Also, this proportion between four segments is a special case of the Babylonian proportion as was the proportion formed by the four strings of the tetrachord of Philolaus. For the two quaternions happens also that the second term is the arithmetic median of the extremes and the third the harmonic median. In the case of the tetrachord of Philolaus, the law of determination was that the first term was twice the quarter: in this case, the law of formation is that each term is the golden part of the previous.

In conclusion: side s5 of the Pythagorean pentalpha is divided by two other sides of the same pentalpha in two intermediate points M and N such that A E: A N = A M: M N which are respectively equal to

In this proportion, each segment is the golden part of the previous, and it happens like the proportion of the four strings of the tetrachord that the second segment is the arithmetic median of the extremes and the third is the harmonic median of extremes. Moreover, like the Pythagorean range is achieved with the law of the fifth of the tetrachord of Philolaus, and so each term of the chain of equal relations is obtained by taking the gold part of the previous term, that is, through the division of the circle into ten and five equal parts.

This law extends indefinitely the fifth tetrachord and the octave in successive octaves and extends the chain of equal relationships between the side of a pentalpha and that of the respective Pentagon and the side of the consecutive pentalpha and pentagon. In conclusion, the pentalpha bears imprinted in the distribution of its sides a natural law of harmony because like the A string that is the harmonic median di so the side of the pentagon is the harmonic median between the entire side of pentalpha and the part of it comprise between the two other sides of the pentalpha.

On the other hand the last of the five regular Pythagorean and Platonic polyhedra, the regular dodecahedron, has twelve faces that are regular pentagons and calling the apothegm of this polyhedron, that’s to say with 2 *a* the height of the dodecahedron or the distance between two parallel faces, it can be shown that the planes parallel to the two parallel bases, intermediate between them and passing respectively for the five vertices of the dodecahedron next to that base, they divide the height 2 *a *of the dodecahedron into two points M and N such that, indicating with A B the height

the segment A N = B M is the golden part of A D, the segment A M = B N is the golden part of AN and the intermediate segment M N is the golden part the segment A M. These four segments form a Tetractys similar to that formed by the four segments of the side of pentalpha written in the pentagonal face of the dodecahedron. To use a term of magic it can be said that both the dodecahedron and its face bear the signature of the same harmony, the harmony of pentalpha coincides with the harmony of the dodecahedron.

On the other hand, it can be shown that the golden part of the height 2′ s equal to the side s10 of the decalpha inscribed in the face of the pentagonal dodecahedron (decalpha obtained by bringing together a ten-point division of the circumference into ten equal parts of four in four), one can also demonstrate that the radius of the circumscribed circumference is the golden part of the side s10 of the inscribed decalpha and in the end, we know that the side l10 of the inscribed decagon is the golden ratio of the radius r. So the Tetractys of the four segments marked on the height of the dodecahedron consists of four segments: 2 a, s10, r, l10 which, therefore, constitutes the geometrical ratio

2 a : s 10 = r : l 10

wherein, each term is the golden part of the previous, and then the second term is the arithmetic median of the extremes while the third term, that’s to say the radius r, is the harmonic median of the extremes. The dodecahedron has, therefore, the property: the radius of the circumference circumscribed to the face of the dodecahedron is the harmonic median between the height of the dodecahedron and the side of the regular decagon inscribed in the face.

This third Babylonian proportion between the Tetractys of the four above indicated elements on the dodecahedron is also connected with the number of sides of the pentagonal face and with the number 12 of the polyhedron faces; as in the case of the tetrachord, the Babylonian proportion was connected with the law of fifth, with the five black keys of the piano and the black and white twelve keys of the octave. If we are planning to conduct the twelve parallel planes to the twelve faces of the dodecahedron and pass through the next five vertices, they determine in the interior of the dodecahedron another regular dodecahedron for which there are the same properties, and so on indefinitely.

Since in the Pythagorean science the seven liberal sciences were closely related and closely linked to the various arts, it can be expected that in the various arts we can see the evidence of the importance the Pythagoreans admitted to the golden part and the harmonic median. In fact, the canon of Polycleto’s statuary connects to the consideration of the harmonic median (3), whereas the golden part has great importance in architecture before Pericles (4).

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- A. Reghini, Per la restituzione delle geom. pit;
- The regular pentagon, and thus also the decagon and the pentalpha, can be constructed even without a compass starting from a strip with parallel sides. Just tie it and pull as you do for the tie knot. One can recognize and demonstrate easily that it folds into three equal segments A B, C D, and E A, and the two segments D E and C B result they too equal to the other three. The ribbon comes out from parts of the sides of the pentagon D E and C B and has the shape of a bishop’s miter (bishop, in English, in the game of chess) or even the apron of the companion. The toothed belt or union chain that is wrapped and knotted around the columns of the temple, which are ten taking out the two columns at the temple entrance, shapes ten of these pentagonal nodes, like the ten regular pentagons circumscribed to a regular decagon ;
- See L. Robin, La pensée grecque, page 273 ;
- See M. Cantor, Vorlesungen über Geschichte der Mathematik, 2a ed., I, 178;