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] --