Pythagorean Sacred Triangle and Specular Numbers

He establishes a correspondence between white light, the sacred chord and the Sacred Triangle. Just as the chromatic range of all colors emanates from white light and just as all notes symbolically emanate from the sacred chord, in the same way all qualities expressed by numbers emanate from the Sacred Triangle. It is possible to demonstrate, in fact, that the Sacred Triangle contains within itself the qualitative aspect of every number. If we inscribe a circle in the Sacred Triangle with sides 3, 4 and 5, this will have radius 1 and therefore diameter measuring 2. The area of the Sacred Triangle is 6, alternately summing the sides of the triangle we get 7, 8 and 9. Finally the perimeter of the triangle measures 12. With these 10 Numbers we can symbolically summarize all existing numbers (editor's note, see article). It is not a vain effort to indicate to the initiatic schools of the West the possibility they have to reveal a Pythagorean Cosmogony that binds together again a chromatic symbolism, with the musical and geometric ones. At the origin we place the symbol of Fire in an abstract sense represented by the equilateral Triangle. This triangle can represent Divine Fire, the Science of the Trinity as well as the Ternary itself, from which emanate the three Creative Spheres of Light-Sound symbolized chromatically by the colors Red, Green and Blue, or acoustically by the notes Do-Fa-La, or visualized with the figures of the Triangle, Square and Pentagon. From this celestial manifestation through the figures of Triangle, Square and Pentagon we can descend towards corporeal manifestation operating symbolically on the three polyhedra: tetrahedron, octahedron and icosahedron. The principle of duality is a symbol of change, that is, of the vital pulsation originated from the generative and vibratory center of the Triangle. A graphic representation of such Cosmogonic derivation is found in the Figure.

The Specular Numbers (Tabit)

Among all figured numbers, the specular or oblong ones are particularly important for the type of constructions we will make later. Specular numbers have this name because they can be represented as two triangles facing each other as in a mirror. But since they can also be represented as harmonic rectangles, in the times of the ancient Greeks they were called by the name of promekes which means rectangular or oblong. In fact, the numbers in question can be figured as rectangles having sides in ratio equal to the main musical intervals. The first 10 specular numbers are $\{2,6,12,20,30,42,56,72,90,110\}$ and their sum is equal to $2 + 6 + 12 + 20 + 30 + 42 + 56 + 72 + 90 + 110 = 440$. In a certain sense these same numbers can also be put in relation to the number five being able to be written as the sum of the square of a number plus the number itself:

An important relationship is that which exists between these numbers and $\pi$. They in fact constitute the coefficients of its representation in the form of continued fractions, in fact, defining $\Delta(0) = 1$, we have that:

Besides this interesting result that links $\pi$ to specular numbers, we have another that involves the sum of the reciprocals of specular numbers. The total sum of all reciprocals of specular numbers, in fact, turns out to be unity: $$\sum_{n=1}^{\infty} \frac{1}{\Delta(n)} = 1$$