Mills Mess

The most elegant, bewitching, beguiling, bewildering three ball juggling pattern.

Learn to juggle this well, and you're way beyond just juggling and into artistry as well.

Yes, this is OffTopic, but I frequently use it as an example in work, programming, mathematics, PopularScience, and other fields in which I have an active interest.

Interesting. In what contexts do you use this example?

If you examine only the time aspects and interaction between the hands and balls, it's the same as an ordinary three ball cascade. Appearances can be deceptive, and the simple can be made elegant.
- I explain this in more detail below
If you attach a ribbon to each ball, and pin the other end of the ribbon to a distant wall, can you predict what will happen to the ribbons? (Ideally, that is - in practice, they simply get knotted (so to speak).)
- If you treat juggles as braids, MillsMess is the "unbraid", which comes as a surprise to most.
If you remove one ball the pattern is completely pointless.
- Except for the purpose of practicing

Specifics are more difficult, but these are a few of the points I illustrate with this pattern.

To expand on the first point:

: If you examine only the time aspects and interaction between the hands and balls, it's the same as an ordinary three ball cascade.

Watch the pattern closely and you'll see that it follows this pattern:

Beat : Left hand where? : Right hand where? 1 : Throw Red from right : Catch Blue in middle 2 : Catch Green in middle : Throw Blue from right 3 : Throw Green from right : Catch Red in middle 4 : Catch Blue in middle : Throw Red from left 5 : Throw Blue from left : Catch Green in middle 6 : Catch Red in middle : Throw Green from left

By "Left" and "Right" in this case, I mean the viewer's left and right. Follow the balls and you can see that from the right there are three successive throws: red, blue, green, and then from the left, red, blue green. These are made with alternate hands. All the catches happen roughly in the middle.

But the above timing is exactly that of an ordinary cascade. If you make all the throws from the middle and all the catches on the outside, left hand on left, right hand on right, you get an ordinary three-ball cascade.

Beat : Left hand where? : Right hand where? 1 : Throw Red in middle : Catch Blue on right 2 : Catch Green on left : Throw Blue in middle 3 : Throw Green in middle : Catch Red on right 4 : Catch Blue on left : Throw Red in middle 5 : Throw Blue in middle : Catch Green on right 6 : Catch Red on left : Throw Green in middle

Because the timing is identical and only the positions of throws and catches have been changed, MillsMess can therefore be considered to be an ordinary three ball cascade that has been warped in space, but not time.

Not quite. Not just the position but also the direction of throw is sometimes different as well. Each throw of Red or Green in MillsMess becomes a throw in the opposite direction in the 3-ball cascade, but the direction is the same for each throw of Blue. A similar point applies to the catches.

Hmm. I thought that if the position of a throw, and the position of its corresponding catch have both changed, it would be a good guess that the direction of the throw has to be changed. But for Blue that guess wouldn't be right.

Note that one pattern cannot be continuously deformed into the other. Proof? MillsMess is the unbraid, the cascade, normal or reverse, gives a non-trivial braid. QED

The warping in space creates an asymmetry between the balls. In the above animation, we can see that there's a stage where the blue ball is falling into the middle and passes strictly between the red and green balls. It crosses an imaginary line segment joining the red and green. The green, however, is never strictly between the red and blue, nor is the red ever between the green and blue.

You haven't mentioned it, but is a reverse MillsMess also possible?

It looks like this still hasn't been answered, but yes it is possible, I personally enjoy it. It's a much choppier pattern, and doesn't look as "cool" to a lot of people.

How does one normally start the pattern?

Two ways. Yo can start with red and green in the right hand, blue in the left hand, cross right over left, then

the right hand throws the red from the left back into the middle and the arms uncross
the left hand throws the blue from the left into the middle and the arms cross
- this brings the left hand into the middle where it catches the red ball
the right hand throws the green from under the left arm, back into the middle
- both hands move vaguely to the right, bringing the right hand into the middle where it catches the blue ball
Repeat starting from the other side.

Alternatively, you can transition to it from a standard cascade, although it's easier from a reverse cascade.

The pattern can be practised by juggling with your arms crossed. It can be thought of as changing from juggling crossed right over left to juggling crossed left over right, then back again, repeatedly.

In MillsMess, the standard cascade and the reverse cascade, the balls travel in a figure of eight. In the cascades, the balls are evenly spaced, but in MillsMess they are bunched together on that path. In the standard cascade, they come up in the middle; in both the others, they go down in the middle (the figure of eight is traversed in the opposite direction).

They still all follow identical timing, the only difference being in the spatial positions of the throws and catches. If we assume the hands are full for exactly half the time (and you can juggle these things so that that is true) then on every beat of the metronome one hand throws, the other hand catches.

Why is it called MillsMess?

A one-sided version was invented by RonGraham and SteveMills? together one evening, and then later that evening Steve worked out how to make it two-sided and symmetrical. It's named after him, and since it looks a mess when done badly the alliteration came naturally. It's also inspired BurkesBarrage, RubensteinsRevenge, AndrewsAlliteration and many others. See those first three at:

Here's a side-by-side comparison animation giving MillsMess between standard and reverse cascades -

Notice the perfect symmetry of the cascades - each ball does exactly the same thing.

Would it be possible to modify MillsMess slightly so as to eliminate the spatial bunching you referred to, and hence achieve greater symmetry?

To answer these specific questions:

You can vary the timings of the throws and catches slightly, but in an "ideal" Mills Mess the throws happen at exactly regular, evenly spaced times, and the catches occur at exactly regular, evenly spaced times.
Usually the hands are full for about 60% to 75% of the time, but the exact phase shift can vary slightly. The animation has the hands full for exactly half the time, although immediately after a throw is made the hand is still travelling in that direction a bit, so it looks a little like the ball is still "in" the hand.
The pattern is asymmetrical with regards the balls because the balls have period three, the hand movements have period three, and each throw in a set of three is different.
No, you cannot modify MillsMess slightly so as to eliminate the (spatial) bunching to achieve greater symmetry. It is fundamental to the throw positions.

Thank you for your answers. Further points arise.

You say that the spatial bunching can't be eliminated. Why can't it be eliminated? What goes wrong if one tries to do so?

The bunching is an integral part of what Mills Mess is. Some describe it as three consecutive throws from the left, then three consecutive throws from the right. That gives a kind of fountain, or bunch, of three from the left, then the same effect from the right. These throws are done with the first and third on each side with the arms crossed, with the throwing arm on top on the first of the three, and underneath on the third. If you remove the bunching, it's just not Mills Mess.

The six throws can be described in terms of two groups, but why must there be bunching, rather than even spacing?

If the balls are thrown strictly in turn, and three are thrown consecutively from the same side, they will be on the same half of the figure of eight trajectory.

Also, the ball that passes strictly between the other two is always the middle one of the bunch.

You mentioned "each throw in a set of three is different". This needs slight elaboration to clarify what constitutes a difference (or to state explicitly what differences you meant).

The three throws in Mills Mess are

Right arm crossed over left, throw with the right hand from the left
Arms uncrossed, throw with the left hand from the left
Right arm crossed under the left, throw with the right hand from the left

Then repeat from the other side:

Left arm crossed over right, throw with the left hand from the right
Arms uncrossed, throw with the right hand from the right
Left arm crossed under the right, throw with the left hand from the right

Does it really matter which arm is higher when crossing arms (other than crossing them as described is much easier than not doing so)?

Yes, it really, really does matter which arm is higher. Crossing the arms the other way up for each throw is called the "Funky Mess", is much harder, and in my opinion ugly. It simply doesn't flow nicely. Crossing them the right way gives a natural flow to the movements, where each throw natural leads to the next arm position.

When you throw with the arm that is crossed over the top, it is easy and natural to continue and uncross the arms. When you throw with the arm that is crossed underneath it is not. This is much easier demonstrated than explained, but when you do it properly it is manifestly true. It's part of the definition of MillsMess that it happens this way and not that.

I accept that, especially your point that crossing the arms the opposite way at any stage when the arms are crossed is much more difficult. However, I would regard the type of crossing used as being just that, and would not regard changing it as changing the throw, even though performing the throw becomes more difficult. In other words, I regard the throw as being what the throwing arm does, without regard to what the other arm is doing, even though that may affect the difficulty of carrying out the throw. I accept that the type of crossing used in a juggling pattern (that involves crossing of the arms) is an important part of the full description of the pattern, but I see it as a separate aspect of the full description from the specification of the throw.

Yes, you're absolutely right. In particular, the hand movements and the relative timings of the balls can be divorced from each other and studied separately. Different hand movements, though, give different juggling tricks.

I think I can now devise appropriate timings for MillsMess, and hence diagram the critical moments. Your frame by frame analysis will help me check my diagrams are correct. I would like to identify the constraints that exist (to ensure the juggling pattern is physically possible), and hence make a complete list of all basic juggling patterns with three balls and regular timing (or prove that only the patterns mentioned so far are possible).

This is how I see the standard 3-ball cascade -

Step : Left hand Position : Right hand Position --------------------------------------------------------- 1 : Throw Red in middle : 2 : Catch Green on left : 3 : : Throw Blue in middle 4 : : Catch Red on right 5 : Throw Green in middle : 6 : Catch Blue on left : 7 : : Throw Red in middle 8 : : Catch Green on right 9 : Throw Blue in middle : 10 : Catch Red on left : 11 : : Throw Green in middle 12 : : Catch Blue on right

This is what you get if you concentrate on the juggle as exchanges, not juggling as flow. You get 75% dwell time and it's really jerky. If you focus on juggling as flow you get smoother juggling with a dwell time around 60% - much closer to what the animations are doing. When you concentrate on flow, one hand throws at about the same time as the other hand catches. For each hand there's a longer pause between the throw of one ball and the catch of the next. It's more like this ...

Step : Left hand Position : Right hand Position --------------------------------------------------------- 1 : Throw Red in middle : 2 : 3 : Catch Green on left : 4 : : Throw Blue in middle 5 : 6 : : Catch Red on right 7 : Throw Green in middle : 8 : 9 : Catch Blue on left : 10 : : Throw Red in middle 11 : 12 : : Catch Green on right 13 : Throw Blue in middle : 14 : 15 : Catch Red on left : 16 : : Throw Green in middle 17 : 18 : : Catch Blue on right

Thanks. I used "steps" rather than "beats" to avoid implying regular timing, though I did (incorrectly) get the impression of regular timing for this juggling pattern.

See MillsMessFrameByFrame

For the MillsMess animation, it is more difficult to see just what is happening and hence how to represent it, but without use of "empty" steps, I get a 12-step representation, as follows -

Step : Left hand Position : Right hand Position Frame No. ------------------------------------------------------------------------ 1 : Throw Red from right : : 08 2 : Catch Green in middle : : 16 3 : : Throw Blue from right : 16 4 : : Catch Red in middle : 24 5 : Throw Green from right : : 24 6 : Catch Blue in middle : : 32 7 : : Throw Red from left : 32 8 : : Catch Green in middle : 40 9 : Throw Blue from left : : 40 10 : Catch Red in middle : : 00 11 : : Throw Green from left : 00 12 : : Catch Blue in middle : 08

I've included a timing column to give the frame number from your analysis. Events with equal frame numbers are given separately since precise synchronization is impossible in practice and visually indetectable. It's now clear that a 3 to 1 gallop is not implied.

Here is the 12-step sequence for the reverse 3-ball cascade -

Step : Left hand Position : Right hand Position Frame No. ------------------------------------------------------------------------ 1 : : Throw Red from right : 32 2 : : Catch Green in middle : 40 3 : Throw Blue from left : : 40 4 : Catch Red in middle : : 00 5 : : Throw Green from right : 00 6 : : Catch Blue in middle : 08 7 : Throw Red from left : : 08 8 : Catch Green in middle : : 16 9 : : Throw Blue from right : 16 10 : : Catch Red in middle : 24 11 : Throw Green from left : : 24 12 : Catch Blue in middle : : 32

The above sequences have obvious similarities, but sufficient differences to make the visual patterns very different. There's a spatial distortion in a sense, but an asymmetrical one which involves reversals of direction.

Consider this: after the red ball is thrown, it is caught in the other hand (catching it with the same hand would be rather banal), so the hand that threw the red ball catches another ball, the green one, say, and then throws that one. Where does the blue ball come in? Presumably, it is the next one caught by the hand that just threw the green one. Continuing this analysis, one finds that there isn't much one can do to vary the pattern (changing the colours doesn't count; arm crossing does, so there are distinct patterns). Hence, it's hardly surprising that ANY two juggling patterns involving three balls are either highly irregular or capable of being regarded, on your terms, as spatial distortions of each other (possibly with one reversed). That's why I don't find the spatial distortion concept useful. Also, it's not a continuous mapping.

It's not a continuous mapping in 2D space, correct. That much should be obvious from the assertion that MillsMess is the unbraid, but the cascade in either direction is not the unbraid.

Your analysis about the effective requirement that the balls be thrown strictly in sequence is flawed. There are perhaps over 50 regularly performed tricks with three balls that have different timing from the 3 ball cascade and provably cannot be spatially mapped to a three ball cascade. To get you started:

These are neither trivial nor banal, and they are different from each other as well. There's also 531, 441, 4440, 504 and others. See http://www.solipsys.co.uk/new/SiteSwap.html for an explanation.

You're right that these other patterns aren't trivial and are periodic, but each period combines some trivial steps with considerable irregularity. Consequently, both look rather inelegant. I would apply "highly irregular" to both.

I deduce from this that by "regular" you mean "every throw the same". Yes, you're then correct, everything is the three ball cascade but with throws and catches in different positions. You seem to be saying that since every regular juggling trick is just a three ball cascade with throws and catches in different positions then it's obvious that Mills Mess is just a three ball cascade with throws and catches in different positions. Well, you're right, but when you look at the MillsMess animation, is it obvious? You seem to have spent a long time arguing that it's not true!

By "regular", I meant "incorporating a high degree of symmetry, with the balls following identical routes or routes closely related by symmetry demands".

If both hands do the same thing, every throw is the same and every ball travels the same trajectory you have either a cascade or a reverse cascade. Unless you have an even number of balls, in which case you get a fountain or reverse fountain. If you insist that every juggling trick is like that, then there aren't many. How much symmetry would you like? There's a near continuous line here.

In the first, a trivial throw (the ball goes straight up then down) is followed by two normal throws (although possibly the ball is caught by the same hand, which would be inelegant), and then another straight up throw (or is it a lift?), and the ball is then carried all the way across the juggle-space at maximum height before being dropped. Clearly, each period involves a high degree of irregularity. I don't count carrying a ball in a straight line for an "unnatural" visual effect (the "up, across and drop" could be replaced by a higher, natural throw) as a proper part of juggling, as the move is ugly - it's just a way of lengthening the time the ball carried takes to go from one side of juggle space to the other without increasing the height of the space. If you allow carries, how does one limit the extent to which they are used? Would one allow the ball to be lifted up by one hand, transferred to the other and carried across, and then put down into the original hand without being unsupported at any stage? For natural, graceful juggling, holding of a ball needs to be limited to a short move which is smoothly blended into the next throw.
In the second, the red and blue balls are thrown trivially (straight up and down, to be caught by the same hand), while the green ball is thrown from hand to hand. The irregularity involved here is that each ball follows a different route. These routes could be varied independently (subject to minor constraints to prevent collisions and allow enough time for the hand movements).

You may not find the concept of spatial distortion useful; fine, but many jugglers find it interesting, useful and instructive to know that MillsMess (the three ball version) is, as far as the ball-hand relationship and timing are concerned, the same as the three ball cascade.

Only if "ball-hand relationship" doesn't include direction of throw, which is a somewhat arbitrary restriction.

In some ways, MillsMess is more like the reverse cascade than the standard cascade, but your "spatial distortion" concept applies best to the standard cascade (since only the positions column changes). However, changing the positions to obtain the standard cascade involves reversing the direction in which the figure-of-eight route is traversed, and reversing the direction of the hand movement used to perform the throw. It is this hand movement reversal which I would not describe as spatial distortion, and which I would consider part of "interaction between the hands and balls". So it is those specific points that make the differences amount to more than just spatial distortion, even when timing is synchronized.

You have still to comment on what happens to MillsMess as you gradually change the timing.

Nothing substantial. It looks a little more jerky and a little less fluid. Do you want me to make that animation as well?

... and why do you think doing so can't simultaneously achieve roughly equal spatial intervals, and hence no spatial bunching.

MillsMess requires that the balls come up on the outside, down in the middle, and consists of three consecutive throws from each side in turn. It also requires a specific pattern to the crossing of the arms, but I'll ignore that for now. The balls describe a figure of eight, as is clearly seen in the top section of MillsMessFrameByFrame. Since there must be three consecutive throws from each side, the three balls must at some time all be on the same lobe of the figure of eight, so they are spatially bunched.

Cannot one gradually change just the timing so that the throw positions and directions and the routes traversed are unchanged, but the third throw from the left occurs after the first throw from the right? ...

No, you can't. You have to do three consecutive throws from the left, then three from the right. If the ones from the left are Red, Blue, Green, then the ones from the right have to be Red, Blue, Green. You're asking that the green is thrown from the left after the red is thrown from the right, and now you no longer have three consecutive throws on each side.

I had in mind the juggling pattern below, which would be awkward, but not impossible to perform. The steps are not required to be at equal time intervals, and the throw heights are not all the same.

Step : Left hand Position : Right hand Position ----------------------------------------------------------- 1 : : Catch Blue in middle 2 : Catch Green in middle : Throw Blue from right 3 : : Catch Red in middle 4 : : Throw Red from left 5 : Throw Green from right : 6 : Catch Blue in middle : 7 : Throw Blue from left : Catch Green in middle 8 : Catch Red in middle : 9 : Throw Red from right : 10 : : Throw Green from left

I might not give the above pattern a new name. If you think it does deserve a new name, I assume such a name already exists, in which case it deserves a mention, since it's a pattern much closer to MillsMess than the standard cascade. By using a suitable degree of timing change, wouldn't one eliminate spatial bunching? I would be interested to see the corresponding animation, know what name it has, and know the timing of the throws and catches. I haven't yet constructed tables to see what happens when one tries this, so it may be that some problem arises.

So you're simply theorizing without trying things for yourself. Soon I'll stop doing your homework for you.

Fair point. I'll try to construct a table for it, and then remove the question if I fail.

Achieving exactly 50% dwell time is impossible in practice - one cannot catch with one hand at exactly the same time as throwing with the other. Hence my use of 12-step tables.

Your 12-step tables are wrong. Here is the animation for your 12-step tables. See how the hand flicks comparatively rapidly while empty as compared with when it's full, and compare that against the animations on MillsMessFrameByFrame or at the top of this page.

That's interesting, and I grant that I asked to see it. When I gave 12 steps, however, I didn't say they were equally spaced in time - that's why I called them steps, not beats. I gave 12 steps because I thought there were twelve steps in the animation on this page. You assumed that I was implying a 75% dwell time, but that was not the case.

[Not to be confused with TwelveStepPrograms? ... :P)

On the animation at the top of the page, and those on MillsMessFrameByFrame, the hand moves at about the same speed whether full or empty. If your 12-step table has equal time spacings, then according to your 12-step table the hands don't move at equal speed, and shows that your 12-step table cannot show equal timed steps. That means it shows spatial positions of throws and catches and very little else.

That is correct - the steps were never intended to imply equal timing. The intention was to list the key events in the correct order. I separated the events which didn't seem to be simultaneous (although they are simultaneous if performed exactly as diagrammed).

In particular, with a dwell time of 75% we get this 12-step table with equal timed steps

Step : Left hand Position : Right hand Position ----------------------------------------------------------- 1 : Throw Red from right : 2 : Catch Green in middle : 3 : : Throw Blue from right 4 : : Catch Red in middle 5 : Throw Green from right : 6 : Catch Blue in middle : 7 : : Throw Red from left 8 : : Catch Green in middle 9 : Throw Blue from left : 10 : Catch Red in middle : 11 : : Throw Green from left 12 : : Catch Blue in middle

With dwell times of 25% we get this 12-step table with equal timed steps

Step : Left hand Position : Right hand Position ----------------------------------------------------------- 1 : Catch Red in middle : 2 : Throw Red from right : 3 : : Catch Blue in middle 4 : : Throw Blue from right 5 : Catch Green in middle : 6 : Throw Green from right : 7 : : Catch Red in middle 8 : : Throw Red from left 9 : Catch Blue in middle : 10 : Throw Blue from left : 11 : : Catch Green in middle 12 : : Throw Green from left

Same entries, different relative timings between the hands. Now - did you actually intend to show accurate relative timings? Regardless, you didn't because the dwell time is 50%. See MillsMessFrameByFrame.

That is again based on a false assumption that I intended a 75% dwell time.

It isn't. Just because a ball is still close to a hand it does not mean the hand is holding the ball.

Note that the animation at the top of the page does not have the balls landing in the exact centre. They land slightly further across from centre, making their arcs slightly wider than would otherwise be the case. The animations further down and on the MillsMessFrameByFrame page are adjusted to have the balls caught (or thrown in the case of the normal cascade) exactly in the middle.

That's an interesting point. It would avoid confusion, since you have the frames for both animations, if the frame-by-frame analysis didn't incorporate the adjustment you mention, so that the individual frames given for MillsMess correspond exactly (but showing the arm positions) to the original animation given on this page.

I'll reorganize the discussion to avoid duplications and misconceptions. I've added a frame number column to the tables, so it's now evident that I didn't intend "steps" to imply equal timing. I've altered comments that could be misinterpreted as over-critical.

I suggest you examine MillsMessFrameByFrame and convince yourself that, despite your repeated assertions, the dwell time really is 50%. Okay, I've revised my comments to accept the timing, and concentrate on other aspects.

I look forward to that.

I look forward to getting this page cleaned up ...

For anyone who's interested, there is a wiki dedicated to juggling at http://www.jugglingdb.com/jugglewiki/

To the deleter: many programmers juggle. Also, SiteSwap? notation has direct relevance to SaulAmarel's work or reformulations and the CannibalsAndChristians? problem. Plus this page has signal on it, so go find some crap to delete instead.

Deleted by the originator, etc, etc.

Thanks for originating this page. It's gathered signal. But there's no special privileges in being the first to write a page. Since this one is far from OffTopic in several folks minds, you have no business deleting it. If it's no longer interesting to you, why not move on?

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