Re: [Phys-L] new puzzle

[Brian Whatcott <betwys1@sbcglobal.net>]

 I too have happy memories of Scientific American in the sixties. This was the heyday of C.L.Stong, a Westinghouse engineer, who described how to make a number of high tech science instruments in "The Amateur Scientist" using thje pen then pencil drawings of these devices by Roger Hayward. I enjoyed making the proton spin precession gradient magnetometer. "Take a small polythene bottle and wind it with several hundred turns of copper wire in several coaxial bobbins. Fill with water...."   Then there was the gas chromatograph with flame ionization detector. "Make an electrolytic hydrogen generator to maintain a small flame on the tip of a hypodermic needle insulated with a PTFE collar, which ionizes the gas evolved from the end of a small bore copper column packed with fuller's earth..."Particle accelerators were described, and HV generators. Clair Stong died in 1975
Brian W    On Thursday, May 5, 2022, 10:44:24 PM CDT, David Bowman <david_bowman@georgetowncollege.edu> wrote:  
 
 Regarding CB's fond memory:

> I enjoyed it when Martin Gardner included it in one of his Mathematical Games columns - Way Back When -

I don't quite remember the particular column topic of JSD's puzzle, but I do remember Gardener's Mathematical Games column as always being a not-to-be-missed highlight of each issue of Sci Am containing it.  I don't think that magazine has ever fully or adequately recovered since he ceased publishing there.

Probably my favorite Gardener puzzle/problem is the so-called napkin-ring problem ( https://en.wikipedia.org/wiki/Napkin_ring_problem ) which has the screwy result that the volume of a spherical ball with a circular cylindrical tunnel drilled through and removed from it (such that the ball and its tunnel are concentric) depends *only* on the length of the tunnel.  This result seemed so counterintuitive and compelling that it motivated me to try solving the problem in an arbitrary number of dimensions to see if the result was a general result holding in any dimension of at least 2, or was just a weird result only holding in just 3 dimensions.  As I discovered, it ends up that it is the latter, and the general formula for the hyper-volume in D-dimensions is messy and does depend on other parameters than just the tunnel length.  Only when D=3 do things simplify to the point where the volume only depends on the tunnel length.

In any event JSD's puzzle/problem motivated me to propose here one of my own.  This one is a simple trigonometry problem.  In HS or jrHS students typically are introduced to the definitions of ordinary trig functions, and those students also learn their exact values for the special cases of angles that are integer multiples of π/4 rad (= 45°) and integer multiples of π/6 rad (= 30°) .  Besides these angles, (and perhaps a few that are accessible by half-angle formulas like 22.5° or 30°) pretty much for nearly all other angles students are led to believe that the exact values of trig functions are not exactly representable in terms of rational numbers and radicals.  But this is not actually the case.  There *are* other angles whose trig function values are exactly representable in terms of radicals and rational numbers.  For instance, angles that are integer multiples of π/5 rad (= 36°) are also so representable.  So here is the problem:

Find cos(π/5) (=cos(36°)), cos(2π/5) (=cos(72°)), cos(3π/5) (=cos(108°)), & cos(4π/5) (=cos(144°)) explicitly in terms of the Golden Mean, φ = (1 + √(5) )/2, and show your work.

This problem probably doesn't say much about the structure of the world, but a generalization of it may have something to say about the structure of the unit circle in the complex plane.  (Also it might not be only mere coincidence that these angles also are relevant in quasi-crystalline order.)

David Bowman



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