---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh@gmail.com> Date: Wed, Jun 18, 2014 at 2:17 AM Subject: Re: [math-fun] Spirography To: Bill Gosper <billgosper@gmail.com> I don't think it can be made to lock. The arc described by any point on a tooth of the inside gear will have curvature of constant sign—i.e. it makes a little convex loop when it comes in to touch the annulus (see picture). It looks like the shape of the loop is such that you can't force locking, but can come fairly close with rounded, convex inner teeth and spiky, concave outer teeth. Of course, the precise shapes would be peculiar to the radii of the disks involved, which would make for fewer combinations. I think the best thing that could be done would be to use three dimensions to effectively add more teeth, by having several layers of teeth, making the teeth slope, or replacing the teeth with an undulating ridge-into-slot mechanism. Caveats: Nothing I said above was rigorous; it was based entirely on intuition from a model of a very special case. The model is also inaccurate, as it assumes that the inner gear rolls along a circle (so that the pen draws a hypotrochoid) rather than pivoting about successive contact points (approximating the hypotrochoid by a series of small arcs). The latter may render my intuition invalid, especially for small numbers of teeth (for lots of teeth you would probably need to make the teeth very short to see the effect), as it was based on looking at hypotrochoids. Actually, I made a model of the small-number-of-teeth version, and the pictures it makes are a neat-looking variation on spirographs: Module[{n = 8, r0 = 6, r1 = 4, r2 = 1, \[Theta]0 = 0}, Manipulate[ParametricPlot[Block[{m = Round[2*\[Pi]*t/(2*(\[Pi]/2 - \[Pi]/n - ArcCos[r0*Sin[\[Pi]/n]/r1]))], t0 = Round[2*\[Pi]*t/(2*(\[Pi]/2 - \[Pi]/n - ArcCos[r0*Sin[\[Pi]/n]/r1]))]*(1/2 - 1/n - ArcCos[r0*Sin[\[Pi]/n]/r1]/\[Pi])}, ri[r0*e[m/n] + r1*e[-(t - t0) + 1/2 + m/n] + r2*e[(t0 - t) + (\[Theta]0 - t0)]]], {t, 0, t1}, Axes -> None, PlotRange -> {{-1.1*r0, 1.1*r0}, {-1.1*r0, 1.1*r0}}, ImageSize -> 400], {t1, 1, 30}]] with e[z_] := Exp[2*I*\[Pi]*z], ri = ({Re[#], Im[#]} &), r2≤r1≤r0, and r1>r0*Sin[π/n]. Julian, who keeps typing "teetch" On Mon, Jun 16, 2014 at 4:40 PM, Bill Gosper <billgosper@gmail.com> wrote:
Allan Wechsler>I want Gosper's young friend Julian's opinion on this. I'm still betting that there are locking solutions for moderate numbers of teeth, say, eight. In fact, I suspect there are locking solutions for all numbers of teeth, but that for larger numbers the grip would be extremely tenuous. On Mon, Jun 16, 2014 at 3:57 PM, meekerdb <meekerdb@verizon.net> wrote:
On 6/16/2014 10:14 AM, Allan Wechsler wrote:
The patent office calls similar devices "rose engines". As for locking gears, I have an intuition that the thing is possible with teeth that have concave shanks and a widened head, which can fit between the heads of the opposing teeth when at an angle, but not when straight on. (One would connect them by using the third dimension, dropping a new gear onto the surface to intermesh with one already there.) My intuition also says that the thing becomes much more delicate when there are lots of teeth.
I doubt that can be made to work. As the teeth unmesh they are pulled out almost orthogonally. You could make gears that resisted unmeshing by embedding magnets in the teeth. Brent Meeker
gosper.org/Spirographgears.png