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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