CoincidentalTaxonomy2

Seven months ago in CoincidentalTaxonomy (19 Oct 2001) I tried to reconstruct some thoughts that a friend (JB) and I came up with in the mid-1980's. By pure coincidence (^_^) as I was cleaning out my old office this week I found my notes from an informal lecture that we gave on that subject to some friendly colleagues in August 1984.

JB & I invented a 3x3 matrix of coincidence, with "Class" on the horizontal axis and "Type" on the vertical. Our three "Classes" were Deep, Definitional, and Mere. The three "Types" of coincidence we named were Events, Form, and Quantity. Our matrix looked like:

Then JB & I talked about examples of various of these coincidence categories:

• "Mere Quantity" (or maybe "Deep Quantity"?): the ratio of masses of the proton and the electron are very close to 6*(pi)⁵, and the Fine Structure Constant 2*(pi)*e²/hc is approximately 1/137
• "Mere Quantity": the number of amino acids coded for by DNA is equal to the number of three-letter comma-free codes of four symbols, the DNA base pairs (it fooled Linus Pauling)
• "Deep Form": the shapes of the continents South America and Africa fit together like jigsaw-puzzle pieces (which led Wegener to hypothesize continental drift)
• "Deep Events" (or "Mere Events"?): Arp's observations of the associations of quasars with peculiar galaxies
• "Deep Quantity" (or "Mere Quantity"?): the Bode/Titus law for the distances of the planets from the Sun in the solar system
• "Mere Quantity": (pi)³ is approximately 31
• "Mere Events": no supernovæ have been seen in our galaxy for several hundred years
• "Deep Events": gravitational mass equals inertial mass (which suggested General Relativity to Einstein)
• "Deep Form": all electrons are identical
• "Deep Quantity": all elementary particles have identical (or multiples of identical) charges
• "Mere Quantity": the Sun and the Moon have almost the same apparent diameter in the sky
• "Deep Form": electromagentism and gravity both obey an inverse square law (approximately)
• "Deep Quantity" (or "Mere Quantity"?): Dirac's large number hypothesis about the cosmos
• arguably "Deep" (or "Definitional"?) concerning Events, Form, and Quantity: the Riemann hypothesis, that the nontrivial zeroes of the zeta function all have real part equal to 1/2

My housekeeping frenzy also uncovered a letter from another friend (DLM) dated 15 August 1984, provoked by the above, in which he described an amazing example of computational coincidence:

The Drude Model (pronounced Droo-dah and affectionately referred to at my undergraduate institution as the Crude Model) of thermal and electrical conduction in metals brings up a couplet of coincidences which clearly facilitated the bearing of fruit in the early days of solid state physics. Drude, through a classical model relying heavily on the kinetic theory of gases, was able to derive the Wiedeman-Franz Law (a proportionality between the ratio of thermal to electrical conductivity and temperature) in terms of fundamental ideal gas law parameters. The ratio which follows from Drude's model is about half the observed value. Drude, however, in his original calculation of the electrical conductivity erroneously found half the correct result --- placing his prediction, coincidentally, in extraordinary agreement with the observed value. But this wasn't half the coincidence (no pun intended). In fact, Drude's success was due to two errors of about 100 which in the ratios coincidentally compensated for each other. At room temperature the electronic contribution to the specific heat derived with quantum mechanics is 100 times smaller than the classical prediction, but the mean square electronic speed is 100 times larger. The fruitful aspect of this Comedy of Coincidences is that the impressive apparent success of the model drew many researchers into the field and probably accelerated the rate at which the true solution was found.

A highly readable account of this coincidence, ad the Drude Model, can be found in Ashcroft and Mermin's Solid State Physics, pps. 20-25 especially.

This reminds me of Johannes Kepler's decade-long struggle to compute the shapes of planetary orbits based on Tycho Brahe's data. At various stages along the way Kepler had derived the equation for an ellipse ... but he didn't recognize it as such. So he decided to throw out all of his work and assume an elliptical orbit --- at which point everything miraculously fell into place! Lucky man, eh?

TopicScience - TopicPersonalHistory - 2002-05-14

(correlates: YouAreExtraordinary, CoincidentalTaxonomy, AppropriateUnits, ...)