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

Reghini, in this part of Sacred Numbers in Masonic Pythagorean Tradition, gives a rational explication for letter G and a deep one on masonic symbolism. As well as a summarize 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 tetrahedron, from the** **greek κυρ Fire (the right translation is Sun). Each face is divided by the three diameters of the circumscribed circumference leaded to the vertexes of the face into six triangles rectangles equal to each other, and, considering the tetrahedron which have 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 and the icosahedron consists of 120 tetrahedra which them as 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 esahedron, 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, has 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 basis and have as vertex the polyhedron’s centre. Now 360 is the number of divisions of the twelve signs of the zodiac, and is the number of days of the Egyptian year.

The 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 centre of the face for common vertex) each of which is composed of the other six (determined by a diameter and by two sides of pentalpha). A total of 360 triangles. Plutarch, in turn (2), having 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 by 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 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 dodecahedron and consequently is the number of the vertexes of polar polyhedron, that’s to say the icosahedron. Twelve is also the number of the edges of 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 centre of “omotetia” is obtained in the usual Pythagorean way the following dodecahedral numbers. The formulas of the regular polyhedral numbers (except of the tetrahedral number) were determined the first time by Descartes, and are found in a manuscript of his remained unpublished for over a century; in particular the n ° dodecahedral number is given by the formula