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 Post subject: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 3:03 pm 
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Hi Non-Twisty Puzzles fans,

Looney Gears is a strange eccentric planetary gearing system. Whereas normal planetary gear systems are symmetrical, this one has no symmetry whatsoever.

This particular set of gears was the result of a challenge by Oskar to Andreas Röver in 2006. Andreas found this nice set of all-primary-numbers gears which is about 0.01% exact.

At this moment (20 July 2012), it is not known whether there exist exact solutions for this type of eccentric planetary gear systems. Oskar's conjecture is that no exact solution exists. It would make a nice math graduation project to prove the conjecture or find a counter example.

EDIT: Bill Somsky proved Oskar's conjecture wrong. It is actually quite easy to find exact solutions, "Somsky Gears".

Watch the YouTube video of Looney Gears.
Watch the animation.
Watch the YouTube video of Somsky Gears.
Buy the object at my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar

P.S. My Californian friend George Miller asked me to make a video in pouring rain. Such situation is much more rare in my country than you would expect. Finally, the right conditions were there: pouring rain and a water-proof puzzling object ready for demonstration.
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Looney-Gears-v2---view-4.jpg
Looney-Gears-v2---view-4.jpg [ 47.25 KiB | Viewed 8990 times ]

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Last edited by Oskar on Sun Dec 23, 2012 6:28 am, edited 3 times in total.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 3:15 pm 
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Hey Oskar, that's really crazy. Great idea using herringbone gears to keep them stable and held in. Isn't the answer to your question again the LCM(7,11,13,17) and since they are all co-prime that's just 7 * 11 * 13 * 17 == 17017?

When you say these gears are about 0.01% exact do you mean that there is no tooth shape / size that makes each gear mesh perfectly? That is there is some tiny amount of "fudging" going on?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 3:26 pm 
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bmenrigh wrote:
Isn't the answer to your question again the LCM(7,11,13,17) and since they are all co-prime that's just 7 * 11 * 13 * 17 == 17017?
Wrong. Were you just guessing?
bmenrigh wrote:
When you say these gears are about 0.01% exact do you mean that there is no tooth shape / size that makes each gear mesh perfectly? That is there is some tiny amount of "fudging" going on?
Indeed. The amount of fudging is actually so tiny, that I did not actually fudge the design at all when I CADded it.

Oskar

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 3:32 pm 
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Oskar wrote:
bmenrigh wrote:
Isn't the answer to your question again the LCM(7,11,13,17) and since they are all co-prime that's just 7 * 11 * 13 * 17 == 17017?
Yeah I was just guessing. I see at least one mistake now though and that is assuming that you have to turn the center gear over
13 times which is obviously not right. If I can change my guess I'd go with 7 * 11 * 17 == 1309.

I think I see now why it's hard to create an asymmetrical planetary gear system. The tooth size is fixed so that all of the teeth mesh together. So the number of teeth instead influences the radius of each gear. You have to find a way to pack circles with radius that are an integer ratio to each other inside of another circle with an integer ratio to them. Seems hard.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 3:35 pm 
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bmenrigh wrote:
If I can change my guess I'd go with 7 * 11 * 17 == 1309.
Wrong again. Like all my YouTube-fan guesses so far. Instead of just randomly selecting numbers and multiplying them, you could also try and do some reasoning ... :D

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 4:01 pm 
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Oskar wrote:
bmenrigh wrote:
If I can change my guess I'd go with 7 * 11 * 17 == 1309.
Wrong again. Like all my YouTube-fan guesses so far. Instead of just randomly selecting numbers and multiplying them, you could also try and do some reasoning ... :D
Yeah it's more complex than I assumed :oops: Watching your video over and over, after you do 1 turn with the 13 gear, if the gears were fixed in space then the 11 gear would have made more than one revolution. Because of the interaction with the outer gear though it ends up going through less than a full revolution. I need to figure out the rule that governs what rate an outer gear does a revolution which seems dependent on both the central gear and the outermost enclosing gear.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 5:44 pm 
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(13*17)*(13*11)*(13*7)/13 = 221221 turns of the central cog per complete cycle.

Note the 37 cog does not factor into the equation, but its circumference has to be rounded ("fudged") slightly to the nearest integer number of cogs.

Is that right?

EDIT: Sorry, forget this post - I forgot that the planets rotate about the sun as the sun turns within the stationary frame. I have recalculated, and posted the correct answer below.

What a challenge this has been! :D

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Last edited by KelvinS on Fri Jul 20, 2012 7:40 pm, edited 1 time in total.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 7:13 pm 
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The more I've thought about this, the less sure I am of anything.

I believe the function that describes how much an orbiting gear turns for every turn of the 13 gear is non-linear.

The best I've been able to come up with is:
f(x) = (13 / x) * 2 * (13 / (37 + x))

Which reduces to:
f(x) = 338 / (x^2 + 37*x)

Unfortunately this function doesn't exactly match (but it's rather close) to Oskar's video. I don't know why this is such a hard problem. I must be approaching it wrong.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 7:37 pm 
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OK, forget my last post. The answer is:

7*11*17*13/(13+37) = 340.34 turns of the central cog to re-align all 4 internal cogs with each other. However, to get rid of the two decimals (about 1/3 of a turn) and make all these cogs vertical again, you need to multiply this by 3, to give:

1021.02 turns of the central cog per complete cycle of all 4 internal cogs, so that they are perfectly aligned and *almost* in their original vertical starting position.

Or if you want an *exact* solution, then you have to multiply 340.34 by 50 instead of 3, which gives 17017 turns as calculated above by bmenrigh - so his answer was technically correct, even if it was just a guess.

:D

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 8:31 pm 
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I'm sleepy, but I found this interesting so I gave it a go. Here's my answer: 1769

Explanation: Ignore the red wheel for now. Look at the other 3 meshing on the outer wheel. They are always the same distance apart. After moving them by 7*11*17=1309 cogs in a particular direction, they will all be upright again. However, the 13 gear won't be, so we multiply by 13 to get 17017 cogs moved to get all upright. But the red cog hasn't rotated 17017 times. Consider the 7 cog and the 13 cog, fixed relative to each other. The 7 cog moves by 17017 cogs, so the 13 cog rotates 1309 times. Then, keep the 13 cog in place, and drag the 7 cog round it by the number of times the 7 cog goes round the 37 cog. This is approximately calculated at 460, and I assume I can round this due to the fudge factor involved. Adding these rotations gives the answer of 1309+460=1769.

I think my reasoning is fairly sound, although lack of alertness may have introduced some silly errors ...


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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 8:55 pm 
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The trouble I'm having with any straightforward analysis is that the assumption that the rate at which the 7 or 11 or 17 gear turns is just some fixed ratio with the 13 or 37 gear.

For example, by looking at Oskar's video, after one turn of the 13 gear it appears that:

  • The 7 gear turns 1.125 times
  • The 11 gear turns 0.625 times
  • The 17 gear turns 0.3125 times

These are not a linear relationship to their number of teeth. Their circular motion due to the interaction with the outer 37 gear is affecting the 13:N ratio in some way.

It seems like the ratio that each gear spins compared to the 13 gear has in some way to do with the ration between the 13 and 37 gear.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 9:04 pm 
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bmenrigh wrote:
It seems like the ratio that each gear spins compared to the 13 gear has in some way to do with the ratio between the 13 and 37 gear.
Yes, 13/(13+37), hence my calculation above.

See: http://en.wikipedia.org/wiki/Epicyclic_gearing

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 9:15 pm 
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KelvinS wrote:
bmenrigh wrote:
It seems like the ratio that each gear spins compared to the 13 gear has in some way to do with the ratio between the 13 and 37 gear.
Yes, 13/(13+37), hence my calculation above.

See: http://en.wikipedia.org/wiki/Epicyclic_gearing

So using this math, how many times does the 7 gear spin when the 13 gear spins once?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 9:53 pm 
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well the 13 gear in the middle doesnt actually rotate on its axis 360 degrees. it gets part of its rotation from its changing position and it gets the other part of its rotation be actually spinning on its own axis. I hope this picture explains what I mean.

The first picture shows the gear getting its rotation from its location changing. the second picture shows the rotation the gear gets from spinning on its axis. thats why the gears dont move 13 teeth in a single rotation. if you could figure out how much it spins when you make a rotation that might help.


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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Fri Jul 20, 2012 11:48 pm 
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bmenrigh wrote:
KelvinS wrote:
bmenrigh wrote:
It seems like the ratio that each gear spins compared to the 13 gear has in some way to do with the ratio between the 13 and 37 gear.
Yes, 13/(13+37), hence my calculation above.

See: http://en.wikipedia.org/wiki/Epicyclic_gearing

So using this math, how many times does the 7 gear spin when the 13 gear spins once?

Sorry it's a bit more complicated than I first thought, but now I've managed to figure out the math. Here goes:

Each full turn of a central cog with s teeth within a fixed outer annual cog with a teeth will result in n turns of a bridging cog with p teeth, where:

n = (a - p) / p * s / (s + a)

In this case s = 13, a = 37, and p = 7, 11 or 17, so:

For p = 7: n = (37 - 7) / 7 * 13 / (13 + 37) = 1.114286
For p = 11: n = (37 - 11) / 11 * 13 / (13 + 37) = 0.614545
For p = 17: n = (37 - 17) / 17 * 13 / (13 + 37) = 0.305882

Note that the lowest common denominator required to give a whole number in each case is:

7*11*17*5*5 = 32725 turns

So 32725 full turns of central cog with 13 teeth gives:

32725 * 1.114286 = 36465 full turns of bridging cog with 7 teeth
32725 * 0.614545 = 20111 full turns of bridging cog with 11 teeth
32725 * 0.305882 = 10010 full turns of bridging cog with 17 teeth

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 12:43 am 
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KelvinS wrote:
Sorry it's a bit more complicated than I first thought, but now I've managed to figure out the math. Here goes:

Each full turn of a central cog with s teeth within a fixed outer annual cog with a teeth will result in n turns of a bridging cog with p teeth, where:

n = (a - p) / p * s / (s + a)
[...]

Great job, I could not figure that equation out. I tried many close / similar ideas but I was mostly guessing.

I don't understand the (a - p) / p term. Can you explain how you derived this the full formula?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 12:50 am 
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bmenrigh wrote:
KelvinS wrote:
Sorry it's a bit more complicated than I first thought, but now I've managed to figure out the math. Here goes:

Each full turn of a central cog with s teeth within a fixed outer annual cog with a teeth will result in n turns of a bridging cog with p teeth, where:

n = (a - p) / p * s / (s + a) [...]

Great job, I could not figure that equation out. I tried many close / similar ideas but I was mostly guessing.

I don't understand the (a - p) / p term. Can you explain how you derived this the full formula?

Yes, assume that the bridging cogs are all fixed in position and spin round their axes so that the outer ring turns as you rotate the central cog, then you can easily calculate rotation of the outer ring as well as each bridging cog for each turn of the central cog. However, you then have to subtract anticlockwise rotation of the outer ring from anticlockwise rotation of the bridging cog, and add anticlockwise rotation of the outer ring to clockwise rotation of the central cog, so that all rotations are measured relative to a fixed outer ring. The complexity arises from using the wrong frame of rotation reference: If you try to think about the outer ring being still from the start then things get confusing because they are moving as well as rotating. It's a bit like trying to calculate the path of the moon by usinig the sun as the frame of reference, rather than the earth. Therefore you have to imagine the bridging planetary cogs as being fixed in position but spinning, do the basic calculations, and only then switch the frame of reference. The addition and subtraction terms come from that last step.

You can re-write the formula above as follows, to see where it comes from:

n = ( s / p - s / a) / ( 1 + s / a )

Think of this as the number of bridging cog turns ( s / p ) per central cog turn ( 1 ), after both of these have been adjusted by the number of outer ring turns per central cog (+/- s / a, depending on relative direction).

It's a great problem, much trickier than I thought, has kept me entertainted for hours. :lol:

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Last edited by KelvinS on Sat Jul 21, 2012 11:51 am, edited 2 times in total.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 10:20 am 
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KelvinS wrote:
...
7*11*17*5*5 = 32725 turns
...

That is 100% correct! Congratulations for being the first person solving this problem. The two drawings below show the effect of those 32725 turns. If you want to get all gears also back to their original position, you'll have to do another 32725 turn, which makes 65450 turns in total.

Oskar

Start position
Attachment:
Looney Gears - start.jpg
Looney Gears - start.jpg [ 94.32 KiB | Viewed 8547 times ]

After 32725 turns
Attachment:
Looney Gears - after 32725 turns.jpg
Looney Gears - after 32725 turns.jpg [ 94.3 KiB | Viewed 8547 times ]

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 10:42 am 
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Oskar wrote:
That is 100% correct! Congratulations for being the first person solving this problem. The two drawings below show the effect of those 32725 turns. If you want to get all gears also back to their original position, you'll have to do another 32725 turn, which makes 65450 turns in total.

Oskar


Oskar, could you please make a demonstration video of the cycle please ?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 10:55 am 
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RubixFreakGreg wrote:
Oskar, could you please make a demonstration video of the cycle please ?
If you make that video and show it unedited on YouTube, then I shall refund your purchase. Note that I expect you to be counting turns out loud.

Oskar

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 11:29 am 
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Oskar wrote:
KelvinS wrote:
...
7*11*17*5*5 = 32725 turns
...

That is 100% correct! Congratulations for being the first person solving this problem. The two drawings below show the effect of those 32725 turns. If you want to get all gears also back to their original position, you'll have to do another 32725 turn, which makes 65450 turns in total.

Oskar

Thanks.

Now who can work out how many possible permutations there are to assemble all four gears within the outer ring, with each cog in any position and orientation? That should tell us what % of all permutations can be reached within 65450 turns (and the probability of ever being able to align all the numbers) from any given starting position.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sat Jul 21, 2012 1:27 pm 
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Oskar, can I speed up the video to make it under the youtube limit as I am not a partner ?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Jul 23, 2012 12:56 pm 
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Would it be possible to make this in acrylic for cheaper? I suppose instead of using herringbone gears, it could just sit flat.

-d


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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Jul 23, 2012 1:32 pm 
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darryl wrote:
Would it be possible to make this in acrylic for cheaper?
The original 2006 version was laser-cut in MDF.

Oskar
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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Jul 23, 2012 1:37 pm 
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darryl wrote:
Would it be possible to make this in acrylic for cheaper? I suppose instead of using herringbone gears, it could just sit flat.
Nice idea.

I was also wondering about more complex versions, with more cogs...

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Jul 23, 2012 1:57 pm 
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KelvinS wrote:
I was also wondering about more complex versions, with more cogs...
Like this?
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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Jul 23, 2012 2:06 pm 
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Oskar wrote:
KelvinS wrote:
I was also wondering about more complex versions, with more cogs...
Like this?
Exactly, but asymmetric... and you may as well ask the same question (turns per cycle?), but make it a bit more difficult this time. :lol: :shock:

Also, I see room in there for 3 more smaller cogs. :wink:

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sun Aug 05, 2012 3:16 am 
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Bill Somsky surprised me with an analysis of the Looney Gears.
    1) He found an algorithm to construct exactly fitting Looney Gears. Let's call those Somsky Gears.
    2) He found an exact solution for the Looney Gears set of 37+13+17+11+7, and managed to find an exact fit for an additional 13 gear.
    3) He found a set of Somsky Gears having six planetary gears.
Amazing!

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Attachment:
37-13-07-13-17-11-10.36618080.png
37-13-07-13-17-11-10.36618080.png [ 4.33 KiB | Viewed 7718 times ]

Attachment:
G34-18-10-8-6.gif
G34-18-10-8-6.gif [ 264.42 KiB | Viewed 7718 times ]

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 12:48 pm 
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Unbelievable! I'd say this should be patented, but I suppose it needs some sort of real world application other than just to look at in awe.

-d


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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 1:25 pm 
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darryl wrote:
Unbelievable! I'd say this should be patented, but I suppose it needs some sort of real world application other than just to look at in awe.

-d

I agree, however you can't patent something after it has been published, except in the US where you get 12 months grace.

Meanwhile, if there's no intention to patent then I would love to see the solution/equation that determines what configurations are possible, just out of curiosity. I assume any participation of pi in the equation cancels out to give the perfect integer solutions?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 1:46 pm 
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Oskar wrote:
At this moment, it is not known whether there exist exact solutions for this type of eccentric planetary gear systems. Oskar's conjecture is that no exact solution exists. It would make a nice math graduation project to prove the conjecture or find a counter example.
Oskar wrote:
Bill Somsky surprised me with an analysis of the Looney Gears.
    1) He found an algorithm to construct exactly fitting Looney Gears. Let's call those Somsky Gears.
So are we saying Oskar's conjecture has been disproven? Who is Bill Somsky? Do we have a nice dissertation or paper to expect to come out of this work? It sounds very interesting and I'd be interested in reading it if so.

Carl

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 4:28 pm 
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darryl wrote:
I'd say this should be patented, but I suppose it needs some sort of real world application
Strangely, whereas Bill cannot patent his Somsky-gear-generating algorithm (at least not according to European patent law, that sort-of excludes software algorithms), still someone could design a twisty puzzle using Somsky gears, and patent that (as twisty-puzzle mechanisms can be patented). So start designing now!
KelvinS wrote:
you can't patent something after it has been published
Well, Bill has disclosed his algorithm to me, but he hasn't published it yet. You have only seen some verifiable examples. Given these examples, can you guess or reverse-engineer his algorithm? It is quite elementary and elegant.
wwwmwww wrote:
So are we saying Oskar's conjecture has been disproven? Who is Bill Somsky? Do we have a nice dissertation or paper to expect to come out of this work?
Yes, my conjecture has been convincingly disproven. Bill is one of my YouTube fans, which is about all that I know about him. And yes, Bill and I are thinking about writing a paper.

Oskar

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 4:37 pm 
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A paper would be fantastic, though I do think you should seriously consider a patent before publishing the algorithm: have a think about significant applications besides 'just' puzzles, especially in engines, etc. :D

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 4:55 pm 
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Oskar wrote:
Bill Somsky surprised me with an analysis of the Looney Gears.
    1) He found an algorithm to construct exactly fitting Looney Gears. Let's call those Somsky Gears.
    2) He found an exact solution for the Looney Gears set of 37+13+17+11+7, and managed to find an exact fit for an additional 13 gear.
    3) He found a set of Somsky Gears having six planetary gears.
Amazing!

Hi Oskar, I have a few questions that I hope you can clarify.

When I was looking at trying to solve your original problem, I was operating under the assumption that the number of teeth for each gear is the only factor controlling the diameter of the gear. That is the ratio of the radius between an 11 tooth gear and a 13 tooth gear is always constant.

I also assumed that for a given set of gears, say, outer 37, inner 7, 11, 13, and 17 that there would only be one way to place the inner sun gear and outer planet gears.

Finally, I'm also assuming that for your 4-planet animation that extra 13 gear planet isn't necessary and for your 6-planet animation you have 2 of each gear and I assume you can pull out each duplicate and still have a perfectly good solution.

So given those assumptions, how is it that your 37 outer gear, 13 sun gear, {7, 11, 17} planets were not an exact solution but the exact solution you show is the same thing with an extra 13th planet (which I assume can be removed). It seems like there is some hidden difference between your approx solution and this exact one and I don't see it.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 4:59 pm 
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bmenrigh wrote:
So given those assumptions, how is it that your 37 outer gear, 13 sun gear, {7, 11, 17} planets were not an exact solution but the exact solution you show is the same thing with an extra 13th planet (which I assume can be removed). It seems like there is some hidden difference between your approx solution and this exact one and I don't see it.

Check out the relative positions of the planetary gears: they gain or lose space as you move them around the off-center sun.

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Last edited by KelvinS on Mon Aug 06, 2012 5:08 pm, edited 2 times in total.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 5:00 pm 
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KelvinS wrote:
Check out the relative positions of the planetary gears: they gain or lose space as you move them around the off-center sun.
Yes I see that but their relative positions didn't factor into your formula at all. Can Oskar just take out the 4 gears he has printed and put them in differently and have an exact solution?

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Mon Aug 06, 2012 5:02 pm 
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bmenrigh wrote:
KelvinS wrote:
Check out the relative positions of the planetary gears: they gain or lose space as you move them around the off-center sun.
Yes I see that but their relative positions didn't factor into your formula at all. Can Oskar just take out the 4 gears he has printed and put them in differently and have an exact solution?

Yes, I believe so, but let's see what Oskar says. I think the original problem came from assuming that the 3 planetary gears should be set about 120 degrees apart to keep the thing stable, so it was this angle which caused the mis-fit, not the gears themselves.

But, while Oskar could take out the 4 gears he has printed and put them in differently to get an exact solution, it would not be very stable because the sun gear would not be supported on one side, hence the additional gear.

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 Post subject: Re: Looney Gears by OSKAR & ANDREAS
PostPosted: Sun Dec 23, 2012 6:10 am 
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Here is a video of the exact Somsky Gears, and how they are generated. Read more in this new post.

Oskar

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