The Sacred Triangle (3,4,5): Mathematical Symbolism and Cosmic Harmony

The Sacred Triangle $(3;4;5)$ whose legs are divided into segments obtained from the inscription of the circle with unit radius $OH = 1$. The measurement of individual segments reveals the triangle's dependence on Numbers 1, 2 and 3, being $AH = AH' = 1$; $BH = BH'' = 2$ and $CH' = CH'' = 3$, that is, in this case $r_C = 1$. If the fact that the circle inscribed in the Sacred Triangle is precisely a circle with unit radius is little known, even less known is the fact that, if we consider the three tangent points of the circumference inscribed in the Sacred Triangle, we obtain that these three points reveal exactly the occult composition of the Sacred Triangle by dividing the legs and hypotenuse into segments of measure 1, 2 and 3. In other terms, relative to Figure 4, 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. 5). The Ternary 1, 2 and 3 thus remains clearly visible even in the Sacred Triangle as a tripartition of its area through the polygons originating from the tangent points of the circle inscribed in the triangle. Starting from the triple $(3;4;5)$, the Ternary 1, 2 and 3 generates all other Pythagorean Triples 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 $$I_0 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad I_+ = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad I_- = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ We believe to be the first to explicitly write that every Pythagorean triple is of the form $$I_1 T \cdots I_n T \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ with $i \in \{0, +, -\}$ where we have understood $$\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ All existing Pythagorean triples 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 indeed have $5^2 + 12^2 = 13^2$. Without further complications, it is easy to understand how it is possible to obtain not only the famous triple $(3;4;5)$ from the generating principles 1, 2 and 3; but how, from this, it is then possible, through recourse to these same primitive Numbers 1, 2 and 3, to obtain all other Pythagorean triples by means of simple geometric transformations. From a symbolic and qualitative point of view, the Pythagorean Triple of the Sacred Triangle $(3;4;5)$ is the first and natural expression of the Ternary 1, 2 and 3 in the world of becoming and is the only true Pythagorean Triple from which all others derive through simple transformations of the same Ternary, that is, different incarnations or realizations of this unique geometric reality expressed graphically by the Sacred Triangle. Colors, Numbers and Notes of the Sacred Triangle According to the prophets, three spheres emanate from God that fill the three heavens: the first, or sphere of Love, is red; the second, or sphere of Wisdom, is blue; the third, or sphere of Creation, is green. From these three Colors-Light all others derive, and in the same way the musical harmonics of the sacred chord Do:Fa:La generate all others. The three colors Red, Green and Blue find their natural musical equivalence with the notes Do:Fa:La (Fig. 6), whose frequencies are in mutual ratio $3:4:5$. In other words, assuming we have La tuned to 440 Hz, the corresponding Do will have frequency 264 Hz or $3/5$ of the previous one, while Fa will have frequency 352 Hz, i.e., $4/5$ of the frequency of La 440 Hz. The same proportions $3:4:5$ exist for the colors Red, Green and Blue, so that a fundamental Red at 420 THz corresponds to a Green at 560 THz and a deep Blue at 700 THz. From the Triple 3, 4 and 5, secondary Numbers then derive, which become of primary importance for the numerical-symbolic understanding of the Cosmos. Combining the numbers of the Sacred Triangle through the addition operation we have: $$3^1 + 4^1 + 5^1 = 12$$ $$3^2 + 4^2 + 5^2 = 50$$ $$3^3 + 4^3 + 5^3 = 63 = (7^2 + 7^2 + 7^2)$$ while combining them through multiplication we get: $$3 \cdot 4 \cdot 5 = 60$$ $$3^2 \cdot 4^2 \cdot 5^2 = 3600$$ $$3^3 \cdot 4^3 \cdot 5^3 = 631000 = (7^2 + 7^2 + 7^2) \cdot 1000$$ The numbers highlighted by the combination of the Sacred Triple 3, 4 and 5 are those that symbolically regulate the harmonic cyclical development of the Cosmos: the number 12 is the number of the perfect cycle and of the 12 turns of the Sun in the 12 Zodiacal Signs; the numbers 50 and 60 are those of the Pentagram $5^{10}$ and of the Hexagram or Star of David $6^{10}$; 72 is the traditional Number indicating...