ball juggling
patterns!
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Generalization It’s easy to generalize this to the set of
b-ball juggling
patterns! Analytic continuation:
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Generalization It’s easy to generalize this to the set of
b-ball juggling
patterns! Analytic continuation:
Singularities whenever bs = b+1.
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Generalization It’s easy to generalize this to the set of
b-ball juggling
patterns! Analytic continuation:
Singularities whenever bs = b+1. There are b-1 classes of
zeroes, found from roots of
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More Visualization — 3-ball zeta fcn
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More Visualization — 4-ball zeta fcn
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More Visualization — 5-ball zeta fcn
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More Visualization — 6-ball zeta fcn
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References Carsten Elsner, Dominic Klyve, τ. “A Zeta
Function
for Juggling Patterns.” Journal of Combinatorics and Number
Theory 4 (2012), no. 1, pp. 53-65.
Fan Chung and Ronald Graham. “Primitive Juggling Sequences.”
Amer. Math. Monthly 115 (2008), no. 3, pp. 185-194.
Burkard Polster. The Mathematics of Juggling. Springer-Verlag,
New York, 2003.
Email: [email protected]
Thank you!
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