
Juggling by numbers: How notation revealed new tricks - ColinWright
http://www.bbc.co.uk/news/magazine-20728493
======
ColinWright
The writer didn't take one of my suggested edits:

    
    
      > The higher the ball is thrown, the bigger the number,
      > so throwing a four means you are throwing the ball
      > higher than a two. 
    

Originally she said that a 4 is twice the height of a 2, which is simply
wrong. I suggested the above phrase, and then suggested having "The exact
relationship is complicated."

Because the exact relationship ends up predicting the existence of throws that
go backwards in time. When I give my talk I get the audience to make this
prediction, and then I show how it can, in fact, be done (in some
interpretation). I then show that balls traveling backwards in time are the
same as anti-balls traveling forward in time, and it's a lot of fun.

Well, for me, at least.

~~~
oftenwrong
>Because the exact relationship ends up predicting the existence of throws
that go backwards in time. When I give my talk I get the audience to make this
prediction, and then I show how it can, in fact, be done (in some
interpretation). I then show that balls traveling backwards in time are the
same as anti-balls traveling forward in time, and it's a lot of fun.

I don't get it.

~~~
ColinWright
Let's try to help you - here's a revised version of what I said:

1\. The exact relationship between the number used to represent a throw is
complicated.

2\. When you use the relationship, you end up predicting that some throws have
to go backwards in time.

3\. SiteSwap predicts the existence of juggling tricks that use these "Time-
Traveling" throws.

4\. The juggling tricks can, in fact be done.

5\. We interpret a ball going backwards in time as an anti-ball going forwards
in time.

So, which bit don't you get? Ask, and I shall expand.

~~~
madamepsychosis
Could you link us to something explaining the number & the throw? (BTW I saw
your talk as a kid, though before I knew any real maths. Very fun!)

~~~
ColinWright
I've just submitted link to a gentle, but thorough explanation of "Vanilla"
SiteSwaps:

<http://news.ycombinator.com/item?id=4947536>

I don't imagine it will get any upvotes, but it's a place to put some of the
more detailed discussion, if anyone wants it.

In short, we juggle to a fixed rhythm with the hands taking it in turns. For a
given throw, the number is when that ball will next be thrown. Time between
throws is made up of "Flight Time" and "Dwell Time".

Under reasonable, simplifying assumptions, the Dwell Time is always one beat,
so the time of flight is always one less than the number describing the throw.

And thank you! Nice to know you remember it being fun.

------
jpallen
From reading the title I thought that 'juggling' and 'tricks' where metaphors
for how choosing the right notation in mathematics can make a problem trivial,
while choosing the wrong notation can make it nearly impossible. Even though
the article is talking about a concrete example of juggling, I think it can
still be taken as an appropriate metaphor.

Perhaps the oldest and most obvious example of the power of notation is the
invention of algebra itself. Before that, people used to write maths in words.
Fermat's Last Theorem for example was stated as:

    
    
        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.
    

This isn't very easy to understand! With algebra we can just write

    
    
        There are no integers, a, b and c such 
        that a^n + b^n = c^n when n is an integer n > 2.
    

This is much easier to read once you're familiar with algebra, but much more
importantly, the notation is very conducive to being able to manipulate the
expression to derive new results from it.

Perhaps that's the criteria for good notation - it makes additional
manipulations and insights easy to the point of trivial.

~~~
ColinWright
A friend of mine says that the act of "doing math" is the process of inventing
a language (and its associated notation) in which it's easy to talk about the
problem you're working on. Once you have the notation and language, the
problem becomes easy. Well easier.

Certainly the notation for juggling has made a whole slew of juggling tricks
easier to think about, analyse, and learn. A good notation is like a good
programming language, it enables, and then gets out of the way.

~~~
jpallen
Not only does the problem become easier, using the right notation enables us
to do things that we just couldn't do before. Humans (or at least me) can only
hold a few simple things in our head at once. If the problem is too
complicated, we can't deal with it, and the only way to approach it is to
split it down into simpler abstractions that we can hold in our head all at
once. I think this is a large part of what mathematics is - taking concrete
concepts that are hard to hold in our head, and turning them into an
abstraction that describes their important and general properties in a way
that is simple enough that we can keep them in our head. So much of
mathematics is the pursuit of different abstractions and general properties of
things.

I can juggle with 3 balls, and I'm able to do a passable Mills-Mess, but I'm
not sure I have the experience to judge the utility of your juggling notation.
However, it certainly seems to be doing a similar thing. Your notation
contains exactly the same information as the entire pattern but it can
actually be kept in ones head and worked with. Even being able to juggle
Mills-mess, I'm still not really sure what's going on, and certainly struggle
to imagine the whole pattern. I'm not surprised it took you 2 and half pages
to describe!

~~~
ColinWright
You might like my page about it:
<http://www.solipsys.co.uk/new/MillsMess.html?HN0>

And the frame-by-frame analysis:
[http://www.solipsys.co.uk/new/images/MillsAnalysis/Analysis....](http://www.solipsys.co.uk/new/images/MillsAnalysis/Analysis.html?HN0)

------
DanielRibeiro
That was an amazing trick: discovering juggling tricks through math!

Quick meta question: is ColinWright (submitter) the Colin Wright cited on the
article, who did this discovery?

~~~
barking
Never noticed that but you're right
<http://news.ycombinator.com/user?id=ColinWright>

------
ximeng
I thought it was a bit strange to go from mills mess to site swap. Any site
swap can be juggled as a mills mess, they're effectively independent. E.g. see
the first trick in this video for a five ball mills mess 97531 pattern.

<http://www.youtube.com/watch?v=wZx4cLTDWTE>

As the article mentions, site swap doesn't help you to find the interesting
variations. E.g. the 97531 above probably moves too fast for the untrained eye
to appreciate, it just looks like a blur. But a juggler can appreciate the
level of difficulty involved.

For examples of what non-jugglers prefer versus jugglers, compare "Chris
Bliss" and "Chris Bliss Diss" on youtube.

~~~
ColinWright
Stunning juggling - is it you? Or something you found?

~~~
ximeng
Just found it I'm afraid, Mill's mess with 5 no siteswap was about as far as I
got.

------
cytzol
Not only is Siteswap used to turn tricks into sequences, it's used to get them
back out again. Anyone with a few minutes to spare should try Juggling Lab to
see what the tricks look like in practice:
<http://jugglinglab.sourceforge.net/bin/example_gen.html>

~~~
Robin_Message
Fun challenge for anyone: reverse engineer what the numbers mean and the rules
for acceptable combinations by typing them in and observing the juggling
patterns or error messages. Been a good diversion, thanks for sharing the
link.

------
barking
_...it took two and a half sides of A4 and I thought, for a juggling move that
lasts one second there must be an easier way," says Wright.

The system he helped devise became known as Siteswap._

 _When known juggling tricks are written down in [Siteswap] notation form, an
overarching pattern emerges._

Siteswap sounds to be analagous to a high-level programming language.

------
ColinWright
If anyone wants to know more, ask. I'll try not to swamp you with more than
you want.

I am (slowly) writing a book about it.

~~~
tzs
I think it's interesting that such an obscure thing was developed
independently and simultaneously by your group in the UK, and Boppo's group
far away in California.

Is this pure coincidence, or was there some trigger that caused both groups to
start working on the problem at the same time?

~~~
ColinWright
In the late 70s a book was published called "Juggling for the Complete Klutz".
It claimed that if you could find reverse in a Volkswagon Beetle, you could
learn to juggle.

This was given as a joke gift to literally thousands of geeks by people who
thought "Yeah, right, like _that's_ gonna happen." And then it did.

So in the early 80s there were, for the first time, thousands of geeks who
could juggle. At least some were going to take it a bit seriously, and at
least some of them were going to apply scientific processes to trying to
understand the vast and bewildering array of possible tricks.

Paul Klimek of Santa Cruz was chronologically the first, but he didn't really
tell anyone. There's a pre-cursor in an article by Charlie Dancey in Juggler's
World magazine. Boppo and Bengt (in Colorado, not California) started to try
to make it popular, but got some crucial things wrong and didn't explain it
well. They were also _extremely_ enthusiastic, and some people stopped paying
attention, which was a shame.

We invented it independently very shortly after, and our explanations were
tighter, cleaner, and more focused, as you might expect from mathematicians.

It was subsequently invented independently by at least two others that I know
of. It seems to have been "The Right Idea."

~~~
tzs
> Boppo and Bengt (in Colorado, not California)

No, they were in California. Boppo was still an undergrad at Caltech (class of
'87), as was Bengt (class of '88). Colorado is where Boppo went to grad
school, several years later.

> We invented it independently very shortly after, and our explanations were
> tighter, cleaner, and more focused, as you might expect from mathematicians

Actually, the group at Caltech also had a math guy, Bill Banks
(<http://www.math.missouri.edu/~bbanks/vitae.pdf>)

He and Boppo juggled together a lot, but I don't know if Bill made any
contributions to siteswap development.

Another math guy/juggler, Allen Knutson
(<http://www.math.cornell.edu/People/Faculty/knutson.html>), arrived at
Caltech the next year. I don't know if he contributed any improvements or not.

~~~
stib
Wow. I was at Boppo's site swap workshop at some east-coast juggling fest in
the early 90s, where his girl friend would shout out a string of numbers,
often containing 8s and 9s, and he'd demo them. And Allen taught me how to
pass 10 beanbags, although I never made it work with anyone else. Cool to see
where Allen ended up.

I never heard of the Cambridge swaps until now.

------
JonnieCache
Colin, are you familiar with the Turntable Transcription Method? It's a
notation for scratch DJs to describe their hand movements which seems similar
in many ways to what you've been doing here, albeit less mathematical.

<http://ttmethod.com/>

~~~
ColinWright
Cool link, and interesting - thank you. It clearly has a lot in common with
music notation for pure percussive instruments - time along the horizontal
axis, an action indicated vertically.

The thing about all these other notations is that they are descriptive, but
not predictive. The thing that sets SiteSwap apart is that within juggling
there are things that are (with certain axioms) fundamentally impossible. The
notation encodes that in its structure, and then predicts the existence of
things that are possible.

In plain SiteSwap we assume that the juggler throws and catches only one ball
at a time. With that restriction, there are illegal combinations, and SiteSwap
lets us work out what they are so we can avoid them. An example is 543. The
balls throw in that combination all land at the same time, and hence is
illegal in Vanilla SiteSwap tricks.

This has, in turn, led to a lot of people explicitly deciding to break the
rules, and seeing what comes of that.

------
allerratio
Reminds me a bit of knot theory, though knot theory isn't really as practical:
<http://en.wikipedia.org/wiki/Knot_theory>

~~~
ColinWright
We can use Braid Theory - a sub-discipline of Knot Theory - to talk about
juggling. Juggling patterns are usually in a single plane, so imagine juggling
while walking backwards with the balls leaving glowing trails. Those trails
form a braid.

<http://en.wikipedia.org/wiki/Braid_theory>

There are "invariants" for braids, which can then be used to distinguish
juggling patterns. That can be used to show that the 3 Cascade is in some
sense fundamentally different from Mills Mess. Mills Mess is actually the
"Unbraid", something that's rather non-obvious.

------
Aardwolf
I don't get it, first it says the numbers mean how many BEATS it's in the air,
next it says the numbers mean how HIGH it's in the air. Does the same number
mean both?

E.g. it says using 3 balls is 333. So does juggling with 3 balls but throwing
them higher in the air become 444 then? That makes no sense, because it also
said that even numbers mean the ball lands in the same hand.

~~~
ColinWright
I can't find anywhere that it says the number is how high it goes. It says
clearly:

    
    
        > These sequences encoded the number of beats of each
        > throw, which is related to their height and the hand
        > to which the throw is made.
    

It doesn't say that a "4" is twice as high as a "2" (although an earlier draft
did - I managed to get that fixed).

When you throw a ball, the number is how many beats later the ball will next
be thrown. Simplifying, some of that is in the air, and one beat (or so) is in
the hand, so the flight time is one less than the number shown. That would
mean that a "5" is in the air for four beats, and thus will go 4 times higher
than a "3", which is in the air for two beats, or half the time.

The truth is more complicated, because your hands aren't full for half the
time (which is the one beat assumption above), and the dwell time (percentage
time your hand is full) actually varies (although is usually around 65% to
70%).

For the numbers to be used it's the beat time that matters, and you can throw
higher by slowing the beat, but that still leaves a "3" as a "3" and that
doesn't convert it into a "4".

All these details are covered in the gentle introduction, but this specific
question is dealt with in even more detail in the "Technical Notes" section:

<http://www.solipsys.co.uk/new/SiteSwap.html#technotes>

------
jlengrand
haha, reminds me a lot of things from my childhood !

What I love with Siteswap is also that it can predict whether you want to do
is possible. I remember trying to do something for two days before noticing
that it won't work . . . :)

------
lurker14
If there were ever a thread that was useless without pics, this is it.

~~~
ColinWright
See also:

<http://www.solipsys.co.uk/new/JugglingTalk.html?HN1>

<http://www.solipsys.co.uk/new/SiteSwap.html?HN1>

