Principles of Nature: towards a new visual language
© copyright 2003-2008 Wayne Roberts. All rights reserved.
Resonances among the parametric equations for generating Pythagorean, Eutrigon, and Co-eutrigon Triples
This web page is reproduced from the author's book (W. Roberts, 2003, p.148), edited and reformatted for the web.
One of the first things that struck me after working out the parametric equations for the Eutrigon and Co-eutrigon theorems was the notion that there existed these ‘equations behind the equations’ and that they only appeared to reveal themselves from a Diophantine approach—finding whole number solutions for the equations and thus generating triangles in integral side-lengths.
More interesting still, some fascinating synchronicities appear,
- Equations with two integers as inputs are generating three integers as output (‘Triples’ for right triangles, Eutrigons, or Co-eutrigons in integral sides). This seems to resonate with how the exponents of terms fall or rise by one unit in the processes of differentiation or integration in the Calculus.
- Fascinatingly, the hypotenuse c exhibits a fractal or recursive quality in that it recapitulates the same formula for both c2 and c (but with different integral inputs).
- Firstly, in the Pythagorean Theorem , c2 = a2 + b2, and then, in the parametric equation for c (which generates an integer for each side in a right triangle), we have c= p2 + q2. Thus not only is c2 the sum of two squares a2 + b2, but in the Pythagorean Triads, c itself is the sum of two other squares p2 + q2.
- Secondly, in the Eutrigon Theorem, we have the hypotenuse c (the side opposite the 60° angle) given by c2 = a2 + b2 – ab. The same formula reappears in the parametric equation for c (in generating a eutrigon triple): c = p2 + q2 – pq, yet again with two different integers as input.
- Similarly, in the Pythagorean equation, one of the legs has a fractal-like quality— a difference of squares: a2 = c2 – b2. Reappearing in the Pythagorean parametric equation for side a we again find that it is, in turn, a difference of two other squares, a = p2 – q2. The parametric form for the Eutrigon Theorem (in which all sides must be integers) has a side (leg) which is also the difference of two integral squares.
These findings suggest that further research in this area—looking at what might best be described as the ‘harmonics of integers’—and of the interconnections between and among the various Triples (relating to right triangles, eutrigons, and co-eutrigons) will likely yield many new connections between geometry and number, and assist towards the development of a syntax of forms within a visual and musical language.