On 2015-06-26 10:35, James Propp wrote:
Here are some questions related to my next blog post:
1) Who first came up with the idea of a square-wheeled bicycle/tricycle, and who first built one?
2) Did Martin Gardner ever discuss it? (And if so, where?)
3) Did Gardner ever discuss the fact that when a flange-wheeled train ( https://en.wikipedia.org/wiki/Train_wheel [1]) is speeding down a track, a part of the wheel is actually going backwards?
4) Did Gardner ever discuss the following puzzle?: "Rotate one dime around another, without slipping. How many turns does the rolling dime make as it goes around once?"
5) Did Gardner ever discuss the fact that there are only 23 hours and 56 minutes in a day, or the related fact that in the course of a year, the Earth spins on its axis 366.25 times?
6) Did Gardner ever discuss the following puzzle?: "Cut a round hole into a piece of plastic so that the diameter of the hole is twice the diameter of a dime, and roll a dime around the inside of the hole without slipping. What path is traveled by a point on the perimeter of the dime?"
7) Does anyone know of a really elementary solution to the preceding puzzle? (Mine using analytic geometry. I could recast it in terms of vectors. But I'm hoping for a more accessible solution.)
8) Does the answer to the preceding puzzle have applications that involve converting linear motion to circular motion and vice versa, e.g. in steam-drive trains?
9) Did Gardner ever discuss other puzzles about circular motion, in the same vein as the ones given above? (I know about the two-screws puzzle, but no others come to mind.)
10) Do any of you know of any other puzzles of this kind that might belong in the same post? (No curves-of-constant-width stuff, please; that'll be a whole different blog-post.)
I plan to send a draft of the essay to math-fun in a week or so. Thanks, Jim Propp Hopefully the essay format permits mgifs. This seems like a good time to remind of Julian's http://gosper.org/sidereal.gif [2] and gosper.org/tri-penta.gif illustrating that a quadricuspid epicycloid is a pentasectrix and a quadricuspid hypocycloid (astroid) is a trisectrix. The angle being sected is between the short, green segment (r=1/4) and horizontal. There are five solutions (plus three mysterious bogons) to the pentasection, and three solutions (plus five mysterious bogons) to the trisection. --rwg Somewhat related is 1. Rotational Symmetry in http://gosper.org/fst.pdf [3] which shows, given the Fourier series for an arc in the complex plane, how to get the Fourier series for that arc repeated m times around a regular m-gon. Links: ------ [1] https://en.wikipedia.org/wiki/Train_wheel [2] http://gosper.org/sidereal.gif [3] http://gosper.org/fst.pdf