The Museum of Science and Industry in Chicago has an example of a pair of "square gears". They remain precisely meshed as they are rotated. A movie of a less precisely meshed pair can be seen here: http://kmoddl.library.cornell.edu/resources.php?id=1768 After looking at the various pages linked to the one above, I am convinced that the pair made at Cornell are based on the following approximation: Start with a square 2a on a side. Replace the corners such with quarter circles such that that the length of each remaining straight section is equal to the length of each curved segment. This can be done by setting the radius of the quarter-circles r = a/(1 + pi/4) If you make a pair of rollers with this cross section and roll one around the other, a curved segment is always in contact with a straight segment. Furthermore, the distance between the centers of the two rollers varies from the average by at most plus and minus 0.031%. So a pair of gears based on this shape needs only a very slight amount of slop to work. An odd thing is that if you try the same trick with other regular polygons as the starting point, you get much poorer results. A pair of rounded hexagons wobble by 0.11%. You have to have a 15-sided figure before you get results superior to a rounded square. This seems really counter- intuitive. My question is, what is the ideal shape that would do this "for real" instead of relying on slop? (Think rollers rather than gears for this.) I have just demonstrated numerically that the supercircle (x^4+y^4=a^4) does not work for this. -- Please reply to: | "Any sufficiently advanced incompetence is pciszek at panix dot com | indistinguishable from malice." Autoreply is disabled |