Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time? Only Adam Goucher took up my challenge of discovering a relationship that was actually meaningful. The three orbital periods TI=1.769137786 TE=3.551181041 TG=7.15455296 when turned into frequencies fI=1/TI=0.565247098 fE=1/TE=0.281596457 fG=1/TG=0.139771137 can be "explained" by just two numbers: fI=4f1-f2 fE=2f1-f2 fG=f1-f2 f1=0.14182532 f2=0.00205418 The 4:2:1 pattern is the "Laplace resonance" and the tiny correction, or large period 1/f2=486.8 days, is probably related to the periapsis precession you get from a slightly non-spherical gravitational field (an oblate source such as Jupiter). Adam's continued fraction method is also how I first approached the problem. Veit On Apr 12, 2012, at 6:22 AM, Adam P. Goucher wrote:

...

A puzzle I gave my mechanics class:

The average orbital periods of the Jovian moons Io, Europa and Ganymede are:

TI=1.769137786 TE=3.551181041 TG=7.15455296

These are taken from Wikipedia; the time unit is days.

1. (math) On the basis of just these numbers, infer the existence of a much longer period (on the order of hundreds of days).

That's a nice puzzle. The first thing I did was to enter TE/TI, TG/TE and TG/TI into a continued fraction calculator, to produce best rational approximations. The first few convergents for TE/TI are:

2/1 275/137 * 3027/1058 3302/1645

And those for TG/TE are:

2/1 137/68 * 3153/1565 3920/1633

Finally, the convergents for TG/TI are:

4/1 89/22 93/23 275/68 *

The asterisked convergents suggest a ratio of:

TI : TE : TG = (1/275) : (1/137) : (1/68)

This means there is a large period of 275 TI = 137 TE = 68 TG. For each of the orbital periods you have provided, this results in the following approximations to the large period:

275 TI = 486.51289115 days 137 TE = 486.511802617 days 68 TG = 486.50960128 days

= approx 486.51 days.

(Technically, a better method than using continued fractions is the LLL lattice reduction algorithm. However, since the data are so precise, the continued fraction method was adequate.)

Sincerely,

Adam P. Goucher

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The reason we know that perpetual motion is impossible is that so many clever people have tried & failed. The reason we know that heavier than air flight is impossible is that so many clever people have tried & failed. Distinguishing these two cases is difficult. Rich ---- Quoting Veit Elser <ve10@cornell.edu>:

Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time?

On 4/29/2012 12:36 PM, Veit Elser wrote:

Isn't this obsession -- with looking for relationships where there almost certainly aren't any -- a waste of time?

Only Adam Goucher took up my challenge of discovering a relationship that was actually meaningful.

How are the transient approximate relationships between the orbits of random inconsequential chunks of space rock more meaningful than a permanent approximate relation between fundamental constants? But seriously, as far as wasting time goes, here's is a summary of my contributions to mathematics: - I worked on the OEIS. - I did a little work on Eric Weisstein's Treasure Trove of Mathematics. - I found a relationship between the Collatz conjecture and regular expressions. Jeff Shallit wrote a paper for me, "The 3x+1 Problem and Finite Automata", which gave me Erdos number 2. The proof in the paper is elementary to anyone versed in the subject, I think it ended up as a textbook problem. - I asked a question about prime divisors that sent John Conway and some associates to the blackboard for a few minutes. The result was the discovery of "Twin Peaks" (which see on MathWorld). - I found density waves in the Ulam sequence. Don Knuth actually contacted me about this. - John Conway was looking for a name for a polyhedron with holes in all its faces, and I came up with "holyhedron." This got me a mention in Pickover's "The Math Book", but it took true geniuses to actually construct one of these things. - I found the fifth taxicab number, which had in fact been published 3 years earlier. - I may have salvaged pi^4 + pi^5 ~= e^6 from the USENET dustbin. Doubtless it would have been rediscovered. - I made up an arguably clever graph for deciding if a number is divisible by 7, which is on Tanya's site. All in all, I think I've overachieved, but in all honesty, my greatest achievements barely rise above the level of a waste of time. Twenty minutes after I'm gone, I'll be about as noteworthy as any Sumerian bean farmer. So I don't worry much about wasting my time, given my talents, that's pretty much a foregone conclusion.