I don't believe self-interlocking gears can be made to work under the following conditions: 1. It "works" in 2D 2. Both gears turn at a constant velocity 3. Interlocking is by a single tooth, when centered on the line between the circles of the gears, being gripped by the two surrounding teeth of the other gear because its width is greater than the gap from the gripping teeth. 4. Both gears are the same size. The reasoning is as follows. For each gear, draw a circle through the "fattest" part of each gear, where the highest percentage of the circle is covered by the teeth, and such that the circles so drawn intersect. The "occupancy" of a gear on a circle is equal to the fraction of the circle covered by the teeth. Consider one of the two points at which those circles intersect. In order for the gears to mesh and spin without clashing, with both spinning at a constant rate, the occupancy of one gear plus the occupancy of the other must sum to 1 or less. Now consider the case when one tooth from either gear is centered on the axis between the two centers. If the occupancy is 1 or less, then the chord from where the edges of the tooth intersect that gear's circle, must be shorter than the chord from where the edges of the gap from the other gear intersect *that* gear's circle. Thus, that single tooth cannot be gripped by the gap from the other gear. This argument might be able to be generalized for the case that the gears are different sizes. So either there's got to be something going on in 3D, or the gears need to stutter, or the interlocking has to be something more complicated then just "grabbing" of a single tooth. -tom On Thu, Jun 19, 2014 at 3:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
No. The gears shown in the picture, gosper.org/Moregears.png while as they are both can always be pulled apart orthogonally and don't work physically due to very slight intersection, can probably be modified so that they work physically and in some phases can't be pulled apart orthogonally. Combining this with the multilayer gear idea (so that you're always in such a phase), it would be possible to make a pair of gears which can't ever be separated orthogonally. They could probably still be separated without using the third dimension, though. Assuming that those modifications work. I might give it a try.
Julian
On Wed, Jun 18, 2014 at 4:54 PM, rwg <rwg@sdf.org> wrote:
-------- Original Message -------- Subject: Re: [math-fun] Spirography Date: 2014-06-18 13:38 From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>
Gosper, is Julian all pessimistic because the classic spirograph has the rotor rolling around the inside of the stator? Does the problem persist if the rotor rolls around the outside?
On Wed, Jun 18, 2014 at 4:23 PM, Steve Witham <sw@tiac.net> wrote:
Date: Mon, 16 Jun 2014 11:54:40 -0400 [in issue 29]
From: James Propp <jamespropp@gmail.com>
Has anyone designed a mechanism that permits the spirographer to focus on circumferential force?
The knockoff product I have, has holes in both parts, suitable for pushing pins through, so in theory you only have to push on the pen tip. Rather than a wax tray the first thing I'd think of would be corrugated cardboard & the second, cork.
I'm imagining something like a latchable/unlatchable zipper.
That's a fascinating idea in itself. It could be used in funicular
railways, for instance.
Although the zipper tooth design is asymmetrical, I have a coat that unzips from both ends.
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