Mike Naylor is a norwegian math professor and artist, who has been ...
### TL;DR
In this paper, Mike Naylor describes a simple method...
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The mathematical art of juggling:
using mathematics to predict, describe and create
Mike Naylor
Norwegian center for mathematics education (NSMO)
Norwegian Technology and Science University (NTNU)
7491 Trondheim Norway
Email: abacaba@gmail.com
Abstract
Mathematics has the power to describe, predict and create patterns, a power that is very well demonstrated in the pattern-
rich world of juggling. In this paper we examine a simple method of describing juggling patterns using mathematical
notation, and then use this notation to predict new juggling patterns. We conclude with a demonstration of how mathematics
has been used to create beautiful patterns that did not exist before these mathematical methods had been used, and how
mathematical names are now used by jugglers worldwide – a powerful demonstration of mathematics advising the arts.
Introduction
Mathematics is the science of patterns, and nowhere can we see that better
than in the field of mathematical art. By recognizing, extending and
creating patterns we can create artistic representations of stunning beauty,
either visually as in prints and sculpture, acoustically as in music or
soundscapes, or dynamically as in dance or performance pieces. One of
the most powerful examples of this interplay between mathematics and art
can be demonstrated in the pattern-rich world of juggling.
The power of mathematics lies in that mathematical tools and
thinking enable us to observe phenomena and use these observations to
describe, predict and create. This power is well demonstrated in the world
of juggling, where mathematics can be and is used to describe juggling
patterns, predict which sequences of notation describe viable patterns, and
then use notation to create new and beautiful patterns that have never
existed before. Perhaps most surprising is that the mathematic notation
and names of some of these patterns have entered the common language
of juggling and are used by non-mathematician jugglers worldwide.
Figure 1 lends a fanciful introduction into our mathematical
exploration of juggling. Titled Nothing but Jugglers, it displays the
inevitable result of creating a scene with nothing but jugglers. In order to
properly represent jugglers, the people in the scene must be juggling
something, but what can those objects be if the only objects allowed are
jugglers? Mathematicians are quick to conclude the scene must be
recursive with the number of jugglers growing exponentially to infinity as
their size decreases to nothing. (An animated version is available at
www.mike-naylor.com).
Mathematicians often find juggling appealing, perhaps because juggling is rich in patterns and
combinations. In fact it was computer programmers who first discovered some of the mathematics of juggling in
their efforts to create juggling pattern simulators. The first notation was created by Paul Klimek in Santa Cruz in
1981 and further developed by others in the following years [1]. Many juggling notation systems have been
created since then, including systems involving multiple jugglers, but none match the elegance and simplicity of
the original system known as “Vanilla Siteswap Notation,” or more simply as “siteswap.”
Figure 1: Nothing but Jugglers; a
juggler juggling jugglers juggling
jugglers... (image by author)
Bridges 2011: Mathematics, Music, Art, Architecture, Culture
33
Vanilla siteswap describes juggling patterns using only a sequence of integers that signifies the length of
time each object is in the air. In performance, a juggler may throw objects behind the back, under the leg, or
from the back of the hands or elbows. Items may be bounced off the forehead or knees, thrown with palms
upwards or downwards, bounced off the floor, or rolled across the chest. Multiple objects may be caught and
thrown with one hand, and hands may throw asynchronously or simultaneously. Siteswap notation ignores these
tricks, making the assumption that each hand throws just one object at a time, alternately in a simple arc.
Despite this severe limitation, this system describes a surprisingly large number of juggling patterns and has
been used to create new juggling moves that are now enjoyed by jugglers worldwide.
Encoding throws and basic patterns
A juggler engaged in a repeating pattern moves to the steady beat of the throws and the steady rhythm of the
balls slapping into the juggler’s hands like a metronome. In siteswap notation the unit of measurement is this
“beat”; each beat corresponding to a point in time in which a throw can be made. Left and right hands alternate,
so that on the first beat, the left hand throws, on the second beat, the right hand throws, and so on. Each throw in
a juggling pattern is labeled by the number of beats which transpire until the thrown object is thrown again. In
practice there are about 3 or 4 beats per second.
Figure 2: Three-ball cascade with path visualizations
The basic three-ball juggling pattern is known as the cascade because it evokes images of water tumbling over
itself (many juggling patterns are given names inspired by water-images). In the three-ball cascade, the balls
travels in arcs of identical height, from right hand to left and left to right. Since the juggling takes place in a
plane parallel to the front of the juggler, we could represent the entire pattern, including the element of time, by
stacking snapshots of the location of the balls to create a three-dimensional (four-dimensional?) woven structure
as shown in Figure 2. (An interactive animation of this structure can be found at www.mike-naylor.com). A
simplified version of a type which we will use to analyze juggling pattern is shown in Figure 3. Each string of
arcs represents the path of one ball through time as it travels from hand to hand. Every throw lands three places
forward of where it originates in this diagram; each of these places corresponds to a beat. In this case each throw
has a duration of three beats. If we were to count “1, 2, 3, 1, 2, 3, 1, 2, 3...” in time to the beats as this pattern
were juggled, the same ball would be thrown on the same numbered count every time.
Figure 3: Simplified three-ball cascade space-time diagram
We can encode this pattern by writing down the number of beats of each throw: “3-3-3-3-3-3- ...” By adopting a
rule similar to repeating decimals and designating the pattern solely in terms of its repeating digits, the above
pattern can be simplified to a single digit: “3.”
Naylor
34
Figure 4 shows the pattern and the space-time diagram for the basic four ball juggling pattern. Every
throw is a “4,” and the pattern name in siteswap notation is therefore also written as “4.” Although the paths
appear to be crossing in this diagram, the balls do not cross paths at all – every ball thrown from the left hand
returns to the left hand and likewise for the right hand. Because of the parity of this system, even numbered
throws return to the same hand while odd numbered throws change hands. This pattern is known to jugglers as
“the four-ball fountain.” (The basic patterns with an even number of objects are called fountains because the
balls seem to spout like water from the center of the pattern and fall outwards.)
Figure 4: Four-ball fountain and its space-time diagram
In general, a one-digit pattern notation denotes the basic juggling pattern for the same number of objects. A “5”
is the 5-ball cascade, a “6” is the 6-ball fountain, etc. (See Table 1). Consider yourself lucky if you get the
chance to witness a “7,” “8,” or “9.” Very few jugglers have mastered the 7-ball cascade, while perhaps less than
a dozen can juggle nine or more.
#
as a throw
as a pattern
0
empty Hand
both hands empty
1
fast hand-off
one ball shuttled back
and forth between
2
hold
one in each hand, no
motion
3
chin-level arc from
one hand to other
three-ball cascade
4
head level throw
returning to same
hand
four-ball fountain
5
above head-level
throw from one hand
to the other
five-ball cascade
6
high throw returning
to same hand
six-ball fountain
7
. . . etc . . .
Table 1: Digits, throws and patterns
The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create
35
Throws corresponding to digits less than 3 warrant some explanation. A “0” is used to denote an empty hand for
one beat. A “1” is a quick hand-off to the other hand. A “2” would, in theory, leave the hand with a very short
toss, and be caught immediately with that hand taking no action between throw and catch. In practice, however,
this is almost never done. Instead, a 2 designates a pause; the hand holds the ball for that beat without throwing
it. The pattern designated by the single digit “2,” then, would be right-hand pause, left-hand pause, right-hand
pause, left-hand pause ... in other words, this pattern is simply holding one ball in each hand and not doing
anything – a juggling pattern most everyone can do!
Two-Digit Patterns
Showers. Most non-jugglers believe that the proper way to juggle three balls is in a “circle,” one hand throwing
the balls in a high arc, the other hand catching and passing the balls quickly to the throwing hand. This pattern is
called “the shower” (another water name!) and it is actually quite a bit more difficult to perform than the
cascade. The space-time diagram for the three ball shower is shown in Figure 5.
Figure 5: Space-time diagram for the three-ball shower
In this case, the left hand is doing the high throws, the right hand is quickly shuttling each ball to the left hand
where it is immediately thrown again. The balls that leave the left hand land five beats later, and the balls that
leave the right hand land one beat later. This patterns requires two digits to encode as there are two different
types of throws being made. It is encoded as the repeating unit of digits, that is, “5-1.”
By convention, we usually name the pattern with the greatest digit first, although this is not necessary.
One could easily say that a “1-5” is the same pattern as a “5-1” but in the opposite direction. Owing to the
ambidextrous nature of juggling and the arbitrary choice of which hand to start the pattern with, we will not
make a distinction and instead refer to patterns and their reflections by the same name.
By increasing the height of the high throw, we “make room” for additional balls. A four-ball shower would
be a “7-1,” a five-ball shower is a “9-1,” and so on. You may wish to verify these with space-time diagrams. The
record for objects juggled in a shower pattern is eight – an incredible “15-1” pattern performed by a woman on
the island of Tonga in the Pacific [3]. In Tonga, juggling is an activity for girls, many of whom juggle limes or
tui-tui nuts. When Western jugglers first came to Tonga, islanders thought it was very funny to see men juggling!
Half-Showers. Another class of juggling patterns are the half-showers. They are similar to showers in that one
hand throws the balls over the top of the pattern, but instead of the other hand making a quick hand-off, the ball
is sent across in short arc. Half-showers are somewhat slower and easier to perform than showers.
The diagram for the four-ball half-shower is shown in Figure 6. Here, the left hand always makes high a
throw to the right hand, which returns the ball to the left in a shorter arc. This pattern is a “5-3.”
Figure 6: Four-ball half-shower
Naylor
36
One-Hand Patterns. Two balls in two hands is just not challenging enough for most jugglers; when we speak
of a two-ball juggle, we generally mean two balls in one hand as shown in Figure 7.
Figure 7: Two balls, one hand
The zero denotes the empty hand; this pattern is encoded as “4-0.” Notice how similar the path diagram is to the
four-ball fountain. In effect, the paths of the two balls in the right hand have been removed, and a zero entered
as a place-holder. Similarly, three balls in one hand (an impressive feat!) is encoded “6-0,” and four balls in one
hand is an “8-0.”
Suppose we were juggling two balls in one hand as above (the “4-0”), and picked up a ball in the other
hand and simply held it. This three-ball pattern would be no more difficult than the “4-0,” but it would have a
different siteswap name: a “4-2” (see Figure 8). Likewise, juggling three in one hand and holding a fourth in
the other hand would be designated “6-2.”
4 2 4 2 4 2 4 2 4 2 4 2 4 2
Figure 8: Two in one with the other hand just holding a ball
Patterns of Patterns. The two-digit patterns have been organized in Table 2, and the “patterns of patterns”
within the table have been extended both downwards and to the right to include such patterns as the “6-4,” a
five-ball pattern in which one hand juggles three by itself, the other hand juggles two.
1 hand
shower
1 hand
+ hold
1/2
shower
1 hand
+ 2 in 1
high
shower
1 hand
+ 3 in 1
super
shower
1 hand
+ 4 in 1
1 ball:
2-0
2 ball:
4-0
3-1
3 ball:
6-0
5-1
4-2
4 ball:
8-0
7-1
6-2
5-3
5 ball:
10-0
9-1
8-2
7-3
6-4
6 ball:
12-0
11-1
10-2
9-3
8-4
7-5
7 ball:
14-0
13-1
12-2
11-3
10-4
9-5
8-6
8 ball:
16-0
15-1
14-2
13-3
12-4
11-5
10-6
9-7
9 ball:
18-0
17-1
16-2
15-3
14-4
13-5
12-6
11-7
10-8
Table 2: Two-Digit Patterns
This listing quickly extends beyond the capabilities of human jugglers, but it accommodates all possible two-
digit siteswap patterns. Notice that the sums of the digits of each of the two-digit patterns is twice the number of
objects being juggled – a property of these patterns we will generalize shortly.
The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create
37
Three Digit Patterns
A “quick and dirty” way to generate some three-digit patterns is to remove one or two paths from the three-ball
cascade. Removing one ball from the pattern gives us the two-ball pattern “3-3-0” (see Figure 9).
3 3 0 3 3 0 3 3 0 3 3 0 3 3
Figure 9: Two balls in three-ball cascade pattern
Removing one more path reduces it to a one-ball pattern: “3-0-0” (see Figure 10).
3 0 0 3 0 0 3 0 0 3 0 0 3 0
Figure 10: A very simple 3-digit pattern
Notice that the “3-3-0” is a two-ball pattern with a digit sum of 6, and the 3-0-0 is a one-ball pattern with a digit
sum of 3 – both sums are three times the number of objects being juggled. Incidentally, the “3-3-0” and the
“3-0-0” are excellent patterns to practice when one is learning the 3-ball cascade.
Let’s replace the zero in the “3-3-0” with a different digit. This digit must represent a throw that will land
in an empty hand, so throws of 1, 2, 4 or 5 will not work; these throws would result in two balls landing in a
hand. A six, however will work - six spaces to the right of a zero in the space-time diagram is another zero
which represents an empty hand and thus a potential landing spot for this additional object. Changing the “0” to
a “6” actually creates two new paths in the pattern, since each new set of arcs connects every other empty space
in the pattern (see Figure 11). The “3-3-6” or equivalently “6-3-3”, is a four-ball pattern wherein two balls
cascade back and forth between hands while the other two balls stay one in each hand in alternating high throws.
This pattern has a digit sum of 12, again, 3 times the number of objects. If instead of changing the zero to a six,
we change it to a nine, there will be room for an additional object in the pattern. A 9-3-3 is a five-ball pattern,
and the digit sum is exactly three times the number of objects.
Figure 11: The four-ball 6-3-3 pattern
Properties of Legal Patterns
As we have seen, one digit patterns are equal to the number of objects juggled, in two digit patterns the digit sum
is twice the number of objects in the pattern, and in three-digit patterns the digit sum is thrice the number of
objects being juggled. In fact, the mean of the digits in a siteswap pattern is exactly the number of balls in the
pattern. If the mean of the digits is not an integer, the pattern will not work. A 2-1, for instance, tells the juggler
to hold a ball in the left hand, make a quick pass from the right, hold the left, quick pass from the right ... the
right hand never receives any balls, and the left hand never gets rid of any balls, an impossible situation.
The following proof that the mean of the digits equals the number of objects being juggled is based on a
clever argument by Warren Nichols of Florida State University [4]:
Naylor
38
Assume we have a legal pattern of n digits and a digit sum of m. In the example below we consider the
legal pattern 6-3-0-2-4; this pattern has 5 digits and a digit sum of 15. Arrange these digits in a circle and
construct a space-time diagram which, instead of being linear, wraps around the circle to join itself in a
continuous manner (Figure 12). Since this is a legal pattern, the paths are uniformly distributed around the
circle, that is, there could not be a place in the space-time diagram where there were, say 3 paths, and
another place where there were 4 paths, as this would indicate that the number of objects being juggled
changes mid-pattern. The circumference of this circle is n beats since there are n digits, and the total
length of the paths is the sum of the digits or m beats, so the paths wrap around the circle m/n times. The
paths are uniformly distributed, so at every point in time during this pattern there are m/n paths, which
means there are m/n objects being juggled.
The wrapped space-time diagram for 6-3-0-2-4 appears below. The circumference is 5 beats, the total
length of the paths is 15, indicating that the paths must wrap around the circle 15/5 = 3 times. 6-3-0-2-4
is thus a three-ball pattern.
Conversely, if m/n is not an integer, then the paths cannot wrap around the circle an integral number of
times. The number of objects being juggled must change during the pattern, and therefore the pattern is
not legal.
Figure 12: The total length of the paths is 15 and the circumference is 5;
the paths wrap around the circle 15/5 = 3 times.
The fact that the mean of the digits must be an integer is a necessary but not sufficient condition for the digits to
represent a legal pattern. One can create strings of digits which average to an integer, but which are impossible to
juggle.
5-4-3, for example, has an integral average of 4. The space-time diagram in Figure 13 shows what
happens when this pattern is attempted.
L R L R L R
5 4 3
Figure 13: Collision!
All three balls land at the same time in the same hand! While in real-life, this would be an impressive feat, in
siteswap notation this situation is referred to a collision, and it is not allowed. In order for a pattern to be legal,
only one object may land in a hand at one time.
The problem with this pattern is not the choice of digits, but rather the order of the digits. If we permute
the digits as “5-3-4,” the pattern works just fine as shown in Figure 14.
Figure 14: Collision fixed
The Mathematical Art of Juggling: Using Mathematics to Predict, Describe and Create
39
Construction of a space-time diagram is not necessary to determine if a pattern is legal. Other mathematical tools
can be found in a paper included in the references [4].
The Lost 3-Ball Pattern and Other Patterns
We now have all the tools necessary to create and test legal patterns. All that is left it to use these tools to
creatively explore the space of possible juggling patterns. One of the truly great powers of mathematics is
exemplified in the story of siteswap notation. As jugglers studied pattern notation, they discovered patterns that
were beautiful and unknown. One of the most striking is the “4-4-1” pattern (Figure 15). This pattern consists of
two staggered throws straight up, then a quick hand-off which is immediately sent up as the first of a different
pair of staggered fours (Figure 18). This pattern is beautiful to see and satisfying to perform, and was not
discovered until mathematics showed the way!
Figure 15: The 4-4-1 (“The lost 3-ball pattern”) is referred to by jugglers worldwide as the “4-4-1”.
While many juggling patterns have fanciful names such as the cascade, the shower, and the fountain, in juggling
circles the “4-4-1” pattern is called by its siteswap name. The mathematical terms have entered into juggling
culture worldwide – a powerful testament to the interplay between mathematics and the arts. Other siteswap
pattern names broadly used by jugglers include the 5-3-1 and the 5-5-5-0-0. Juggling is one of the few arts in
which practitioners regularly use names derived from mathematical notation!
These pattern names exemplify the true power of mathematics. We can observe a phenomenon and create
mathematical notation to describe it, look for patterns and generate rules within our notation, and then use these
rules to create new symbolic patterns which correspond to new discoveries within the natural world. Not only
does mathematics have the power to describe – mathematics has the power to create!
Finale
Several siteswap juggling pattern simulators are available for free on the internet, but siteswap notation is more
than just a powerful tool for creating electronic jugglers. Siteswap can aid jugglers while practicing complicated
patterns by focusing attention on individual throws, and the notation has been used to discover new patterns
which were previously unknown anywhere. Incidentally, there are 11 legal three-digit three-ball patterns –
finding them all and visualizing what they look like is a nice challenge.
Regardless of whether or not a juggler understands or uses siteswap, juggling is still a sport of kinesthetic
mathematics. One cannot help but be mystified and drawn into the intricate web of translations, inversions,
dilations, and permutations as a skilled juggler weaves patterns upon patterns of order balancing on the edge of
chaos.
References
[1] Polster, B., The Mathematics of Juggling, Springer, 2002.
[2] Knutson, A., Site-Swap FAQ, [Online] Available: http://www.juggling.org/help/siteswap/faq.html [January
12, 2012].
[3] Truzzi, M., On Keeping Things Up in the Air, Natural History, 88, 10, December 1979.
[4] Naylor, M., Patterns of patterns: the mathematics of juggling, The Journal of Recreational Mathematics, 32,
1, 2004.
Naylor
40
Here's a similar animation made by a reddit user
Mike Naylor is a norwegian math professor and artist, who has been focusing in developing creative ways to educate people about math concepts.
The fact that the mean of the digits of a sequence is the number of objects being juggled gives us an easy and quick way to exclude possible juggling sequences.
For example, 5432 can't be a juggling sequence because (5 + 4 + 3 + 2)/4 is not an integer. However, this is a necessary but not sufficient condition to have a valid juggling sequence - for example, 321 has this property but is not a juggling sequence.
### TL;DR
In this paper, Mike Naylor describes a simple method of describing juggling patterns using Mathematical notation. He then proceeds to infer a simple set of rules that tell if a juggling sequence is valid or not.
Using these rules, Mike shows that it is possible to generate all the valid juggling sequences and ends up finding a missing pattern: 4-4-1!
The notation invented by Paul Klimek in 1981 is called site-swap. It represents the order in which props are thrown and caught in each cycle of the juggle, assuming throws happen on beats that are equally spaced in time.
Site-swap notation uses the time between tosses to characterize the pattern
If you want to learn more about the capabilities of human jugglers you should [read this paper](https://fermatslibrary.com/s/how-many-objects-can-be-juggled) by Jack Kalvan that has been annotated on Fermat's Library.
Jack is a mechanical engineer and avid juggler, tries to answer the following question: What is the maximum number of objects a human can juggle?
The analysis is centered around the maximum velocity of the hands, which being a limiting factor, is only one piece of a more complex puzzle that involves throwing accuracy among others.
Kalvan writes that flashing 15 balls (each ball is only thrown and caught once) "doesn't seem too unlikely" and his prediction appears to be not too far from reality - in April 2017, Alex Barron was able to flash 14 balls.
[](https://www.youtube.com/watch?v=WK6AlhTjUbY)
[For more information on Alex's record you can check this [article](https://www.wired.com/story/the-physicsand-physicalityof-extreme-juggling/)]