The CROSS-RATIO from another point of view

Date: 7 Dec 2023 - Sezione: scienze/matematica/

Moebius ((1790-1868) demonstrated that the cross-ratio could be considered as the only invariant of projective geometry, that is, the only stainless steel pillar that supported the building, which the physicist Dirac termed as the most elegant branch of geometry, an informing concept to classify the scattered branches of geometry that had been developed at least up to his time. But for me the cross-ratio always had the flaw of being based on a geometric definition, a formula of which there was not, or I had not been taught, any intuitive motivation. I tried to find such motivation, and I ended up very far from projective geometry. But at least I found a way to almost intuitively deduce the fundamental formula and many properties of the cross-ratio. I found this method by mysellf, and I present it as little more than a mnemonic method. A (not in-depth) search of the sources did not indicate to me any others who have used this method. I would be grateful if one or more of my predecessors were mentioned to me by some unlikely reader, as I have no ambition to be considered the inventor of a method which, due to its simplicity, is certainly well known. As soon as I will know the name, I will simply add it as a reference in a new edition.

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