Gabriele Venturi
Some internal connections between radians
and sexagesimal trigonometry
If you say, with Cioran, that you do not belong to humanity (because you have accepted that they deny your reason for being), in this way you offend – in an unbearable way – all those who have excluded you from humanity: you offend them in a mortal way by accepting your exclusion (and so you project yourself outside their value system), partly because they sense how much contempt is implicit in this, and partly because they cannot welcome you back into the order to show you your place again, in this case “within humanity” (on the extreme margin, at the bottom left).
Imre Kertész
1.
In order to carry out our work, we first need to explain to non-Italian readers the concept of what we have defined as “harmonic equation” in the Italian section of the website. These are equations of the form « f(x) = x », and therefore, in theory, something very simple. However, those familiar with our work know that, in practice, the structure of these equations can vary enormously, reaching very high levels of complexity. However, since in this article we will be using rather simple structures, we can start working on a very simple example involving a 60° angle. This angle, through a factorial function of its trigonometric parameters, can return to itself after passing through that of 120°.
x = 60°
y = cos x√tg x = 3°
z = y!! : y! = 120°
cos x = x : z = 1/2
x = 60°
This example is, we repeat, entirely trivial, and may even seem insignificant. However, in our opinion, it may be sufficient to begin to comprehend the general form of a harmonic equation. A harmonic equation has, so to speak, the shape of a circle. It begins at the point where it will end, and its beginning and end are to be considered arbitrarily chosen, given that every transition point through which the function returns to its starting point could be chosen as the initial-final point of the function. If we consider the previous harmonic once again, we could start, instead of from the angle of 60°, from its cosine
cos x = x : z = 1/2
x = 60°
y = cos x√tg x = 3°
z = y!! : y! = 120°
cos x = x : z = 1/2
Another harmonic equation that we can construct starting from the angle of 60° – but with a rather different structure – is the one shown below.
x = 60°
y = x! mod 360° = 0°
cos x = (y! + cos y)-1 = 1/2
x = 60°
Moving on to a higher level of complexity, we can take into consideration a harmonic equation that concerns the fine structure constant, which, as we can see, can return to itself through functions that have to do with the hyperbolic sine.
α.1 = 137°,035999084 -4.916658190516151498737180625734066762527453687742465196… × 10^-64
sinh x = [α.1 ∙ (180/π) ∙ 23 ∙ 10-43 ∙ (π/180)14]-1 = 6,5413827881480628455029221349087e+62
x = 145°,33157152480719636538733663222
sinh α.1 = (360° ∙ 453571468503837741947744149633489232065687676807324486688) + x =
= 163285728661381587101187893868056123543647563650636815207825.33157152480719636538733663222
α.1 = 137,035999084 -4.916658190516151498737180625734066762527453687742465196… × 10^-64
We are using very simple examples because we believe we are addressing people who are unfamiliar with our work. Readers who want to gain a clearer and more comprehensive understanding of the complexity and importance of this type of relationship, both for the system of physical constants and for mathematics in general, can visit the Italian section of the website and have the articles translated into English by artificial intelligence.
On the foreign language sections of our website, however, we want to limit ourselves (at least for the moment) to offering readers those parts of our work that are both simple and stimulating. Above all, we are interested in proposing issues that seem to be of general interest, so that as many professional mathematicians as possible can explore and resolve them. In this case, what we are interested in proposing are the internal implications that seem to link natural (or “metaphysical”) trigonometry in radians and one of the infinite trigonometries that can be constructed on the basis of them, namely 360° trigonometry.
