Tuesday, July 19, 2016

Math Toys, Much Enjoyed!


TNG was with me for the talk, and frankly he was quite amused at his first exposure to his father live-tweeting a talk. I said it helped me remember later to look at the tweets; I should have added that seeing my mistakes later keeps me humble. As soon as we walked out of the ballroom, one of the screens showing tweets displayed one of my typos.  I also quickly realized that an equation I had derived on the fly was quite wrong.  Worse, when reviewing the tweets I discovered that while I had apparently avoided the perverse transformations auto-correct was trying to apply to "Tokieda", I succeeded in spelling it "Takieda" in many tweets -- and over time I wasn't even locally consistent, switching back-and-forth.  As someone who learned to correct strangers in the spelling of my surname before I could read, this is highly embarrassing!

Tokieda started off by taking a typical one-handled coffee mug and tapping it with a spoon.  Now, I've done this many a time out of boredom, but unfortunately never with an inquisitive bent.  Tokieda showed how tapping points at 90 degree separation from each other generates the same pitch, though the position relative to the handle changes which pitch.  He illustrated the math behind this, and how the first observation (points on the rim that define a square yield the same pitch) can be modeled without considering the handle, whereas the pitch for a given square's vertexes requires considering the handle.  He also posed the problem of the inverse: in a darkened room, is it possible to infer the location of the handle given only the pitches?  He pointed out not only the obvious problem of being unable to narrow down the handle to a specific quadrant, but that mugs with two (he showed this), three or four handles at compass points are indistinguishable at these levels.  Alas, this simple understanding of the both the power and limits of models is so rarely seen in public discourse.

Tokieda then showed another trick with the mug.  If a small number of balls are placed in the mug and the mug is swirled in a circular fashion, the balls with orbit in the direction of the swirl.  But if enough balls are placed so that the outer ring is packed around a central ball, then the orbit is opposite the swirl!  Tokieda likened the first case to a gas: the interactions of the balls to each other are few and unimportant, whereas the second case is more like molecules in a liquid state, wherein interactions between the balls drive their behavior.   He described this as the model of such a phase transition with the fewest degrees of freedom.  Remarkable!

I think I got lost in trying to tweet the further and nearly missed the next demonstration, in which he unscrewed a nut from a bolt simply by applying a running electric toothbrush to the bolt.  His final slide describes this as a case of chiral symmetry-breaking.  Given that I have all the apparatus at home ,I really need to try this one!

He next produced two heptagonal wheels, seemingly identical.  He assured us that they were completely alike in internal construction and materials.  Yet one rolled easily and the other refused to roll!  Explanation?  The roller has its corners beveled by a nearly imperceptible amount (the number I heard him say was 0.01mm, which is astounding if that is correct).  This tiny change converts an essentially digital fall into just a bit of analog rolling, which enables the wheel to continue.  In the Q&A, he elaborated on the choice of heptagon: it is the ideal compromise in that it looks like it should roll, but really will with the slight change.  He also pointed out that the surface makes a difference; the truly polygonal wheel will roll on surfaces with some give, as that also reduces the digital nature of the fall.

A truly bewildering demonstration followed, after Tokieda had asked the audience to predict the result.  He had cylindrical jars about one quarter and three quarters full of dry rice: how would they roll?  The nearly full one rolls down a slight slope relatively easily, but the one quarter full one resists rolling, stopping even if pushed.  Huh???  He talked about, but did not demonstrate, jars filled with similar amounts of water -- they will always roll.  He also mentioned that jars with very small amounts of rice will roll.  The answer lies in the behavior of the pile of rice in the jar.  If there isn't enough rice to pile up, then the jar rolls. As a pile forms, the shape of the pile is determined by the angle of repose, which in turn depends on the shape of the grains, not their size or material.  If piles are driven only by angle of repose, then the center of  gravity is uphill of the point of contact.  On the other hand, once the pile is so large that the jar's walls help drive its shape, then the center of gravity moves to below the point of contact, and so a torque is generated that drives the jar down the slope.  However, the jar accelerates more slowly with rice than with water; the grains' interaction with the jar walls makes it more like a viscous liquid.  Tokieda asserted that with an extremely viscous liquid, such as honey, the 2/3 full jar will actually roll uphill for a moment when starting!

Next, a demonstration of a startling bit of origami.  Startling in particular because apparently doing similar things with compass-and-straight-edge is quite hard.  I never got that stuff well; the teacher I had for Geometry had the singular distinction of being the only teacher for whom I established, and retain, an utmost loathing (I try not to hold grudges, but Mr. H. was an awful bully).  Classical geometry probably wouldn't have been my thing in any case, but that toxic personality didn't help matters.  

Anyway,  here is the question: how can you form a regular polygon with an odd number of sides?  Well, to make the pentagon simply take a thin rectangular strip and tie a loose overhand knot in it, then carefully tighten and flatten it.  Even my clumsy hands were able to make a decent pentagon (once the excess strips were torn away).  For a heptagon, go around twice (the basis for the surgeon's knot variant on a square knot).  Nonagons require three passes.  Etc.  Tokieda mentioned some "obvious" way to get the even numbered polygons given this trick, but that escaped me.  He also mentioned the obvious case of the triangle: tie the know with zero passes.

The penultimate demonstration consisted of Tokieda taking a deflated paper balloon and hitting it repeatedly from multiple angles.  Before our very eyes, the balloon expanded into a sphere.  As Tokieda pointed out, this is a remarkable feet: by repeatedly applying local concavity, creating an object which is globally convex!  The effect relies in part on the properties of the paper -- the collisions are not fully elastic.

The final demonstration again got me mentally slapping myself for observing something many times, but never thinking about it very hard.  Tokieda took a large disk (might have been a CD) and set it spinning on his table.  I've spun many a top, dreidel and coin in my life, and have noticed the progression.  At first the top spins on its axis, but ultimately it falls over and starts flapping off-axis -- depending on the nature of the spinning object, this flapping phase can be quite prolonged.  If so, then it is quite noticeable that the pitch generated by the flapping rises.  Tokieda mentioned that a while back this became the subject of quite a bit of back-and-forth in prominent journals.  What controls the flopping?  Why does it rise in pitch?  He showed that this change in frequency is exquisitely modeled by a simple equation (had a t^1/3 in it).  Some papers tried to claim that the viscosity of the air between the disc and the tabletop was driving this behavior.  Other groups tried to rule that our by running tests in a vacuum and showing the effect held -- though Tokieda pointed out that unless the vacuum is very, very good the viscosity of air can still have effects (he gave an explanation involving path lengths of molecules that went past me -- another hole in my physics background).  But, the killer for the viscosity arguments is that the exact same decay equation holds for spinning rings instead of spinning disks. He also showed that a pair of strong, oblong magnets tossed to collide in the air generate the same tell-tale rising pitch   I left with the impression that the jury is still out on this.

If you Google, you can find videos of Tokieda demonstrating many of these.  But as he answered in the Q&A, while those can be fun, nothing beats trying these out on your own.  And with the exception of the heptagonal wheels (which could probably be 3D-printed), almost any household has all of the required items -- or they can be easily and inexpensively obtained at very ordinary stores. 

That typo I saw on exiting the auditorium?  It was in an Aristotle quote Tokieda offered: "in all natural phenomena there is something of the marvelous" .  With gusto and verve, Tokieda reminded that everyday people with everyday objects can go in search of the miraculous.

1 comment:

James@cancer said...

Sound's like a wonderful talk...lots of things to try with the kids over the Summer holidays. Enjoy yours Keith.