Juggling Sequences with Number Theory & “A Tale of Two Kingdoms”

Juggling Sequences with Number Theory Stephen Harnish Professor of Mathematics Bluffton University Miami University 35 th Annual Mathematics & Statistics Conference: Number Theory September 28-29, 2007

Theorem 1: (Euler) The sequence has no equal initial and middle sums. Theorem 2: (Dirichlet) The sequence has no equal initial and middle sums. Classical Results

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) ( ) = 15 = (7 + 8) ( ) = 21 = ( ) ( ) = 21 = ( )

Sequence Sums Definition: For the sequence an initial sum is any value of the form for some integer k and a middle sum is any value of the form for some integers j and k, where the length of a middle sum is.

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) ( ) = 15 = (7 + 8) ( ) = 21 = ( ) ( ) = 21 = ( )

Results Theorem 3: Every initial sum of the sequence is equal to some middle sum Conjecture 1: Given the sequence then for each integer k, some initial sum is equal to some k-length middle sum.

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) ( ) = 4 ( ) = 7 ( ) = 12 ( ) = 20

Results Conjecture 2: Some initial sum of the sequence is equal to a k-length middle sum for each k. Theorem 4: The Fibonacci sequence has only two instances of equal initial and middle sums. Namely, middle sums (1) and (2). (Hint: use the fact that and compare with the magnitude of each middle sum of length 1, 2, 3, etc.)

Juggling History 1994 to 1781 (BCE)—first depiction on the 15 th 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 Beek and Lewbel’s “The Science of Juggling” Scientific American, Nov. 95. Burkard Polster’s The Mathematics of Juggling, Springer, Juggling Lab at

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

A story relating juggling with number theory…

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… The Pact (1400 C.E.)

A Tale of Two Kingdoms (first studied by E. Tamref) Values of Culture 1 (Onom) 1.Annual Juggling Ceremony Values of Culture 2 (Laud) 1.Annual Juggling Ceremony

A Tale of Two Kingdoms (first studied by E. Tamref) Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 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) 1.Annual Juggling Ceremony 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. Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 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.

A Tale of Two Kingdoms (first studied by E. Tamref) Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 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. 4.Individuality— Monistic presentation: 1 performer per ceremony Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 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. 4.Complementarity— Dualistic presentation: 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.

Year One For example, at the end of the first year, the solo juggler of Onom performed all period-1 juggling patterns with 0, 1, 2, 3, and 4 balls, while the juggler duet from Laud first performed all period-1 patterns with 0 and 1 ball and then 0, 1, and 2 balls. (Total number of patterns for each: 5)

Year Two

Year Two Also, at the end of the second year, the following were performed at the banquet—the solo juggler of Onom performed all period-2 juggling patterns with 0 to 4 balls, while the juggler duet from Laud first performed all patterns with 0 to 2 & then 0 to 3 balls. (Total number of patterns for each: 25) 0 balls1 ball 2 balls 3 balls4 balls

A Tale of Two Kingdoms (first studied by E. Tamref) Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 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. 4.Individuality— Monistic presentation: 1 performer per ceremony Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 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. 4.Complementarity— Dualistic presentation: 2 performers per ceremony

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-3 Juggling Patterns 0 balls1 ball2 balls…

Period-1 # of Balls: # of Patterns:11111

Year Two Also, at the end of the second year, the following were performed at the banquet--the solo juggler of Onom performed all period-2 juggling patterns with 0 to 4 balls, while the juggler duet from Laud first performed all patterns with 0 to 2 & then 0 to 3 balls. (Total number of patterns for each: 25) 0 balls1 ball 2 balls 1 pattern3 patterns 5 patterns 3 balls 4 balls 7 patterns 9 patterns

Period-2 # of Balls: # of Patterns:13579

Period-3 Juggling Patterns Where have we seen these numbers before? 0 balls1 ball2 balls…

Period-3 # of Balls: # of Patterns:

Again, Period-2 Patterns per ball are odd numbers A balanced juggling performance: ( ) = 25 = (1+3+5) + ( ) Recall: (the sum of the first k odds) = So: = Initial sum = Middle sum ( ) = 16 = (7+9)

Pythagorean Triples Initial & Middle sums for

Sequence: … Examples: Initial Sums:1, 8, 27, 64, 125,… Middle Sums:7, 26, 63, …19, 56,117,…37,… Euler: No initial and middle sums are equal. (proven in the equivalent form of has no solutions in non-zero integers a, b, and c)

The future of the “Two Kingdoms” is resolved through number theory T.F.A.E.: For the specific sequences of the form (initial sum) = (initial sum) – (initial sum) (initial sum) = (middle sum)

Conclusion Theorem 5: (Graham, et. al., 1994) The number of period-n juggling patterns with fewer than b balls is. 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.”

Thus ends our exercise in: Juggling Sequences with Number Theory & “A Tale of Two Kingdoms” Stephen Harnish

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Another story-line from the 14 th C Earlier in 14 th 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 neo- foundationalists successful in finding such equal middle and initial sums of juggling patterns? (Answer: Only for years 1 and 2).