Pythagorean Sacred Triangle and Celestial Harmonies

The Pythagoreans certainly knew this. If we consider the three tangent points of the inscribed circle in the Sacred Triangle, we obtain that these three points reveal exactly the hidden composition of the Sacred Triangle by dividing the legs and hypotenuse into segments of measure 1, 2 and 3. In other words, relative to Figure 5, we have $AB = 3 = 1 + 2 = AH + HB$; $AC = 4 = 1 + 3 = AH' + H'C$; $BC = 5 = 2 + 3 = BH'' + H''C$. But that's not all! If we connect the tangent points with the origin, we obtain a tripartition of the Sacred Triangle through a square of area 1, a quadrilateral of area 2 and a quadrilateral of area 3 (Fig. 6). The Ternary 1, 2 and 3 thus remains well visible also in the Sacred Triangle as tripartition of its area through the polygons originated by the tangent points of the inscribed circle in the triangle. Starting from the triplet (3;4;5) the Ternary 1, 2 and 3 generates all other Pythagorean Triplets through the use of a transformation $T$ that can be represented matricially using only the Ternary 1, 2 and 3, i.e. $T = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 3 \end{pmatrix}$; and three other transformations $I_0, I_+, I_-$ given by $I_0 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$; $I_+ = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$; $I_- = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. We believe we are the first to explicitly write that every Pythagorean triplet is of the form $I_{i_1}T \cdots I_{i_n}T \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, with $i \in \{0, +, -\}$ and where we have intended $\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$. All existing Pythagorean triplets can be written in this form by appropriately choosing the coefficients $i$. For example if we choose a single $i = -$ we have $\begin{pmatrix} 5 \\ 12 \\ 13 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 3 \end{pmatrix} \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, for which we have indeed $5^2 + 12^2 = 13^2$. We don't want to further complicate the article, but what has been said is enough to understand how it is possible to obtain not only the famous triplet (3;4;5) from the generating principles 1, 2 and 3, but how from this it is then possible, through recourse to these same primitive Numbers, to obtain all other Pythagorean triplets by means of simple geometric transformations. From a symbolic and qualitative point of view, the Pythagorean Triplet of the Sacred Triangle (3;4;5) is the only true Pythagorean Triplet and all others are nothing but different transformations, different incarnations or realizations of this unique geometric reality gracefully expressed by the Sacred Triangle. The Harmony of the Spheres: The Circle, Harmonics and Fourier Series (Part 2) (Solis) In the preceding article we showed how any excitation (which we mathematically modeled through a smooth function) defined on a compact and abelian topological group can be decomposed into Fourier series, that is into harmonics. A special case was that of the violin string with the two ends fixed. Being the two ends fixed, they could be joined thus allowing to consider the string as a closed circle which, once excited or plucked, produced the natural harmonics, whose nodes were the regular polygons (Fig. 7). Now, however, we want to take advantage of the mathematical generalization we have enunciated of the phenomenon and find a slightly more abstract application to a group that is no longer spatial, but temporal: the Zodiac. The word cycle derives from the Greek κύκλος which means circle and it is evident that, although here we intend a temporal circle instead of a spatial one, the mathematical laws that regulate its development and excitations remain the same. The fact that time is cyclical is a given of terrestrial existence, dominated by the times and vital solar impulses rhythmed by a motion of revolution and a motion of terrestrial rotation around its own axis that repeat cyclically over the centuries. To these motions are added other cyclical motions decennial, secular and millennial such as nutation, precession of the equinoxes and variation of orbital eccentricity. However, since ancient Egypt 4 fundamental temporal cycles for human existence had been identified that were analogous to one another and that were: