13 ms·

How Many Objects Can Be Juggled (1997)

75 points by mgdo 8y ago

taco_emoji 8y ago
Here's the same paper without the maddening UX: http://www.juggling.org/papers/limits/ http://www.juggling.org/papers/limits/
osteele 8y ago
Also see (Claude) Shannon's Juggling Theorem: https://en.wikipedia.org/wiki/Juggling#Mathematics https://en.wikipedia.org/wiki/Juggling#Mathematics Shannon also built juggling robots. (But not, to my knowledge, machines to measure human juggling.) Here's a video: https://www.youtube.com/watch?reload=9&v=sBHGzRxfeJY https://www.youtube.com/watch?reload=9&v=sBHGzRxfeJY
- imron 8y ago
  > Shannon also built juggling robots. ( I'm disappointed this Kickstarter project never received funding: https://www.kickstarter.com/projects/958028900/human-scale-prototype-for-bugjuggler-a-car-jugglin https://www.kickstarter.com/projects/958028900/human-scale-p...
avmich 8y ago
> I hate to break it to you aspiring numbers jugglers, but no human will ever juggle 100 balls. I hate it when the in first sentence author makes a categorical sentence and then half a page later cites "a - acceleration of gravity = 9.81 m/s^2" as if it's a some sort of fundamental constant :) . Here is an example of juggling under simulated different gravity: https://blog.ted.com/athletic-machines-raffaello-dandrea-at-tedglobal-2013/ https://blog.ted.com/athletic-machines-raffaello-dandrea-at-...
- iiv 8y ago
  Also, he wrote: "g = acceleration due to gravity = 9.81 m/s^-2" m/s^-2 is not the correct unit for acceleration.
  - deleted 8y ago
    [deleted]
  - mark_edward 8y ago
    Yes it is. That's the SI unit. https://en.wikipedia.org/wiki/Acceleration https://en.wikipedia.org/wiki/Acceleration
    - jcranmer 8y ago
      The text was "m/s^-2" (i.e., "m·s²") not "m/s^2".
    - iiv 8y ago
      No it isn't. That's not the SI unit. https://en.wikipedia.org/wiki/Acceleration https://en.wikipedia.org/wiki/Acceleration TIP: Look closer =)
      - mark_edward 8y ago
        Oops, thanks, guess I have mental autocorrect turned on
        iiv 8y ago
        Haha, no problem. It's easy to miss =)
- kybernetikos 8y ago
  > Here is an example of juggling under simulated different gravity I've also seen juggling in VR where gravity can be modified.
jw1224 8y ago
There's a fascinating video on this, which features the current world record holder, Alex Barron — who holds the records for most balls juggled (11) and most balls flashed (14). See https://www.youtube.com/watch?v=7RDfNn7crqE https://www.youtube.com/watch?v=7RDfNn7crqE
qop 8y ago
Can more objects be juggled on the moon then on earth?
- Someone 8y ago
  I don’t think so. The moon’s smaller gravitational acceleration helps, but on earth, you can cheat by juggling objects that have a low terminal velocity and/or that are only just heavier than the air they displace (as long as they are small and light enough so that you can push them upwards reasonably fast, the fact that they won’t go high once you let them go doesn’t matter much)
- rtkwe 8y ago
  Yes, check the definitions of the author's theoretical max ball calculation on the first page gravitational acceleration determines the 'hang time' of a ball for a given value of hand acceleration. So holding everything else the same the same person with the same ability to throw 9 balls at g=-9.8m/s^2 would throw the balls higher on the moon giving them more room in their pattern for more balls.
nsxwolf 8y ago
I can't juggle. Not even in slow motion, with scarves. I've tried many times. I'm convinced some people just can't "get" it.
- patrickmay 8y ago
  I found scarves harder to learn. You can learn to juggle three balls in 30 minutes: 1. Hold one bean bag in your dominant hand while standing, facing a wall, about a foot away. 2. Drop the bag. There, that's over with. (You'll be doing it a lot.) 3. Pick up the bag. 4. Toss the bag from one hand to the other, keeping your elbows loosely at your sides, until you consistently have it arcing at about eye level before descending to the other hand. 5. Do that 100 more times. 6. Hold one bag in each hand (drop them once, if you like). 7. Toss from your dominant hand and, when the bag is at the top of its arc in front of your eyes, toss the other bag. Catch them both. 8. Repeat without pausing another 100 times. 9. Hold two bags in your dominant hand and one in your other hand. Toss the first, wait for the top of the arc and toss the second, wait for the top of the arc and toss the third. 10. You're juggling. Drop all the bags to celebrate.
  - jay-anderson 8y ago
    Definitely. I've had a hard time convincing people to do the required repetitions. They end early and say that they can't juggle. The few people that have, successfully juggle in a relatively short amount of time. Their form isn't great and they can't keep it up for a long time, but they have a great start. A couple other exercises/tips worth mentioning: - Start with your non-dominant hand for 2 balls as well (alternate which hand you start with). - Stand over a couch or bed to make it less costly to drop a ball. - Stand in front of a wall to notice when you're moving forward. - For the advanced: try two in one hand (much harder than 3). Will make 3 ball juggling easier.
    - Kiro 8y ago
      > try two in one hand (much harder than 3). Will make 3 ball juggling easier. What does this mean?
ColinWright 8y ago
One thing that needs to be discussed is the accuracy, which is independent of gravity. You can juggle at about 6 to 8 throws per second, so if you want to juggle, say, 50 objects, each one needs to be in flight for about 7 seconds. That gives 7 seconds of time to drift from the intended landing location (time and space) and that's a long time. Even if you don't have to throw things really high, their "beaten area" grows with time, and that's a quantity I'd be interested in seeing explored.
cridenour 8y ago
If you're looking for a great long form article around juggling, I highly suggest Grantland's article[1] as one of my favorite "why did I just spend an hour reading about that?" [1] http://grantland.com/features/anthony-gatto-juggling-cirque-du-soleil-jason-fagone/ http://grantland.com/features/anthony-gatto-juggling-cirque-...
ErikAugust 8y ago
I came here looking for a paper about OO and having too much state. But this was interesting.
tambourine_man 8y ago
I'm fascinated by these seemingly useless yet remarkably mind grasping problems. I remember reading that Feynman had a profound insight while calculating the wobbling of plates being thrown on a ship. You never know where a fertile mind can be taken by those aimless thoughts.
- pvg 8y ago
  It was just the cafeteria at Cornell, slightly more prosaic than a ship.
logicallee 8y ago
Could a styrofoam or other ball fall more slowly (like a feather) and therefore allow more of them to be thrown?
- jcranmer 8y ago
  The acceleration due to gravity is independent of mass. A 1cm ball made of lead and a 1cm ball made of wood would both fall at the same speed. What does make a difference is shape, due to differing amounts of air resistance and lift.
  - tzs 8y ago
    The force from air resistance at a given velocity will be the same on the 1 cm lead ball and the 1 cm wood ball because they have the same shape and size. The acceleration from air resistance is the force divided by the mass, and so the 1 cm lead ball has much less acceleration from air resistance than does the 1 cm wood ball. As you note, the acceleration due to gravity is the same on both balls. The net acceleration, therefore, is larger for the 1 cm lead ball than for the 1 cm wood ball, and so the lead ball falls faster. The difference in acceleration between the lead ball and the wood ball might not be large enough to easily see if you just drop them a couple meters. Try a ping pong ball and a lead ball of the same diameter (4 cm). That lead ball would weigh 372 g. A ping pong ball is 2.7 g. At a given velocity, the acceleration from air resistance would be almost 140 times greater on the ping pong ball than on the lead ball. That should greatly reduce the height you need to drop from in order to see the difference.
    - ctchocula 8y ago
      This doesn't sound correct to me. The acceleration due to gravity is the same on both balls. However, since the lead ball weighs more than the wood ball, the force of gravity on the lead ball is stronger than the force on the wood ball (F_g = mg). The effect of air resistance can be modeled by a different force. It is typically modeled as a linear function of velocity rather than mass, and it models the behaviour that the faster an object is traveling the more it is affected by air resistance which acts in the opposite direction as the direction the object is traveling in (F_a = -kv). By adding the two forces together, you get a second-order differential equation that describes how the object behaves (F_g + F_a = F_net = ma). [1] https://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html https://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.h...
lanius 8y ago
I'm surprised that robots have not yet surpassed human jugglers. https://en.wikipedia.org/wiki/Juggling_robot https://en.wikipedia.org/wiki/Juggling_robot
spiralx 8y ago
What I find fascinating is how there's a whole mathematical formalism for describing juggling patterns called siteswap which allows patterns to not only be described, but discovered. It's based on throw heights, with the basic 3-ball cascade being represented by 3 - a sequence of balls thrown so they land 3 beats later on, repeated indefinitely. 531 indicates one ball thrown high, one mid-height and then one passed directly across from one hand to the other. Basic intro: http://www.twjc.co.uk/siteswap.html http://www.twjc.co.uk/siteswap.html Much more in-depth treatment here including synchronous and multiplex patterns, ladder and causal diagrams and a bunch of proofs relating to the notation: https://www.jugglingedge.com/pdf/BenBeeversGuidetoJugglingPatterns.pdf https://www.jugglingedge.com/pdf/BenBeeversGuidetoJugglingPa... but you can always go here and try a few patterns such as 3, 441, 531, 504, (4,4) or in windmill/Mill's Mess mode, 423 http://www.gunswap.co/ http://www.gunswap.co/

75 points by mgdo 8y ago

Here's the same paper without the maddening UX: http://www.juggling.org/papers/limits/ http://www.juggling.org/papers/limits/

osteele 8y ago

Also see (Claude) Shannon's Juggling Theorem: https://en.wikipedia.org/wiki/Juggling#Mathematics https://en.wikipedia.org/wiki/Juggling#Mathematics Shannon also built juggling robots. (But not, to my knowledge, machines to measure human juggling.) Here's a video: https://www.youtube.com/watch?reload=9&v=sBHGzRxfeJY https://www.youtube.com/watch?reload=9&v=sBHGzRxfeJY

imron 8y ago
> Shannon also built juggling robots. ( I'm disappointed this Kickstarter project never received funding: https://www.kickstarter.com/projects/958028900/human-scale-prototype-for-bugjuggler-a-car-jugglin https://www.kickstarter.com/projects/958028900/human-scale-p...

avmich 8y ago

> I hate to break it to you aspiring numbers jugglers, but no human will ever juggle 100 balls. I hate it when the in first sentence author makes a categorical sentence and then half a page later cites "a - acceleration of gravity = 9.81 m/s^2" as if it's a some sort of fundamental constant :) . Here is an example of juggling under simulated different gravity: https://blog.ted.com/athletic-machines-raffaello-dandrea-at-tedglobal-2013/ https://blog.ted.com/athletic-machines-raffaello-dandrea-at-...

iiv 8y ago
Also, he wrote: "g = acceleration due to gravity = 9.81 m/s^-2" m/s^-2 is not the correct unit for acceleration.
- deleted 8y ago
  [deleted]
- mark_edward 8y ago
  Yes it is. That's the SI unit. https://en.wikipedia.org/wiki/Acceleration https://en.wikipedia.org/wiki/Acceleration
  - jcranmer 8y ago
    The text was "m/s^-2" (i.e., "m·s²") not "m/s^2".
  - iiv 8y ago
    No it isn't. That's not the SI unit. https://en.wikipedia.org/wiki/Acceleration https://en.wikipedia.org/wiki/Acceleration TIP: Look closer =)
    - mark_edward 8y ago
      Oops, thanks, guess I have mental autocorrect turned on
      - iiv 8y ago
        Haha, no problem. It's easy to miss =)
kybernetikos 8y ago
> Here is an example of juggling under simulated different gravity I've also seen juggling in VR where gravity can be modified.

jw1224 8y ago

There's a fascinating video on this, which features the current world record holder, Alex Barron — who holds the records for most balls juggled (11) and most balls flashed (14). See https://www.youtube.com/watch?v=7RDfNn7crqE https://www.youtube.com/watch?v=7RDfNn7crqE

qop 8y ago

Can more objects be juggled on the moon then on earth?

Someone 8y ago
I don’t think so. The moon’s smaller gravitational acceleration helps, but on earth, you can cheat by juggling objects that have a low terminal velocity and/or that are only just heavier than the air they displace (as long as they are small and light enough so that you can push them upwards reasonably fast, the fact that they won’t go high once you let them go doesn’t matter much)
rtkwe 8y ago
Yes, check the definitions of the author's theoretical max ball calculation on the first page gravitational acceleration determines the 'hang time' of a ball for a given value of hand acceleration. So holding everything else the same the same person with the same ability to throw 9 balls at g=-9.8m/s^2 would throw the balls higher on the moon giving them more room in their pattern for more balls.

nsxwolf 8y ago

I can't juggle. Not even in slow motion, with scarves. I've tried many times. I'm convinced some people just can't "get" it.

patrickmay 8y ago
I found scarves harder to learn. You can learn to juggle three balls in 30 minutes: 1. Hold one bean bag in your dominant hand while standing, facing a wall, about a foot away. 2. Drop the bag. There, that's over with. (You'll be doing it a lot.) 3. Pick up the bag. 4. Toss the bag from one hand to the other, keeping your elbows loosely at your sides, until you consistently have it arcing at about eye level before descending to the other hand. 5. Do that 100 more times. 6. Hold one bag in each hand (drop them once, if you like). 7. Toss from your dominant hand and, when the bag is at the top of its arc in front of your eyes, toss the other bag. Catch them both. 8. Repeat without pausing another 100 times. 9. Hold two bags in your dominant hand and one in your other hand. Toss the first, wait for the top of the arc and toss the second, wait for the top of the arc and toss the third. 10. You're juggling. Drop all the bags to celebrate.
- jay-anderson 8y ago
  Definitely. I've had a hard time convincing people to do the required repetitions. They end early and say that they can't juggle. The few people that have, successfully juggle in a relatively short amount of time. Their form isn't great and they can't keep it up for a long time, but they have a great start. A couple other exercises/tips worth mentioning: - Start with your non-dominant hand for 2 balls as well (alternate which hand you start with). - Stand over a couch or bed to make it less costly to drop a ball. - Stand in front of a wall to notice when you're moving forward. - For the advanced: try two in one hand (much harder than 3). Will make 3 ball juggling easier.
  - Kiro 8y ago
    > try two in one hand (much harder than 3). Will make 3 ball juggling easier. What does this mean?

ColinWright 8y ago

One thing that needs to be discussed is the accuracy, which is independent of gravity. You can juggle at about 6 to 8 throws per second, so if you want to juggle, say, 50 objects, each one needs to be in flight for about 7 seconds. That gives 7 seconds of time to drift from the intended landing location (time and space) and that's a long time. Even if you don't have to throw things really high, their "beaten area" grows with time, and that's a quantity I'd be interested in seeing explored.

cridenour 8y ago

If you're looking for a great long form article around juggling, I highly suggest Grantland's article[1] as one of my favorite "why did I just spend an hour reading about that?" [1] http://grantland.com/features/anthony-gatto-juggling-cirque-du-soleil-jason-fagone/ http://grantland.com/features/anthony-gatto-juggling-cirque-...

ErikAugust 8y ago

I came here looking for a paper about OO and having too much state. But this was interesting.

tambourine_man 8y ago

I'm fascinated by these seemingly useless yet remarkably mind grasping problems. I remember reading that Feynman had a profound insight while calculating the wobbling of plates being thrown on a ship. You never know where a fertile mind can be taken by those aimless thoughts.

pvg 8y ago
It was just the cafeteria at Cornell, slightly more prosaic than a ship.

logicallee 8y ago

Could a styrofoam or other ball fall more slowly (like a feather) and therefore allow more of them to be thrown?

jcranmer 8y ago
The acceleration due to gravity is independent of mass. A 1cm ball made of lead and a 1cm ball made of wood would both fall at the same speed. What does make a difference is shape, due to differing amounts of air resistance and lift.
- tzs 8y ago
  The force from air resistance at a given velocity will be the same on the 1 cm lead ball and the 1 cm wood ball because they have the same shape and size. The acceleration from air resistance is the force divided by the mass, and so the 1 cm lead ball has much less acceleration from air resistance than does the 1 cm wood ball. As you note, the acceleration due to gravity is the same on both balls. The net acceleration, therefore, is larger for the 1 cm lead ball than for the 1 cm wood ball, and so the lead ball falls faster. The difference in acceleration between the lead ball and the wood ball might not be large enough to easily see if you just drop them a couple meters. Try a ping pong ball and a lead ball of the same diameter (4 cm). That lead ball would weigh 372 g. A ping pong ball is 2.7 g. At a given velocity, the acceleration from air resistance would be almost 140 times greater on the ping pong ball than on the lead ball. That should greatly reduce the height you need to drop from in order to see the difference.
  - ctchocula 8y ago
    This doesn't sound correct to me. The acceleration due to gravity is the same on both balls. However, since the lead ball weighs more than the wood ball, the force of gravity on the lead ball is stronger than the force on the wood ball (F_g = mg). The effect of air resistance can be modeled by a different force. It is typically modeled as a linear function of velocity rather than mass, and it models the behaviour that the faster an object is traveling the more it is affected by air resistance which acts in the opposite direction as the direction the object is traveling in (F_a = -kv). By adding the two forces together, you get a second-order differential equation that describes how the object behaves (F_g + F_a = F_net = ma). [1] https://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html https://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.h...

lanius 8y ago

I'm surprised that robots have not yet surpassed human jugglers. https://en.wikipedia.org/wiki/Juggling_robot https://en.wikipedia.org/wiki/Juggling_robot

spiralx 8y ago

What I find fascinating is how there's a whole mathematical formalism for describing juggling patterns called siteswap which allows patterns to not only be described, but discovered. It's based on throw heights, with the basic 3-ball cascade being represented by 3 - a sequence of balls thrown so they land 3 beats later on, repeated indefinitely. 531 indicates one ball thrown high, one mid-height and then one passed directly across from one hand to the other. Basic intro: http://www.twjc.co.uk/siteswap.html http://www.twjc.co.uk/siteswap.html Much more in-depth treatment here including synchronous and multiplex patterns, ladder and causal diagrams and a bunch of proofs relating to the notation: https://www.jugglingedge.com/pdf/BenBeeversGuidetoJugglingPatterns.pdf https://www.jugglingedge.com/pdf/BenBeeversGuidetoJugglingPa... but you can always go here and try a few patterns such as 3, 441, 531, 504, (4,4) or in windmill/Mill's Mess mode, 423 http://www.gunswap.co/ http://www.gunswap.co/