Arturo Reghini’s fourth chapter ends: Plato was silent on the number twelve, but facts and reasons validate the choice of the dodecahedron as a symbol of the universe.

Reghini, in this part of Sacred Numbers in Masonic Pythagorean Tradition, gives a rational explication for the letter G and a deep one on masonic symbolism. As well as a summary of what said so far.

Matila G. Ghyka called the golden part the “Nombre d’Or ” ( golden number in french) and this is the title of his major work devoted to the study of sacred architecture of all time. Music, sculpture, and architecture, all the arts, conform to the law of universal harmony based on the properties of sacred numbers.

To fully understand how important and meaningful should be for Pythagoreans what we found about the dodecahedron, one should remember that for them and for Plato the dodecahedron was the symbol of the universe and that the five regular polyhedra, namely the cosmic figures, were the symbol of the four elements and the universe. If we want to see why there is only to read Plato’s Timaeus, the Pythagorean dialogue par excellence.

The regular tetrahedron with its four triangular faces, four vertexes, and six edges, was the symbol of fire: and it may be that this correspondence has been determined by the shape of the solid whose vertex recalls the tip of the flame, and has been confirmed by the erroneous etymology of the word “pyramid” used by the Greeks, instead of the tetrahedron, from the** **greek κυρ Fire (the right translation is Sun). Each face is divided by the three diameters of the circumscribed circumference leads to the vertexes of the face into six triangles rectangles equal to each other, and, considering the tetrahedron which has for vertex the common center of the regular tetrahedron and for basis the 24 equal triangles in which is divided the surface, the tetrahedron consists of 24 equivalent tetrahedra. In this way the octahedron has eight faces that are equilateral triangles, six vertexes, and 12 edges, so the surface of the octahedron is divided into 48 equal triangles, and correspondingly the polyhedron consists of 48 equivalent tetrahedra. In a similar way, the icosahedron is made up of twenty faces that are equilateral triangles, twelve vertexes, and thirty edges: and its surface is divided into 120 equal right triangles the icosahedron consists of 120 tetrahedra which as a basis and have as common vertex the center of the polyhedron. Each regular polyhedron has a polar polyhedron for which the numbers of faces and vertexes are exchanged, while the number of edges remains unchanged. The tetrahedron is self-polar, the polar polyhedron of the octahedron is the cube which has six square faces, eight vertexes, and 12 edges. Philolaus saw the image of harmony in the cube because the number of its vertexes is the harmonic median of the numbers of the faces and edges, which of course is also true of the octahedron. Each face of the cube is divided by the diameter of the circumscribed circumference passing through the vertexes in four equal isosceles triangles, so the surface of the cube is divided into 24 equal triangles and the cube, or hexahedron, consists of 24 tetrahedra whose vertex is equivalent to the center of the cube. After having attributed to each of these four polyhedra the correspondence with the element fire, air, water, and earth, Plato silences Timaeus who only says: “So it remains to us still a form of composition which is the fifth of what it has helped God for the design of the universe”. We observe that Plato and the Pythagoreans knew that the regular polyhedra are five and five only, as it is demonstrated in a simple way, and we see that also through this way of the cosmic figures one comes to number five.

As for the sudden and unexpected silence of Plato that truncates the exposure of the subject, it also gave the eye to Robin (5), which simply says: “*Au sujet du cinquième Polyedre Regulier, the dodécaedre … Platon est très mysterieux* “, or when arrived to the fifth regular polyhedron, the dodecahedron… Plato is very mysterious”. But he does not attempt to investigate the reasons for the sudden Plato’s silence.

Now the dodecahedron is the polyhedron polar of the icosahedron and thus has twelve faces that are regular pentagons, twenty vertexes, and thirty edges. Applying to it the previous subdivision procedure is that the diameters of the circumference circumscribed to a face, passing through the vertexes, divide it into ten equal right triangles, but if in the face is inscribed the pentalpha, the pentagon is divided by the sides of pentalpha and by the diameters passing through the vertex of the pentalpha in thirty right triangles, which this time are neither isosceles, nor the beautiful right triangles dear to Timaeus (that’s to say with the double hypotenuse of minor cathetus ), nor are they all the same or equivalents. On the other hand, the surface of the dodecahedron is divided thereby into 360 triangles, and correspondingly the dodecahedron decomposes into 360 tetrahedra which have then as a basis and have as vertex the polyhedron’s center. Now 360 is the number of divisions of the twelve signs of the zodiac and is the number of days of the Egyptian year.

This thing is fully confirmed by what two ancient writers say. Alcinous (1), after having explained the nature of the first four polyhedra, says that the fifth has twelve faces as the zodiac has twelve signs. and adds that each face is composed of five triangles (with the center of the face for the common vertex) each of which is composed of the other six (determined by the diameter and by two sides of pentalpha). A total of 360 triangles. Plutarch, in turn (2), has found that each of the twelve pentagonal faces of the dodecahedron consists of thirty right scalene triangles, adds that this shows that the dodecahedron represents both the zodiac and year because it is divided in the same number of shares of them. Plutarch alludes manifestly the to Egyptian year composed of 12 months each of thirty days, in which the five epagomenal days are not part of the Egyptian year.

To well understand the importance of Pythagoreans and Plato of these mathematical observations it should be noted: 1 – that for them, the triangle is the atom (that’s to say the last indivisible part) superficial because it is the polygon with the number of sides necessary and enough to delimit a plane portion, and that correspondingly the tetrahedron, or pyramid, is the solid atom because it is the polyhedron having the necessary number of faces and sufficient to delimit a portion of space. 2 ° -That, because of the same definition of polygonal number, each polygonal number is always the sum of triangular, and for the definition of pyramidal number each pyramidal number is the sum of tetrahedral numbers. So we came to see that even the five cosmic figures, and in particular the symbol of the universe, were composed of tetrahedra, the entire universe was reduced to a sum of tetrahedral atoms.

The number twelve is the number of the faces of the dodecahedron and consequently is the number of the vertexes of the polar polyhedron, that’s to say the icosahedron. Twelve is also the number of the edges of the cube and of the polar polyhedron, that is octahedron. If we consider the number twelve as consisting of the twelve vertexes of a dodecahedron and we develop this dodecahedral number within one of the angles, by taking the vertex as the center of “omotetia” is obtained in the usual Pythagorean way the following dodecahedral numbers. The formulas of the regular polyhedral numbers (except for the tetrahedral number) were determined for the first time by Descartes, and are found in a manuscript of his that remained unpublished for over a century; in particular, the n ° dodecahedral number is given by the formula

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 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 the dodecahedron. Even in the extension of the tetrachord to the 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 to 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 was the number of members of some priestly colleges in archaic Rome, twelve was 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 the universe.

The dodecahedron is inscribed in the sphere as in 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 shown how the hexahedron or cube can be incorporated into the sphere as well as in dodecahedron, one can easily show how the icosahedron, having as vertexes the centers of the twelve faces of the dodecahedron, is a regular inscribed icosahedron; and similarly for the octahedron with vertexes in the centers 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 is equal to the Decade, and represented by the Delta existing in the sanctuary of Delphi, the 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, is so the one potentially contained in the Tetractys of Philolaus and therefore also in the Tetractys depicted in Delta. Furthermore, we have geometrically arrived at the 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 squares 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 the 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 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 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 the world is the harmony of the spheres and sapient music, that in its turn gets inspired. Iamblichus follows an ancient Pythagorean belief, according to which Pythagoras was 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 the 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 sapient music, the harmony of the spheres and so approaches divine perfection. The cathartic function of music made the Tetractys a doctrine particularly valuable for the contribution that it brought to moral and religious perfection. This explains, according to Delatte, that Tetractys was one of the fundamental theories of the Pythagoreans’ arrhythmological philosophy and religion.

The arithmetic and geometric development of sacred numbers that we have exposed go to the consideration of Delta, or sacred triangle, to that of the dodecahedron. The elements of Euclid, in Euclid’s text, start without a preamble with the consideration of the equilateral triangle and, according to Proclus statement (13), Euclid poses for the 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 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 Pythagorean geometry and according to the return that we have made about ten years ago, 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, according to Freemasons geometry is the master art of building and the science of “sacred numbers” known only to them, while 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.