Juggling
Old Egyptians, Claude Shannon & Seymour Papert
This ancient wall painting appears to depict jugglers.
It was found in the 15th tomb of the Karyssa I area, Egypt.
According to Dr. Bianchi, associate curator of the Brooklyn Museum:
"In tomb 15, the prince is looking on to things he enjoyed in life that he wishes to take to the next world. The fact that jugglers are represented in a tomb suggests religious significance. There is an analogy between balls and circular mirrors, as round things were used to represent solar objects, birth and death."
From: http://www.juggling.org/jw/86/2/egypt.html
By Billy Gillen
More:
Claude Shannon: Mathematician, Engineer, Genius…and Juggler
https://www.juggle.org/claude-shannon-mathematician-engineer-genius-juggler/
By Rob Goodman and Jimmy Soni
"Until Shannon arrived on the scene, no papers had explored the math of juggling".
"Scientific aspects of juggling" C. E. Shannon
https://www.jonglage.net/theorie/notation/ladder/refs/Claude%20Shannon%20-%20Scientific%20Aspects%20of%20Juggling.pdf
And:
"The Science of Juggling"
https://cs.stanford.edu/people/eroberts/courses/soco/projects/1999-00/information-theory/juggling.html
http://www.juggling.org/papers/science-1/
The right-hand parts of these diagrams show how the juggled objects progress from hand to hand with time.
Fig. 2a shows the simplest three-ball cascade where jugglers can vary the height of the throw, the width of the hands and evenreverse the direction of motion.
Fig. 2b is the simplest case i.e. two balls and two hands, where a choice can be made at each toss whether to interchange the balls orkeep them in the same hands.
Fig. 2c (The three-ball shower in) is similar to the three-ball cascade but with different timing - the left to right throw going almost horizontally so that the whole pattern looks more like balls rotatingin a circle.
Fig. 2d shows the fountain way of juggling four balls which are usually done in one of the two ways:
Where the two hands are out of synchronism. Thus, the two balls ineach of the two hands never interchange.
Synchronous movement where the balls can be interchanged or not at each toss.
Fig. 2e shows the pattern for a normal five-ball cascade, a naturalgeneralization of the three-ball
Shannon's Juggling Theorem:
Stated:
(F + D) H = (V + D) B
D = how long a ball is held in a hand (Dwell time)
H = number of hands (Number of hands)
V = how long a hand is empty (Vacant time)
B = number of balls being juggled (Number of balls)
Theorem 1:
(F + D) / (V + D) = B / H
Corollary:
In a uniform juggle with a fixed flight time, the range of possible periods is B/(B-H)
Theorem 2:
If B and H are relatively prime then there is essentially a unique uniform juggle, such that each ball progresses through the hands in cyclical sequence and each hand catches the balls in cyclical sequence.
Theorem 3:
If B and H are not relatively prime with B=np and H=nq (p and q relatively prime).
Then there are as many types of juggles as ways of partitioning n into a sum of integers.
In the common case of two jugglers (H= 4), each with 3 clubs (B = 6), we have n= 2, which can be written as a sum of positive integers in two ways: 2 or 1+ 1.
The case of 2 corresponds to the jugglers starting simultaneously.
Thus, at each toss there is a choice of two possibilities: a self-throw or a throw to a partner.
The case of 1 + 1, corresponds to two jugglers out of synchronism.There is no way to pass clubs from one pair of hands to the otherwithout violating the uniform juggle condition
Example: 3 balls 2 hands:
36/24 = 3/2
(Three Ball Cascade)
100/40 = 5/2
Example: 5 balls, 2 hands:
(Five Ball Cascade)
100/40 = 5/2
Example: 5 balls, 2 hands:
(Five Ball Cascade)
"Juggling Clowns"
(Heinz Nixdorf - Museums Forum)
In 1982 Shannon built his very own no-drop juggling diorama.
The display features three animated clowns, representing the three great jugglers Ignatov, Rastelli and Virgoaga, juggling the record numbers of props. "Ignatov" juggles 11 rings, "Rastelli" ten balls and "Virgoaga" seven clubs.
The clowns move as if they are actually juggling.
In 1982 Shannon built his very own no-drop juggling diorama.
The display features three animated clowns, representing the three great jugglers Ignatov, Rastelli and Virgoaga, juggling the record numbers of props. "Ignatov" juggles 11 rings, "Rastelli" ten balls and "Virgoaga" seven clubs.
The clowns move as if they are actually juggling.
"Jugglometer"
(Heinz Nixdorf - Museums Forum)
Claude Shannon: Juggling
More:
The Science of Juggling:
By: Peter J. Beek & Arhtur Lewbel
https://www2.bc.edu/arthur-lewbel/jugweb/sciamjug.pdf
Seymour Papert:
Mindstorms:
Fragment about Juggling:
Animations from Wikipedia:
3 ball cascade
5 ball cascade
From: https://en.wikipedia.org/wiki/Cascade_(juggling)
Animations from Juggling Lab:
https://play.google.com/store/apps/details?id=com.jonglen7.jugglinglab
Animations from Juggling Lab:
https://play.google.com/store/apps/details?id=com.jonglen7.jugglinglab
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