Spelling: Sonsky --> Somsky. Despite Somsky's 6-planet animations, I do not believe they necessarily represent exact solutions and in fact I already emailed Somsky a calculation about irrational angles indicating one of his animations was in fact NOT an exact solution, but rather contained some phase mismatches. My latest theorem I posted, note, only proves existence with approximate phase angle matches, which in general are not exact. The claim Rokicki has that if all gears have integer radii, then exact meshability happens, I do not anymore believe. (I'd "proved" it, but Somsky showed my proof involved a sign error.) As Somsky himself pointed out, this is a necessary but not sufficient condition; the question of whether suitable "phase angles" exist then must be answered. But Rokicki's argument about planets with one pair of them being "Somsky flexible" indeed makes sense to me, and indeed what it tells me is, in my proof about approximate and exact meshability, whenever you have an unexpected extra degree of motional freedom -- such as Somsky's sun+antisun+2 planets flexible configuration where the sun, amazingly, can move its rotation axis -- you get an extra amount of exact meshability. Specifically, my latest proof I posted allegedly indicates that if the planar bipartite graph G has 1 interior face, then we can get exact meshability in the Euclidean plane, and if it has 2 interior faces, then we can get exact meshability in nonEuclidean planes. But if we now add Somsky-esque flexibility to the mix, that should enable boosting "1 face" and "2 faces" to "2" and "3" respectively, whenever we can embed a suitable subgraph of G in an "unexpectedly flexible" way. The sun+antisun+N planets configuration corresponds to a graph G which is bipartite with 2 pink vertices (sun & antisun) and N blue ones (planets) and N-1 interior faces (each a quadrilateral) and 1 exterior face (also a quad). So my general purpose theorem applied to this planets situation, predicts exactly meshing solutions should happen in Euclidean plane when N=2, and when the first 2 planets happen to obey Somsky's flexibility condition, then when there are N=3 planets. Further, for bevel gears (spherical geometry) my general purpose theorem predicts exact meshing should happen when N=3, and if the first 2 planets happen to yield a flexible situation (which I'm not sure is possible) then when N=4. Rokicki seemed to think he could add a pair of planets at once, but that is not clear to me, maybe his flexing-based argument only allows him to add 1 planet. (Rokicki curx paragraph which may be wrong: "At this point you just insert the larger planets of the two Somsky gears; they are guaranteed to mesh with the sun and the outer planet, so you are done." Really?)

Rokicki seemed to think he could add a pair of planets at once, but that is not clear to me, maybe his flexing-based argument only allows him to add 1 planet. (Rokicki curx paragraph which may be wrong: "At this point you just insert the larger planets of the two Somsky gears; they are guaranteed to mesh with the sun and the outer planet, so you are done." Really?)

--on second thought, I think Rokicki is justified in this claim. So therefore, sun+antisun+4 planet scenarios with exact meshability are indeed available in profusion if BOTH the two planet-pairs obey Somsky's flexibility condition individually. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

Pleasure to hear from you, William! Ahh, so full circle back to Oskar's original question: maybe the problem with the offset problem (after making the sun larger and the outer and planet gears smaller) is that we can't get the symmetrical planets to mesh for the new size. (That is, placing the *third* small planet fails to mesh.) So, William, do you have a symbolic solution to the 34-18-10-8-6 case that shows perfect mesh? Or is it just numerically close? -tom On Mon, Jul 13, 2015 at 3:28 PM, William R Somsky <wrsomsky@gmail.com> wrote:

Yes, wrap a belt around any two planetary gears, and push the sun in until it meets the planets, letting them roll along the annulus. You can then add the complementary gears to each planet as per the geometric proof. This gives a 4 planet "somsky" system. In fact, by using different integral lengths for the belt, you can get multiple, geometrically distinct configurations for the same gears.

I used to have (maybe I can find it) a program to tabulate these pairs, listing them and (a numerical calculation of) the displacement of the sun from the center. To find systems w/ more planets, I looked for sets w/ the same annulus, sun and sun displacement. As the displacement is a computed numerical value, I can only say that they are "close", but cannot prove that they are exact matches.

You have to be careful about invoking symmetry, however. In placing the central sun w/ two different sized planets, the sun may end up in a phase that is not symmetric under reflection...

On 07/13/15 13:46, Tom Rokicki wrote:

...

Well, as posted on math fun, if every continuous belt that you can loop around a particular gear set has an integer length (in tooth count), the gears should mesh. (This is what the Somsky proof is saying, as I understand it, for a particular set of teeth and this is what lets it work.)

I claim that any two pairs of Somsky planets can mesh properly. Here's my explanation.

For any single pair of planets, the location of the sun gear is fairly arbitrary. Indeed, you can take the larger of the planets out, move the sun gear arbitrarily as long as it still touches the remaining planet, and always reinsert the remaining planet with everything meshing perfectly. Somsky's proof shows this clearly.

So let's take the sun gear, and the smaller planet from *one* Somsky pair of planets and the smaller planet from *another* Somsky pair of planets. Can we insert the two smaller planets in some way such that the sun meshes properly with both of these, and the two smaller planets mesh properly with the outer gear? I claim yes, and it's pretty easy to visualize; just put in one smaller planet, then the sun gear, then put the other planet against the outer gear and roll it up to the sun gear. When it touches it won't necessarily mesh, but you can move the newly inserted planet closer to the other planet and the sun brushing along the new planet until they mesh (the surfaces move in different directions as you do this so eventually you'll get a mesh).

At this point you just insert the larger planets of the two Somsky gears; they are guaranteed to mesh with the sun and the outer planet, so you are done.

Your case of *three* pairs of Somsky planets is just a situation where one of the pairs has equal-sized planets, which introduces symmetry, which can then be exploited to introduce a mirror of the second pair of planets.

Does this make sense?

On Mon, Jul 13, 2015 at 1:10 PM, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:

...

Hi Tom,

Thank you for accepting the challenge, and for responding so quickly.

...

do you mean, they won't mesh with the same centers as in the original (36-18-10-8-6) ...

That is indeed what I mean. I want to 3D print a Somsky-style contraption, where the sun gear is coaxial with another sun gear, and the same for each other planetary gears and the annulus.

I can follow your reasoning for the individual (35-17-9-9) meshes as does (35-17-11-7). However, I doubt whether your reasoning is correct for both together. How can we check that?

...

Maybe what happens is the two 11's clash?

I am not worrying about clashes. Mathematically, gears can overlap. Moreover, if your reasoning is correct, then there should exist plenty “Rokicki Gears” that don’t overlap. Or am I misunderstanding you?

Best regards,

Oskar

From: Tom Rokicki Sent: Monday, July 13, 2015 9:18 PM To: M. Oskar van Deventer Cc: Warren Smith ; William Somsky ; Julian Ziegler Hunts ; Bill Gosper ; math-fun Subject: Re: New challenge: Offset Sonsky Gears

Oskar,

Thanks for the challenge!

When you say the (35-17-11-9-7) gears don't mesh---do you mean, they won't mesh with the same centers as in the original (36-18-10-8-6), or they don't mesh in *general* (that is, permitting the centers of the gears to move)?

Clearly by Somsky (35-17-9-9) meshes as does (35-17-11-7), and I would think therefore so would (35-17-11-9-7) with only one 11 and one 7, and then by symmetry so would (35-17-11-9-7) with two 11s and two 7s. That is, I would *expect* that the (35-17-9-9) has enough freedom of the sun gear to permit an 11-7 pair to also mesh. Maybe what happens is the two 11's clash?

I'm not doing this mathematically, but strictly intuitively, which of course is always dangerous . . . but I'm curious where my intuition is breaking down.

-tom

On Mon, Jul 13, 2015 at 12:00 PM, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:

...

Gentlemen,

While you are still discussing new theorems about the Somsky Gears (which I am unable to parse as non-mathematician), I would like to take the liberty and pose a new challenge: Offset Somsky Gears.

What Bill Somksy has proven, is that there are plenty of exact solutions for planetary gears where the sun is offset from the annulus gear, with exactly meshing gears. Bill sent me the below 34-18-10-8-6 example mid 2012.

So how about offsetting the generating circle of each gear as shown in the image below? In this example, I offset Bill’s 34-18-10-8-6 geometry into a 35-17-11-9-7 geometry. So the circles fit in this geometry. However, when drawing the corresponding gears, you will discover that they won’t mesh. So offsetting these Somsky Gears does not yield more Somsky Gears.

For regular planetary gears, the classic threefold symmetrical (120-degrees) concentric geometry has many solutions with different gearing ratios that all mesh.

Now, the challenge is to find asymmetric concentric planetary-gear geometry and/or a Somsky geometry that meshes, AND where the above-described offset yields another exactly meshing configuration.

I hope that the challenge is a bit clear. Probably, a professional mathematician can provide a proper definition of the challenge.

Enjoy!

Oskar

-----Original Message----- From: Warren D Smith Sent: Monday, July 13, 2015 3:31 PM To: M. Oskar van Deventer ; William Somsky ; math-fun Subject: Gear topologies main theorem, revised & corrected

I got tired of posting wrong theorem+proofs that get refuted by return mail, so I wrote it more carefully, put it in a file and stuck it on the web via dropbox. You can read the theorem & proof here:

https://dl.dropboxusercontent.com/u/3507527/GearTopologies.txt

-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

-- -- http://cube20.org/ -- [Golly link suppressed; ask me why] --

-- -- http://cube20.org/ -- [Golly link suppressed; ask me why] --

Yes, I do believe that symmetry allows us to state that this is exact. In fact, I believe that it we can then say that (ignoring overlap -- mesh conditions only) that there are one or more exact configurations -- differing in the offset of the sun gear -- for any set of the form: sun = a, annulus = a+2n, 2x"shepherd" planets = n, 2x"minor" planets = b = one of {1,...,n-1} (yes, a one tooth gear is odd, but works in an odd sort of way) , 2x"major" planets = 2n-b. Construction works approximately thus (loose description): set your ring gear w/ phase 0 straight down. Insert the two "shepherd" planets, meshed, at 3 o'clock and 9 o'clock w/in the ring. The sun can now be inserted, meshed, into the center w/ either phase 0 or 1/2 (depends on if a is odd or even) (symmetry grants this phase). Place the two minor planets, meshed, symmetrically into the lower gap between the sun and the ring, and wrap with a "timing belt" of integral length. By symmetry, the mark/phase on the timing belt on the centerline nearest the sun gear will be 0 or 1/2 (depending on if the belt is even or odd length). Select either odd or even to make this match the phase of the sun gear. The sun gear can then be lowered into contact w/ the belt and the minor planets, w/ the shepherd planets herding the sun down symmetrically. Once contact is made, the major planets complementary to the minor planets can be inserted w/ guaranteed mesh due to the geometric proof. This might be what you were saying Tom, or close to. This works by symmetry, but it is still unclear if any more general exact sets greater than 4 planets exist. As I've said, I can demonstrate ones that are numerically very close, but lack any analytic proof confirming or denying exact matches. They seem to come too close numerically to be purely coincidental, but that's far from a proof. - WRSomsky On 07/13/15 16:02, William R Somsky wrote:

In general, I can only get "numerically close" (something like 10^-6), but it's been a while so I don't remember exactly w/ this one. Symmetry may be able to make this one provable exact.

On 07/13/15 15:43, Tom Rokicki wrote:

...

Pleasure to hear from you, William!

Ahh, so full circle back to Oskar's original question: maybe the problem with the offset problem (after making the sun larger and the outer and planet gears smaller) is that we can't get the symmetrical planets to mesh for the new size. (That is, placing the *third* small planet fails to mesh.)

So, William, do you have a symbolic solution to the 34-18-10-8-6 case that shows perfect mesh? Or is it just numerically close?

-tom

On Mon, Jul 13, 2015 at 3:28 PM, William R Somsky <wrsomsky@gmail.com> wrote:

...

Yes, wrap a belt around any two planetary gears, and push the sun in until it meets the planets, letting them roll along the annulus. You can then add the complementary gears to each planet as per the geometric proof. This gives a 4 planet "somsky" system. In fact, by using different integral lengths for the belt, you can get multiple, geometrically distinct configurations for the same gears.

I used to have (maybe I can find it) a program to tabulate these pairs, listing them and (a numerical calculation of) the displacement of the sun from the center. To find systems w/ more planets, I looked for sets w/ the same annulus, sun and sun displacement. As the displacement is a computed numerical value, I can only say that they are "close", but cannot prove that they are exact matches.

You have to be careful about invoking symmetry, however. In placing the central sun w/ two different sized planets, the sun may end up in a phase that is not symmetric under reflection...

I believe the following eight-planet system to be exact: ring = 53, sun = 27, offset = 20.00, planets = 3, 5, 5, 13, 13, 21, 21, 23 Here is an animation w/ the 23-tooth gear omitted (as it overlaps): Seven Planets On 07/13/15 16:02, William R Somsky wrote:

In general, I can only get "numerically close" (something like 10^-6), but it's been a while so I don't remember exactly w/ this one. Symmetry may be able to make this one provable exact.

On 07/13/15 15:43, Tom Rokicki wrote:

...

Pleasure to hear from you, William!

Ahh, so full circle back to Oskar's original question: maybe the problem with the offset problem (after making the sun larger and the outer and planet gears smaller) is that we can't get the symmetrical planets to mesh for the new size. (That is, placing the *third* small planet fails to mesh.)

So, William, do you have a symbolic solution to the 34-18-10-8-6 case that shows perfect mesh? Or is it just numerically close?

-tom

On Mon, Jul 13, 2015 at 3:28 PM, William R Somsky <wrsomsky@gmail.com> wrote:

...

Yes, wrap a belt around any two planetary gears, and push the sun in until it meets the planets, letting them roll along the annulus. You can then add the complementary gears to each planet as per the geometric proof. This gives a 4 planet "somsky" system. In fact, by using different integral lengths for the belt, you can get multiple, geometrically distinct configurations for the same gears.

I used to have (maybe I can find it) a program to tabulate these pairs, listing them and (a numerical calculation of) the displacement of the sun from the center. To find systems w/ more planets, I looked for sets w/ the same annulus, sun and sun displacement. As the displacement is a computed numerical value, I can only say that they are "close", but cannot prove that they are exact matches.

You have to be careful about invoking symmetry, however. In placing the central sun w/ two different sized planets, the sun may end up in a phase that is not symmetric under reflection...