This talking about gears on a sphere almost justifies re-mentioning gosper.org/squeezer.mp4 The file is not corrupt. You'll probably need to download to loop it continuously. The curves were difficult to find--arcs of Lissajous figures. --rwg On Fri, Aug 7, 2015 at 12:56 AM, M. Oskar van Deventer < m.o.vandeventer@planet.nl> wrote:
All,
This talking about non-euclidean space reminds me of another Somsky-gear challenge that I would like to pose to you: gears on a spherical surface.
As you all know, it is possible to have eight gears at the corners of a cube geometry, see e.g. my Gear Shift: https://www.youtube.com/watch?v=PkAlP9W0PM0. Another regular gears-on-sphere geometry is my Skweb Shift: https://www.youtube.com/watch?v=aNoFBTNTrTw. Already less regular is my Hex Shift: https://www.youtube.com/watch?v=GRNhS0eboG0. Note that the latter is fudged, as the geometry is not an exact fit.
So here are my new challenges: -Somsky-gears-on-a-sphere: a sun gears, an anti-sun gear (“annulus”?) and three, four or more planets -Cube-corner-gears-on-a-sphere: eight gears of different sizes spread around the sphere in geometry similar to a cube-corner geometry -Other gears-on-sphere geometries with gears of different sizes.
Enjoy!
Oskar
-----Original Message----- From: Warren D Smith Sent: Friday, August 07, 2015 12:35 AM To: William R Somsky Cc: math-fun@mailman.xmission.com ; M. Oskar van Deventer Subject: Re: "Kepler's Law" for Somsky Planets, and the limited(?) number thereof
--WR Somsky: One thing you need to be very careful of when working w/ gears in non-euclidean spaces is that the circumferences -- which must be integral multiples of the gear tooth-spacing -- are NOT proportional to the radii. In euclidean space, the circumfrences and radii are proportional, so we can get by just working w/ the radii as integral multiples of some unit radius.
--WDS: yes, already knew that. In fact, in S2, circumf = 2*pi*sin(angular radius) = 2*pi*(euclidean radius) where note, conveniently, the euclidean radius is what (in my earlier email) I noted got transformed rationally by inversive maps.
In H2, circum = 2*pi*sinh(angular radius).
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)