Juggling by numbers: How notation revealed new tricks | Hacker News Reader

Juggling by numbers: How notation revealed new tricks

bbc.co.uk

65 pointsColinWright40 comments

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.

>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.

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.

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!)

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.

Looks like it's been edited with your suggestion.

It has removed the bit about a 4 being in the air twice as long as a 2, yes, but has not included the bit where I say the exact relationship is complicated.

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.

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.

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!

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....

Though Gauss famously remarked, "What we need are notions, not notations." :)

That's true, but without notation at one level, it's hard to get the notions at the next level. Familiarity, comfort even, with the notation for the things you understand and the things others have done releases your mind to have the notions at the next level.

After that, you develop the notation for these new ideas, and the cycle repeats.

And in addition, very few are as good as Gauss. It's possible, even plausible, that he didn't need the notations at all. I, for one, certainly do.

For what it's worth, I agree. Ramanujan would also be in that category of people who don't need notation.

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?

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

Yes he is. I have had the pleasure of meeting him during one of his train journey layovers. Very nice guy and very interesting to talk to.

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.

It's how it happened. Mills Mess made it clear how hard it was to write down anything, so we started from scratch to invent a notation. We simplified things to start with, saying "Let's not allow the hands to move about" (thus eliminating Mills Mess initially) and "Let's keep the timing fixed", and so on. Thus SiteSwap was born.

There is a similar notation for restricted types of hand movements called "MMSTD", and there is a general, unrestricted notation called "BeatMap". Not sure how much either have caught on amongst juggers in general, but certainly SiteSwap is well-known in the juggling population. Loved by some, hated by some, understood by some, it's an interesting mix.

With regards the "Chris Bliss Diss" video, note that the soundtrack - including applause and gasps - is from the original video. Jason recorded his "Diss" video in a gym without an audience. It's also not a single take. Not to take away from the incredible juggling - it's stunning. I just wish he had done something for himself, and not this diss.

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

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

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

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.

...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.

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.

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?

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."

> 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.

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.

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/

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.

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

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.

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.

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

No. Juggling 3 balls but throwing them higher is still 333. It's just that the beats are slower and so each beat represents more hangtime and therefore height.

Another possibility is to throw all three balls very high, so that all three are in the air at the same time for a moment, perhaps two extra beats. So each ball lands not three, but five beats after it was thrown. And no throws or catches occur during those two extra beats. Siteswap notation expresses that as 55500. There we go, the notation has helped us realize that this action is identical to throwing the first three balls of a five-ball cascade.

Time spent in the air and height are directly related, so yes, the same number can mean both.

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 . . . :)

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