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 Post subject: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Sun Dec 23, 2012 6:27 am 
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Hi Non-Twisty Puzzles fans,

Bill Somsky invented an algorithm to design exactly fitting Looney Gears, which I would call Somsky Gears. The key understanding is to start with three gears on a diagonal inside the fourth, e.g. 37=7+13+17. Bill proved mathematically that this line of gears can flop around inside the larger circle while retaining an exact fit. Then repeat the process for another set, keeping the outer and central gear the same, e.g. 37=11+13+13. The final step is to overlap the two sets such that the middle gear of one set coincides with the middle gear of another set. This can be achieved only at discrete positions, namely Somsky Gear positions.

The explanation may be a bit hard to follow, so I hope that the video and illustrations help. The Somsky algorithm can help finding exact sets with four planetary gears. Bill even found a set with six planetary gears, which he expects to be very rare.

Watch the YouTube video.
Buy the gear set at my Shapeways Shop. I added the extra 13-gear.
Read more at the Shapeways Forum.
Check out the photos and sketches below.

Enjoy!

Oskar
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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Sun Dec 23, 2012 2:29 pm 
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Hi Oskar, thanks for the follow up, this is pretty cool. The proof doesn't make use of the number of gear teeth, it just makes use of the fact that they all fit together within the circle. I assume based on your video that if three gears in a line sum to the number of teeth of the outer gear then they will fit.

So in keeping with your prime-number-teeth theme, wouldn't 5,13,19 for one line and 11,13,13 for the other (where they share the middle 13 gear) work?

In general, wouldn't pair from the set of partitions of 37 into 3 numbers where the two sets share a number work? For example {12, 12, 13} and {11, 12, 14} ?

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Sun Dec 23, 2012 2:55 pm 
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If I understand the construction properly then this should be an exact solution too:
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I thought for a moment I saw how to fit 8 planetary gears around the center gear but I think I see why that will always have a problem.

EDIT: fixed 28 in image which should have been a 26.

EDIT again: Actually I think 8 planetary gears works. For example:
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EDIT: and probably 10 too:
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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Sun Dec 23, 2012 3:59 pm 
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bmenrigh wrote:
In general, wouldn't pair from the set of partitions of 37 into 3 numbers where the two sets share a number work? For example {12, 12, 13} and {11, 12, 14} ?
That is correct.
bmenrigh wrote:
Actually I think 8 planetary gears works. For example:
...
That is incorrect. You forgot to take the "phase" of the gear teeth into account. This is the reason why your combination {12, 12, 13} and {11, 12, 14} would only work at discrete positions of the center gear. If you add a third set, like {10, 12, 15} then likely none of the discrete solutions for the first two sets would match with the third. The one that Somsky found is an unexpected exception.

Oskar

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Sun Dec 23, 2012 4:09 pm 
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Oskar wrote:
That is incorrect. You forgot to take the "phase" of the gear teeth into account. This is the reason why your combination {12, 12, 13} and {11, 12, 14} would only work at discrete positions of the center gear. If you add a third set, like {10, 12, 15} then likely none of the discrete solutions for the first two sets would match with the third. The one that Somsky found is an unexpected exception.
Hmm so then it would only work in the limiting case where you multiply the number of teeth on each gear by the same factor as that factor goes to infinity. Then they'd be perfectly smooth circles. Obviously this is out-of-bounds for a gear solution though.

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Mon Dec 24, 2012 12:10 am 
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Please excuse the crude analogy but since I haven't done the math to really understand the gears and the angles between them for the phase of the teeth it's the best I have.

Perhaps these are similar to Pythagorean triple triangles like the 3, 4, 5 triangle. There must be certain angles with certain gear counts that allow for more than 4 planetary gears. Is there a formula for the "phase" of the teeth that would indicate where a solution can be perfect or not?

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Mon Dec 24, 2012 3:56 am 
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bmenrigh wrote:
Is there a formula for the "phase" of the teeth that would indicate where a solution can be perfect or not?
Yes. It is not hard to derive. Start with Gear A touching both Gear B and Gear C. "AngleA" is the angle between those touch points and the Gear A center and "TeethA" is the number of teeth of Gear A. Then Gear A adds a phase of AngleA*NumberA. If you have four gears touching in a "square", A-B-C-D, then the accumulative phase should be zero. Hence

AngleA*NumberA - AngleB*NumberB + AngleC*NumberC - AngleD*NumberD = 0

Oskar

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Mon Dec 24, 2012 12:52 pm 
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Oskar, am I correct in understanding that Somsky has come up with a way of finding gearings which work, but there isn't a real proof that there aren't any other gearings which also work? In particular, the requirement that opposite gears all have the same sum is convenient, and I can see it helps in finding solutions, but it isn't clear that it should be an absolute requirement. In fact it's clear that it isn't - you can take the solution with six gears, remove two of them, and presto, a solution with four gears where opposite ones don't have the same sum.


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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Mon Dec 24, 2012 1:03 pm 
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Bram wrote:
Oskar, am I correct in understanding that Somsky has come up with a way of finding gearings which work, but there isn't a real proof that there aren't any other gearings which also work?
That is correct.
Bram wrote:
In fact it's clear that it isn't - you can take the solution with six gears, remove two of them, and presto, a solution with four gears where opposite ones don't have the same sum.
Well, that would still qualify as a Somsky gearing to my modest opinion.

Oskar

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 Post subject: Re: Somsky Gears by BILL SOMSKY and OSKAR
PostPosted: Mon Dec 24, 2012 8:30 pm 
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Come to think of it, when you remove two gears from the six, there's always still an opposite pair which have the convenient Somsky property. Given that if you had eight gears then it would be possible to remove four in such a way that the remaining four didn't have that property it seems very unlikely that such an arrangement can be found.


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