Download Juggling Sequences with Number Theory

Modular Juggling with Fermat
Stephen Harnish
Professor of Mathematics
Bluffton University
[email protected]
Miami University 36th Annual
Mathematics & Statistics
Conference:
Recreational Mathematics
September 26-27, 2008
Archive of Bluffton math seminar documents:
http://www.bluffton.edu/mcst/dept/seminar_docs/
Modular Juggling with Fermat
32  42  52
a n  bn  cn
Classical Results
Theorem 1: (Euler) The sequence (3k)(k+1) +1k0
has no equal initial and middle sums.
Theorem 2: (Dirichlet) The sequence
5k 3 (k  2)  5k (2k 1) 1k 0 has no equal initial and
middle sums.
Initial and Middle Sums of
Sequences
• Note that sequence {1, 2, 3, 4, …} has
numerous initial sums that equal middle
sums:
(1 + 2) = 3 = (3)
(1 + 2 + 3 + 4 + 5) = 15 = (7 + 8)
(1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11)
(1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)
Sequence Sums
Definition: For the sequence x1 , x2 , x3 ,
an initial sum is any value of the form
I k  x1  x2  x3 
, xk ,
 xk
for some integer k  1, and
a middle sum is any value of the form
M j ,k  x j  x j 1  x j  2 
 xk
for some integers j and k, where k  j  1;
the length of a middle sum M j ,k is k  j  1.
Initial and Middle Sums of
Sequences
• Note that sequence {1, 2, 3, 4, …} has
numerous initial sums that equal middle
sums:
(1 + 2) = 3 = (3)
(1 + 2 + 3 + 4 + 5) = 15 = (7 + 8)
(1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11)
(1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)
Initial and Middle Sums of
Sequences--Fibonacci
• Note that sequence {1, 1, 2, 3, 5, 8, 13…} has
the following initial sums:
(1) = 1 = (1)
(1 + 1) = 2 = (2)
(1 + 1 + 2) = 4
(1 + 1 + 2 + 3) = 7
(1 + 1 + 2 + 3 + 5) = 12
(1 + 1 + 2 + 3 + 5 + 8) = 20
Juggling
History
• 1994 to 1781 (BCE)—first depiction on the 15th Beni Hassan tomb of an
unknown prince from Middle Kingdom Egypt.
The Science of Juggling
• 1903—psychology and learning rates
• 1940’s—computers predict trajectories
• 1970’s—Claude Shannon’s juggling machines at MIT
The Math of Juggling
• 1985—Increased mathematical analysis via site-swap notation
(independently developed by Klimek, Tiemann, and Day)
For Further Reference:
• Buhler, Eisenbud, Graham & Wright’s “Juggling Drops and
Descents” in The Am. Math. Monthly, June-July 1994.
• Beek and Lewbel’s “The Science of Juggling” Scientific American,
Nov. 95.
• Burkard Polster’s The Mathematics of Juggling, Springer, 2003.
• Juggling Lab at http://jugglinglab.sourceforge.net/
Juggling Patterns
(via Juggling Lab)
Thirteen-ball Cascade
A 30-ball pattern of
period-15
named:
“uuuuuuuuuzwwsqr”
using standard
site-swap notation
531
Several period-5, 2-ball patterns
90001
12223
30520
14113
A Story Relating Juggling with
Number Theory
A Tale of Two Kingdoms
First Studied by E. Tamref
Values of Culture 1 (Onom)
Values of Culture 2 (Laud)
1. Annual Juggling Ceremony
1. Annual Juggling Ceremony
A Tale of Two Kingdoms
First Studied by E. Tamref
Values of Culture 1 (Onom)
Values of Culture 2 (Laud)
1. Annual Juggling Ceremony
1. Annual Juggling Ceremony
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
A Tale of Two Kingdoms
First Studied by E. Tamref
Values of Culture 1 (Onom)
Values of Culture 2 (Laud)
1. Annual Juggling Ceremony
1. Annual Juggling Ceremony
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
3. Sequential & Complete—
Juggling performances start
with all patterns for 0 balls, then
1, 2, 3, etc.
3. Sequential & Complete—
Juggling performances start with
all patterns for 0 balls, then 1, 2,
3, etc.
A Tale of Two Kingdoms
First Studied by E. Tamref
Values of Culture 1 (Onom)
Values of Culture 2 (Laud)
1. Annual Juggling Ceremony
1. Annual Juggling Ceremony
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
2. Orderly—1 period per year,
starting with 1, then 2, 3, etc.
3. Sequential & Complete—
Juggling performances start
with all patterns for 0 balls, then
1, 2, 3, etc.
3. Sequential & Complete—
Juggling performances start with
all patterns for 0 balls, then 1, 2,
3, etc.
4. Individuality—
4. Complementarity—
Monistic presentation:
Dualistic presentation:
1 performer per ceremony
2 performers per ceremony
The Pact
1400 C.E.

In the first year of the new century when the kings of
Onom and Laud each decreed the annual juggling
period to be 1, a peace treaty was signed.

To strengthen this new union, the pact was to be
celebrated each year at a banquet where each kingdom
would contribute a juggling performance obeying its
own principles. However, to symbolize their equal status
and mutual regard, each performance must consist of
an equal number of juggling patterns.
Onom
Laud
Year One
Kingdom
Kingdom
0
0
0
1
2
1
1
3
4
2
Period-1
# of Balls:
# of Patterns:
0
1
1
1
2
1
3
1
4
1
Year Two
0 balls
1 pattern
1 ball
3 patterns
3 balls
7 patterns
2 balls
5 patterns
4 balls
9 patterns
Year Two Options
(patterns with ball-counts 0-4)
00
33
44
11
42
53
20
24
35
02
51
62
22
15
26
31
60
71
13
40
04
06
17
80
08
Year Two—Onom Performer
00
33
53
11
42
35
20
02
24
62
22
51
31
15
26
13
60
71
06
17
40
04
44
80
08
Year Two—Luad Performers
Performer 1:
00
11
20
02
22
31
13
40
04
Performer 2:
00
33
11
42
20
24
02
51
22
15
31
60
13
06
40
04
Period-2
Patterns per ball are odd numbers
 A balanced juggling performance:
(1+3+5+7+9) = 25 = (1+3+5) + (1+3+5+7)


Recall: (the sum of the first n positive odds) = n2
So:
2
2
2
5 = 3 4
Onom Performer = Laud Performers
Question

Will this harmonious arrangement continue
indefinitely for the Kingdoms of Laud and Onom?

For years 3 and beyond, as the sanctioned
periods continually increase by one, can joint
ceremonies be planned so that each abides by
their own rules and each presents the same
number of juggling patterns?
Period-2 (again)
via initial & middle sums
A balanced juggling performance:
(1+3+5) + (1+3+5+7) = 25 = (1+3+5+7+9)


Subtracting the initial sum (1+3+5) yields:

Initial sum
= Middle sum
(1+3+5+7) = 16 = (7+9)
Period-3 Juggling Patterns
0 balls
1
1 ball
7
2 balls…
19
Period-1
# of Balls:
# of Patterns:
0
1
1
1
2
1
3
1
4
1
2
5
3
7
4
9
2
19
3
37
4
61
Period-2
# of Balls:
# of Patterns:
0
1
1
3
Period-3
# of Balls:
# of Patterns:
0
1
1
7
Period-3

Sequence: 1
7
19
37
61
91 …
• Sums: 1  8  27  64  125 …
13  23  33  43  53 …

Euler’s Theorem
• There are no solutions in positive integers a, b, & c to the
equation:
a b  c
3
3
3
Hence…
The future of the
“Two
Kingdoms” is decided by
number theory
Number Theory
T.F.A.E.:
1.
a b  c
2.
a  c b
n
n
n
n
n
n
3. For the specific sequences of the form (k  1)
(initial sum) = (initial sum) – (initial sum)
(initial sum) = (middle sum)
n
 k n
k 0
Conclusion
Theorem 5: (Graham, et. al., 1994)
The number of period-n juggling patterns
with fewer than b balls is b n .
Theorem 6:
T.F.A.E.:
1. The monistic and dualistic sequential
periodic juggling pact can not be
satisfied for years 3, 4, 5, …
2. F.L.T.
F.L.T.
“It is impossible to separate a cube into two cubes, or a fourth power into
two fourth powers, or in general, any power higher than the second into two
like powers. I have discovered a truly marvelous proof of this, which this
margin is too narrow to contain.”
Fermat/Tamref
Conclusion: “Add one more to your list of applications of F.L.T.”
Last Thread:
Excel spreadsheet explorations of
initial and middle sums
while juggling the modulus
&
topics for undergraduate research
Initial Sums = Triangular Numbers
Initial Sums = Triangular Numbers
Initial Sums = First Powers
Initial Sums = Squares
Initial Sums = Cubes
Initial Sums = Fourth Powers
Modular Juggling
&
Juggling with the Modulus
Modulus 2 Pattern for Cubic I.S.
Modulus 3 Pattern for Cubic I.S.
Modulus 4 Pattern for Cubic I.S.
Other Mathematical Questions
1. Sequence compression
(I.S. seq.)  (base seq.)  (generating seq.)
•
(see Excel)
Generating sequence behind the
base sequence
•
•
•
•
•
{1}
{1,1}
{1,2}
{1,6,6}
{1,14, 36, 24}
HW: What explicit formula derives
these generating sequences?
Hint—Difference operators
Hint—Difference operators
Triangular, Square, Cubic
• Vary IS and Modulus
Other Mathematical Questions
1. Sequence compression
(I.S. seq.)  (base seq.)  (generating seq.)
2. Patterns of modularity for sequences and arrays
A Related Research Topic
Modularity patterns in Pascal’s
Triangle:
• See Gallian’s resource page for
Abstract Algebra (from MAA’s MathDL)
• http://www.d.umn.edu/~jgallian/
And what is this pattern?
I.S. #
Mod
If properly discerned, a special case of FLT
follows (case n = 3).
Other Mathematical Questions
1. Sequence compression
2.
3.
4.
5.
6.
(I.S. seq.)  (base seq.)  (generating seq.)
Patterns of modularity for sequences and arrays
Numerous patterns & properties of IS/MS tables
Explicit formula for middle sums of fixed length
Distribution of IS = MS matches for triangular,
square, cubic, or nth power initial sums
(why or why not?)
Imaginative historical reconstructions—
“What margin indeed would have sufficed?”
Modular Juggling with Fermat
Stephen Harnish
Professor of Mathematics
Bluffton University
[email protected]
Miami University 36th Annual
Mathematics & Statistics
Conference:
Recreational Mathematics
September 26-27, 2008
Archive of Bluffton math seminar documents:
http://www.bluffton.edu/mcst/dept/seminar_docs/
Modular Juggling with Fermat
32  42  52
a n  bn  cn
Website sources
• Images came from the following sites:
http://www.sciamdigital.com/index.cfm?fa=Products.ViewBrowseList
http://www2.bc.edu/~lewbel/jugweb/history-1.html
http://en.wikipedia.org/wiki/Fermat%27s_last_theorem
http://en.wikipedia.org/wiki/Pythagorean_triple
http://en.wikipedia.org/wiki/Juggling
Another story-line from the 14th C
• Earlier in 14th C. Onom, there had emerged a heretical sect called
the neo-foundationalists. They valued orderliness and sequentiality,
but they also had more progressive aspirations—the solo
performer’s juggling routine would be orderly and sequential but
perhaps NOT based on the foundation of first 0 balls, then 1, 2, etc.
These neo-foundationalists might start at some non-zero number of
balls and then increase from there.
• However, they were neo-foundationalists in that they would only
perform such a routine with m to n number of balls (where 1 < m <
n) if the number of such juggling patterns equaled the number of
patterns from the traditional, more foundational display of 0 to N
balls (for some whole number N).
• For how many years (i.e., period choices) were these neofoundationalists successful in finding such equal middle and initial
sums of juggling patterns?
• (Answer: Only for years 1 and 2).