User:Neil Parker/Sandbox/Plimpton

Diophantus II.VIII and Plimpton 322

Diophantus II.VIII describes another technique from the Ancient World (albeit a millennia or so later than Plimpton) for generating ‘Pythagorean’ triples. As is shown in the article, if we embody the core algorithm in modern algebraic form we end up with the generating tuple:

${\frac {t^{2}-1}{2}};t;{\frac {t^{2}-1}{2}}$

This is often referred to as the ‘Platonic Sequence’ and is considered a special case (with q=1) of the Euclidian generator tuple:

$p^{2}-q^{2};2pq;p^{2}+q^{2}\!$

It is interesting that the (currently) definitive thesis on Plimpton 322 (by Eleanor Robson) presents the ancient tablet as a school exercise on reciprocal pairs and apparently dismisses the notion that the numbers represent a set of generated ‘Pythagorean’ triples. We do not for a moment doubt Robson’s evident depth of knowledge in respect of the historical context in which Plimpton 322 was created. Nor indeed her well crafted and thoroughly credible theory of reciprocal pairs. Suffice to note that this theory leads us to the tuple:

${\frac {t-{\tfrac {1}{t}}}{2}};1;{\frac {t+{\tfrac {1}{t}}}{2}}$

It should be apparent that if we scale the above tuple by a factor t, we will have the very same ‘Platonic Sequence’ arrived at by following the Diophantine algorithm.Furthermore if we substitute a rational number m/n in either the Diophantine or ‘Plimptonian’ tuples, we will – upon clearing denominators, arrive at the Euclidian generator!

This relatively simple algebraic manipulation seems to have gone unnoticed quite literally for millennia – in essence what we are saying is that far from being a ‘special case’ of the Euclidian generator tuple, the so called ‘Platonic sequence’ is in fact the core generator of rational triples. And its manifestation in Euclidian form is but Greek ‘smoke and mirrors’ which has sent generations of ‘triple hunters’ off on wild goose chases for ‘coprime’ integer number pairs whilst apparently completely missing the point that rational numbers are – by definition – ‘coprime’ number pairs existing in fractional form.

The authors of Plimpton 322 - chronologically deprived as they were of the benefits of Euclid's 'Elements' - plainly did not suffer any such delusion. They knew precisely how to generate rational triples using the very reciprocal pairs (p/q and q/p) Robson presents us with in order to dismiss the existence of p and q and thereby the ‘generator theory’! Not only that but triples custom designed – it would appear – to populate the trigonometrically significant angular range 30 to 45 degrees. Whilst Robson’s cautions in respect of assuming angular knowledge in the ancient world are noted, we should not for a moment forget that the ‘milieu’ from which Plimpton 322 emerged was the very same which ultimately bequeathed to us the 360 degree circle. It’s clear that at this groundbreaking stage in history there were some extremely astute minds at work. Their understanding may indeed have propagated via 'school exercises' such as that postulated by Robson. In which case the underlying mathematical concepts may considerably predate even Plimpton 322.

We trust the esteemed author of “Neither Sherlock Holmes nor Babylon” will take it in good humour if we make so bold as to conclude with an appalling misquote from Sir Arthur Conan Doyle’s famous detective series:

“Elementary fractions my dear Watson”!