[math-fun] dirac scissors trick + balinese cup trick
(This is the same thing that says you can hold a full cup of coffee in one hand, and without letting go or spilling, turn it around twice and get back in the same position. However, if you turn it around just once, your arm will be in a different, probably awkward, position.)
I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same as a 720 degree one, using strings tied to a pair of scissors, right? My question: where/when did he do it? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org
The 360-720 difference can be illustrated quite dramatically by making a square frame with (as I recall) four strings attached which go through holes in a small block inside the frame. I realize that details here are sparse, but I could probably make a crude one if offered enough money (say twelve million dollars). Steve Gray ----- Original Message ----- From: "Thane Plambeck" <thane@best.com> To: <math-fun@mailman.xmission.com> Sent: Sunday, April 04, 2004 11:47 AM Subject: [math-fun] dirac scissors trick + balinese cup trick
(This is the same thing that says you can hold a full cup of coffee in one hand, and without letting go or spilling, turn it around twice and get back in the same position. However, if you turn it around just once, your arm will be in a different, probably awkward, position.)
I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same as a 720 degree one, using strings tied to a pair of scissors, right?
My question: where/when did he do it?
Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thane: Dirac visited SIUC around 1978 for an Einstein symposium. When I met him, I mentioned to him that I recalled having seen -- around 1965 -- a brief reference to his spinor-related trick in Time magazine. The article suggested that he had discovered it around 1932. He confirmed that that was probably about the right date. I suggest that you ask my physicist colleague Jerzy Kocik (jkocik@math.siu.edu) to fill you in on the details of the spinor connection. He has demonstrated the trick (with flailing arms) and he has expert knowledge of quantum mechanics. I'll send a copy of this email to Jurek (Jerzy). Alan At 01:47 PM 4/4/2004, you wrote:
(This is the same thing that says you can hold a full cup of coffee in one hand, and without letting go or spilling, turn it around twice and get back in the same position. However, if you turn it around just once, your arm will be in a different, probably awkward, position.)
I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same as a 720 degree one, using strings tied to a pair of scissors, right?
My question: where/when did he do it?
Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I've seen this phenomenon described in a nice way, although I haven't done the construction myself. Build a frame, say a cube with 12 rods. Place an object inside the frame, and tie it with strings to the frame, for example 8 strings to the vertices of the cube. The strings need to be slack enough to provide the wiggle room required. Rotate the object 720 degrees. It is then possible to manipulate the strings while holding the frame and object fixed, so as to untwist the strings and restore the original configuration. On the other hand, if the object is rotated 360 degrees, this manipulation cannot be performed. Gene __________________________________ Do you Yahoo!? Yahoo! Small Business $15K Web Design Giveaway http://promotions.yahoo.com/design_giveaway/
Although I haven't tried it, this presumably would be easier to see and do with a small and large tetrahedron rather than a cube. Eugene Salamin writes:
I've seen this phenomenon described in a nice way, although I haven't done the construction myself. Build a frame, say a cube with 12 rods. Place an object inside the frame, and tie it with strings to the frame, for example 8 strings to the vertices of the cube. The strings need to be slack enough to provide the wiggle room required. Rotate the object 720 degrees. It is then possible to manipulate the strings while holding the frame and object fixed, so as to untwist the strings and restore the original configuration. On the other hand, if the object is rotated 360 degrees, this manipulation cannot be performed.
Here's an easy (and I suppose well known) way to demonstrate the double twist phenomenon. Take an ordinary belt. Anchor the "tongue" end, make a double twist and hold the buckle "shiny side up". The aim is to untwist the belt without ever rotating the buckle, i.e. keep it shiny side up at all times so it is always parallel to its original position. It's possible for 720 but not 360 degrees as you will easily see. DG At 07:52 AM 4/7/2004 -0500, you wrote:
Thane:
Dirac visited SIUC around 1978 for an Einstein symposium. When I met him, I mentioned to him that I recalled having seen -- around 1965 -- a brief reference to his spinor-related trick in Time magazine. The article suggested that he had discovered it around 1932. He confirmed that that was probably about the right date.
I suggest that you ask my physicist colleague Jerzy Kocik (jkocik@math.siu.edu) to fill you in on the details of the spinor connection. He has demonstrated the trick (with flailing arms) and he has expert knowledge of quantum mechanics. I'll send a copy of this email to Jurek (Jerzy).
Alan
At 01:47 PM 4/4/2004, you wrote:
(This is the same thing that says you can hold a full cup of coffee in one hand, and without letting go or spilling, turn it around twice and get back in the same position. However, if you turn it around just once, your arm will be in a different, probably awkward, position.)
I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same as a 720 degree one, using strings tied to a pair of scissors, right?
My question: where/when did he do it?
Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's not hard to find examples of graphs mapped to the unit 2-sphere in a way that is unshrinkable, in the sense that there is no homotopy that decreases at least one edge-length without increasing any of the others. For instance, a tetrahedral graph will work. You can think of these physically as net bags that tightly hold a sphere. Puzzle: show that for any such example, there is a two-fold covering space of the graph such that the composed map to the 2-sphere **can** be shrunk, at least slightly. Extra: show that the same holds for graphs mapped to spheres of arbitrary dimension. In at least some cases---for instance, the rounded-out edges of a tetrahedron, octahedron, cube --- this 2-fold covering of the graph can be chosen so the map shrinks all the way to a constant map. The sphere can be extracted from the double cover of the net bag (physically as well as mathematically). I haven't tried constructing these, but they ought to make nice parlor tricks except for the small detail that nobody has a parlor any more. I strongly suspect that for any graph mapped to S^2, some finite sheeted cover can be shrunk all the way to a point. If so, k degree covers are needed, and whether it can always be done in a physically realizable way (i.e. so the strings don't get entangled with each other and you can physically remove the ball from the bag.) Bill Thurston
I recently had the pleasure to try out the Quantian distribution. This is a math-intensive Linux distribution that runs directly off of a CD. For any computer, you can put in the CD, reboot, and you'll be running Quantian Linux in a few minutes (keyboard, mouse, and all else are auto-detected). When done, reboot, remove the CD, and you are back to your current system. Your hard drive will be untouched. It has many math packages I've wanted to try for awhile. Several of them, such as LyX and TeXmacs, have nontrivial set-up procedures. I was delighted to be able to try them all. # Octave, with add-on packages octave-forge, octave-sp, octave-epstk, matwrap and Inline::Octave; # Computer-algebra systems Maxima (including the X11 front-end and emacs support), Pari/GP, GAP, GiNaC and YaCaS; # GSL, the Gnu Scientific Library (GSL) including example binaries; # the QuantLib quantitative finance library including its Python interface; # the Grass geographic information system; # the OpenDX and Mayavi data visualisation systems; # TeXmacs for wysiwyg scientific editing as well as LyX and kile for wysiwyg (La)TeX editing; # various Python modules including Scientific and Numeric Python; # and various other programs such as apcalc, aplus, aribas, autoclass, euler, evolver, freefem, gambit, geg, geomview, ghemical, glpk, gnuplot, gperiodic, gri, gmt, gretl, lp-solve, mcl, mpqc, multimix, rasmol, plotutils, pgapack, pspp, pdl, rcalc, yorick, XLisp-Stat and xppaut. More details are at the website. http://dirk.eddelbuettel.com/quantian.html A free download is at http://www.analytics.washington.edu/downloads/quantian/ It can be bought for $2.29 at http://ulnx.com/product_info.php?products_id=174 More info at http://www.distrowatch.com/table.php?distribution=quantian I'm still going through the disk. I didn't know there was this much free math software available. --Ed Pegg Jr, www.mathpuzzle.com
On Thu, Apr 08, 2004 at 09:49:25PM -0700, Ed Pegg Jr wrote:
I'm still going through the disk. I didn't know there was this much free math software available.
There's more, too. For instance, Axiom, another general-purpose CAS, was recently released under a free software license, but isn't on the disk, probably because it wasn' stable enough at the time the disk was released. Peace, Dylan
participants (9)
-
Alan Schoen -
David Gale -
dpt＠lotus.bostoncoop.net -
Ed Pegg Jr -
Eugene Salamin -
Steve Gray -
Thane Plambeck -
Tom Knight -
William P. Thurston