Towards the end of the 19th century, there were a large number of patents filed regarding translating various forms of motion into other forms. Many would probably be regarded as trivial by mathematicians today, but many are quite clever, and involve interesting mathematical principles. I haven't tried to DuckDuckGo these patents, but perhaps it might be interesting to do so. BTW, back then, you actually had to produce a working mechanical model of your invention, so a lot of these inventions exist in the Patent Office or in the Smithsonian. PS, in those days you also had to send the Copyright Office copies of your works, which I think was a terrific idea, since many works exist *only* in the form that was saved by the Copyright Office -- e.g., some early film clips exist as sequences of B/W images on paper. The nitrate original film subsequently disintegrated, burned, or converted into WWI ammunition. At 10:35 AM 6/26/2015, 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) 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