Golden mean, statistics and probability

It has been said for some time that in today’s science «correlation supersedes causation», and by correlation we obviously mean a statistical correlation. But even since Newton, physics has not been concerned that much with causation, nor could do it, so that more than a radical change we only have a steady increase in the complexity of the variables involved.

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Questions of interpretation – and of principle

According to Proclus, when Euclid wrote the Elements his primary objective was to elaborate a complete geometrical theory of the Platonic solids. Indeed, it has been said on several occasions that behind the name «Euclid» there could be a collective with a strong Pythagorean component. The existence of only five regular solids is possibly the best argument to think that we live in a three-dimensional world.

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Two kinds of reciprocity

The Taijitu, the emblem of the action of the Pole with respect to the world, and of the reciprocal action with respect to the Pole, inevitably reminds us of the most universal figure in physics; we are naturally referring to the ellipse —or rather, it should be said, to the idea of the generation of an ellipse with its two foci, since here there is no eccentricity. The ellipse appears in the orbits of the planets no less than in the atomic orbits of the electrons, and in the study of the refractive properties of light it gives rise to a whole field of analysis, ellipsometry. Kepler’s old problem has scale invariance, and plays a determining role in all our knowledge of physics from the Planck constant to the furthest galaxies.

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Physics and the continuous proportion

We already see that there are purely mathematical reasons for the continuous proportion to appear in the designs of nature independently of causality, be it physical, chemical or biological: in fact the convenience of logarithmic growth is independent even of the form itself, as is the elementary fact of the discrete and asymmetric division of cells.

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POLE OF INSPIRATION – The spark and the thread

Those who like simple problems can try to demonstrate this relationship before moving on. It’s insultingly easy:

φ = 1/φ +1 = φ-1+1 = 1/φ-1

We owe this fortunate discovery to John Arioni. The elementary demonstration, along with other unexpected relationships, is on the site Cut the knot [1]. The number φ is, naturally, the golden ratio (1+√ 5)/2, in decimal figures 1.6180339887…, and φ-1 is the reciprocal, 0.6180339887… . And since its infinite decimal places can be calculated by means of the simplest continuous fraction, here we will also call it the continuous ratio or continuous proportion, because of its unique role as mediator between discrete and continuous aspects of nature and mathematics.

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