Great gifs, but please don't use DropBox. Although the link says ".gif", I initially only get a ".jpeg" still image. I then have to turn on Javascript & go through a who sequence of things (including telling DropBox I don't want to join them & put their buggy & insecure code on my machine) in order to actually download the .gif image. What a pain in the butt. At 01:06 PM 6/26/2015, Veit Elser wrote:
Hypocycloids and the mechanics of âÂÂrigid disksâ come together in an unexpected way  in case you were also llooking for something new.
A rigid circular disk, confined to a congruent hole, has only one mode of motion available: simple rotation about the center of the disk.
So letâs relax the definition of ârigidâ, by allowing the disk to transform by any of the 3-parameter conformal transformations that still have it completely fill the hole. Weâll add some physics by placing masses around the circumference of the disk and assume the motion is completely free, determined only by the kinetic energy of the moving masses.
Iâll spare you the derivation of the Lagrangian mechanics of this new kind of rigid disk, and cut right to a description of the motion. First, we still have the possibility that the the disk rotates rigidly in the ordinary sense of a rigid body. Letâs ignore that and look at the new kind of motion. Below are two links. One shows the motion of a disk that has been painted with the Cornell seal. The second shows the motion of the masses along the circumference.
https://www.dropbox.com/s/qmxdookruimvs0o/conformal_cornell.gif?dl=0
https://www.dropbox.com/s/7hhtcq93uvjg2xz/conformal_masses.gif?dl=0
What about that hypocycloid? It turns out that the motion of the center of the disk (central point of the Cornell seal) describes exactly that curve.
-Veit
On Jun 26, 2015, at 1:35 PM, James Propp <jamespropp@gmail.com> 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