but the dodecahedral n° can even be obtained through a relationship between the n ° pentagonal number and its gnomon. In fact the pentagonal gnomons are the numbers of arithmetic series 1, 4, 7. 10 … so that we have: pentagonal gnomons
pentagonal numbers
and it happens that adding to a pentagonal its gnomon, you get the pentagonal following, and multiplying a pentagonal for the following gnomon one gets the corresponding dodecahedron number. Thus the sequence of dodecahedral numbers is :
dodecahedral
1 20 84 220 816………..;
relationship between pentagons and dodecahedra which arithmetically corresponds to the relationship between the number of sides of the pentagonal faces and the number of the faces of dodecahedron. Even in the extension of the tetrachord to eighth we have seen a connection between five and twelve. Likewise, the Egyptian triangle of hypotenuse 5 has a perimeter given by 12.
The number twelve on his own has traditionally already been sacred and universal. In addition of being the number of the months and signs of the zodiac, twelve was in Greece, Etruria and Rome the number of the allowed gods, twelve the number of members of some priestly colleges in archaic Rome, twelve the number of Etruscan and Roman rods beam, and many surviving Celtic dodecahedra attest the importance that the ancient gave to this number and the dodecahedron. Facts and reasons that confirm the choice of the dodecahedron as a symbol of universe.
The dodecahedron is inscribed in the sphere as in the Pythagorean cosmology cosmos is surrounded by a band, the periékon, and how the universe has in itself and consists of the four elements fire, air, earth, water, so the four regular polyhedra, which are the symbol, can be inscribed within the dodecahedron. In fact it can be show how the hexahedron or cube can be incorporated in the sphere as well as in dodecahedron, one can easily show how the icosahedron, having as vertexes the centres of the twelve faces of the dodecahedron, is a regular inscribed icosahedron; and similarly for the octahedron with vertexes in the centres of the six faces of a cube, and finally how to get from the cube a regular tetrahedron taking as vertexes a vertex of the cube and the vertexes of the cube opposite to it in the three faces of the cube there congruent. The tetrad of the four elements is contained within the cosmos and in this in the band, as the four regular polyhedra are contained in the fifth and in this circumscribed sphere.
Let’s now have a break and look at the taken path. First of all we have come to Tetractys (1, 2, 3, 4), Tetractys equal to the Decade, and represented by the Delta existing in the sanctuary of Delphi, navel of the world. This Tetractys contains in itself the other Tetractys, that of Philolaus (1, 3 : 4, 2 : 3, 1 : 2), in which appear the same elements that appear in the first, and, extending the tetrachord of Philolaus, we have found the law of fifth and we reached to the numbers 5, 7, 12. The eighth, or harmony as the Greeks said, it is so the one potentially contained in Tetractys of Philolaus and therefore also in the Tetractys depicted in Delta. Furthermore we have geometrically arrived at number five in two ways: by means of the Egyptian right triangle that has five as hypotenuse and by the right triangle, with catheti one and two, that has five as square of the hypotenuse (3).
This second path led us to the consideration of the golden part, the division of the circumference into ten and five equal parts, to pentalpha, to dodecahedron, and the harmonic median of the extreme segments of the two Tetractys formed with the elements of these two figures. We have seen that the catechism of Acousmatics places in the sanctuary of Delphi “the Tetractys wherein is the harmony in which are Sirens.” To understand the meaning of this answer of the Pythagorean Acousmatics Catechism, and why they showed so much interest in the subject, there remains to be seen what do Sirens represent, since they are connected in this way with harmony. This symbolism, observes Delatte (4), is completely foreign to the ordinary conception of the Sirens and must be explained by the their identification with the harmony of the spheres and the important role accorded to sacred music in the Pythagorean school. According to Pythagoras (5) are the Sirens who personify this harmony. The same thing happens to Plato (6). Imitating with this sacred music the celestial music, the Pythagoreans (7) hoped to assimilate their souls with divine wisdom and get back among the blessed after death (8).
So Plutarch sees in Ulysses the philosopher hearing this harmony in order to take the wisdom. Plato (9), dealing with the myth of Ero says that the harmony of the spheres is generated by the revolving movement. Plato allegorically explains this harmony assuming that a siren, placed on each of these spheres, makes her voice to be heard, and that the whole of these voices, which accord with each other, produces the harmony of the world. According to Iamblichus (10) the greatest revelation that Apollo-Pythagoras gave to world is the harmony of the spheres and sapient music, that in its turn gets imspired. Iamblichus follows an ancient Pythagorean belief, according to which Pythagoras was an Apollo’s incarnation, who was sacred the sanctuary of Delphi. The Tetractys, writes Delatte (11), seems due to the veneration subject by the Pythagoreans to two causes, from the scientific point of view it explained the laws of celestial and human music, and since the harmony was the great law of universe (12), the Tetractys can be regarded as the source and root of nature, as the oath for the Tetractys. However, it allowed the Pythagoreans to imitate, by means of the sapient music, the harmony of the spheres and so approaches the divine perfection. The cathartic function of music made the Tetractys a doctrine particularly valuable for the contribution that it brought to the moral and religious perfection. This explains, according to Delatte, that Tetractys was one of the fundamental theories of the Pythagoreans aritmological philosophy and religious.
The arithmetic and geometric development of sacred numbers that we have exposed goes to the consideration of Delta, or sacred triangle, to that of the dodecahedron. The elements of Euclid, in Euclid’s text, start without preamble with the consideration of the equilateral triangle and, according to Proclus statement (13), Euclid poses for final goal of its elements the construction of Platonic figures (regular polyhedra). Perhaps from the time of Pythagoras to that of Euclid, the beginning and the end of the geometry remained traditionally unchanged, and the function of Euclid was to introduce his unceremonious postulate, thereby rehashing the demonstrations and substituting such as his test of the theorem of Pythagoras to that of Pythagoras himself, that was certainly another.
According to what remains of the Pythagorean geometry and according to the return that we have made about ten years ago, the Pythagorean geometry was a more general geometry of Euclidean and Archimedean geometry, because that was independent of the postulate of Euclid on parallels and postulate of Eudoxus-Archimedes. The point of departure and arrival were probably the same in the two geometries. But in Euclid the intent is purely geometric; while in Pythagoras, even if the performance was purely geometrical, it was certainly not the intention, since the characteristic of the Pythagorean philosophy was the connection of various sciences always present among them, and in particular of geometry with arithmetic, music and astronomy. For Pythagoreans and Plato geometry was a sacred, esoteric, and secret science, like according to Freemasons geometry is the master art of building and the science of “sacred numbers” known only to them, while the Euclidean geometry, breaking all contacts and becoming an end in itself, degenerated into a magnificent profane science. The wonderful synthesis of all the sciences and arts divined by the genius of Pythagoras disappeared, and began to specialize.
We have highlighted some trace of the deep bond that united music with cosmology and arithmetic, but we believe that the scarcity and rarity of the tracks can be attributed to the importance of the doctrine that was to be one of the secret teachings of the Pythagorean school , and a clue as well as an explanation at the same time are provided by the sudden reserve of Timaeus in Plato’s dialogue of the same name as it comes to talk of the dodecahedron. To reveal this secret it would be a blasphemy, and the Pythagorean legend had that such impiety would sometimes be avenged by daimonion, as happened in the case of the Pythagorean Ippasos that, according to the legend, had died in a shipwreck just to get published the inscription of the dodecahedron in the sphere. Plato had said enough, to say more would have been, if not reckless, outrageous, and Plato remembers μή είναι πρός πάντας πάντα ῥητά, or but is unto all always explicitly.
As for the number seven we are able to reach it only with the extension of the tetrachord to the range and by considering the pyramidal numbers of decagonal basis. There is neither a right triangle which has hypotenuse seven, nor which has seven as square of the hypotenuse, and the same thing happens to the number eleven.