[math-fun] Circular motion
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
Martin Gardner wrote about 'Wheels' in Scientific American, September 1970. It was reprinted in 'Wheels, Life and Other Mathematical Amusements' (1983). At least some of your points are touched on, so it is worth a read.
Hi Jim, For questions about what Martin Gardner wrote about, you'll find the MAA CD of his collected Scientific American columns very useful. It includes a searchable index to the entire collection: http://www.maa.org/publications/books/martin-gardner-s-mathematical-games-th... The first physical square wheel bike I know of is by Stan Wagon. I recommend contacting him to ask if he knows of earlier ones. I'm pretty sure they all had straight tracks allowing only a couple of meters of travel then having to pick them up and turn them around, before I came up with the circular track design for the Math Midway and MoMath. George http://georgehart.com/ On 6/26/2015 1:35 PM, 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
entire collection: http://www.maa.org/publications/books/martin-gardner-s-mathematical-games-th...
Someone who has access to this, please search the term "Soucie" looking for an article on Lines of Action, invented by Claude Soucie.
On 26/06/2015 20:37, Dave Dyer wrote:
Someone who has access to this, please search the term "Soucie" looking for an article on Lines of Action, invented by Claude Soucie.
I don't know whether this is in any way helpful to you, but Sid Sackson's book "A gamut of games" has 6 pages devoted to this game. -- g
At 03:42 PM 6/26/2015, Gareth McCaughan wrote:
On 26/06/2015 20:37, Dave Dyer wrote:
Someone who has access to this, please search the term "Soucie" looking for an article on Lines of Action, invented by Claude Soucie.
I don't know whether this is in any way helpful to you, but Sid Sackson's book "A gamut of games" has 6 pages devoted to this game.
Yes, I was aware. I've collected all available info about this game, and most of the unanswered questions have been answered, but one of the dangling references is a report that there was a mention in some issue of Scientific American.
Dave, I checked and "Soucie" does not appear in the text of the MAA CD of Gardner's SciAm columns. George http://georgehart.com/ On 6/26/2015 7:08 PM, Dave Dyer wrote:
At 03:42 PM 6/26/2015, Gareth McCaughan wrote:
On 26/06/2015 20:37, Dave Dyer wrote:
Someone who has access to this, please search the term "Soucie" looking for an article on Lines of Action, invented by Claude Soucie.
I don't know whether this is in any way helpful to you, but Sid Sackson's book "A gamut of games" has 6 pages devoted to this game.
Yes, I was aware. I've collected all available info about this game, and most of the unanswered questions have been answered, but one of the dangling references is a report that there was a mention in some issue of Scientific American.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dave Dyer on Soucie’s Lines of Action: ".. one of the dangling references is a report that there was a mention in some issue of Scientific American." In November 1969, Martin Gardner highlighted Patterns (reprinted in Mathematical Circus), taken from Sidney Sackson’s Gamut. Toward the end he mentions a couple of other games from the book, but not Soucie’s.
Martin Gardner highlighted Patterns (reprinted in Mathematical Circus), taken from Sidney Sacksonâs Gamut. Toward the end he mentions a couple of other games from the book, but not Soucieâs.
Thanks, I can check that out. There's a high probability that's the one I want.
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
Hypocycloids and the mechanics of “rigid disks” come together in an unexpected way … in case you were also looking 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 let’s relax the definition of “rigid”, by allowing the disk to transform by any of the 3-parameter conformal transformations that still have it completely fill the hole. We’ll 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. I’ll 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. Let’s 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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
Dave, Lines of Action is in *A Gamut of Games* by Sid Sackson. On Fri, Jun 26, 2015 at 4:17 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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 “rigid disks†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 let’s relax the definition of “rigid†, by allowing the disk to transform by any of the 3-parameter conformal transformations that still have it completely fill the hole. We’ll 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.
I’ll 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. Let’s 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (9)
-
Allan Wechsler -
Dave Dyer -
Gareth McCaughan -
George Hart -
Hans Havermann -
Henry Baker -
James Propp -
rwg -
Veit Elser