The triangular who have to order the number of faces of the tetrahedron, octahedron and icosahedron are respectively:
P ( 3, 20 ) = P ( 5, 12 ) = 210 = 3 P ( 5, 7 ) = 2 . 3. 7 . = product of 2 and the first three primes of the decade.
It has also:
We do not know if these properties have already been observed by others, sive Deus, sive Dea (either god, or goddess).
The numbers of Tetractys appear in some formulas that express the cosmic figures as sums of tetrahedra, and also appear in atomic physics in connection with the number of electrons that form the nuclear covering of the rare gas atoms.
We have observed that a pyramidal number can always be expressed as the sum of the tetrahedral numbers. Similarly it can be shown that the same thing occurs for octahedral, cubic, icosahedral and dodecahedral numbers; namely, that a polyhedral of order n is always equal to an additive combination of the three consecutive tetrahedral to order n – 2, n – 1 and n, and are the following identities:
formulas that are easy to verify bearing in mind that the first members are given by the following general formulas of polyhedral numbers:
tetrahedral n, 
cubical n 
octahedral n 
n icosahedral 
n dodecahedral 
In the four preceding identities the coefficient of the average term that’s to say of the tetrahedral (n – 1) ° is respectively 2, 4, 8, 16, that’s to say the power of the two which has as exponent the numbers 1, 2, 3, 4 of the Tetractys .
This would happen according to the Platonic constitution of matter. In atomic physics instead appear the squares of the numbers of Tetractys. And here’s how: If you order the chemical elements according to the laws of Moseley and Mendelejev according to the similarity of their chemical behavior, the first column is occupied by the so-called rare gases, namely helium, neon, argon, krypton , xenon, the emanation of radium. It is found that the number of electrons that constitute the their atomic nucleus covering, in the above written order, which is the natural one according to their atomic weight and the atomic number, is respectively:
2 10 18 36 54 86
The corresponding finite difference or gnomons are then respectively and neatly
2 8 8 18 18 32 that’s to say 2. 12, 2. 22, 2. 32, 2. 42
which is the double of the squares of the numbers of Tetractys.
We observe that the first four triangles given from the Pythagorean formula (see footnote 39) are: ( 3, 4, 5 ), ( 5, 12, 13 ), ( 7, 24, 25 ), ( 9, 40, 41 ), and in them the difference, between the hypotenuse and the odd cathetus has precisely the values 2, 8, 18, 32. These triangles have in fact sides
and the difference between the hypotenuse and the odd cathetus n is
- Gino Loria, Le scienze esatte, second publ., Milano, 1914, page 821;
- Federici Cardinalis Borromaei Archiepis. Mediolani, De Pythagoricis Numeris, Libri tres, Mediolani 1627. See lib. II. chap. XXVI, page 116;
- Theonis Smyrnaei Platonici, Expositio rerum mathematicarum ad legendum Platonem utilium, publ. Hiller, Lipsia, 1878, page 4 and page 100;
- Lidus, De mensibus; publ. Lipsia, 1898; IV, 64;
- Porphyrius, Vita Pythagorae, 51;
- Verg., Aen. I, 94;
- See Delatte, Etudes …, 112. Other passages containing the same association terque quaterque are: Verg., Aen., IV. 589; XII, .155; G. I. 411; G. n. 399; Oratius Car. XXXI, 23; Tibullus, 3, 3. 26;
- Dante, Purg. VII, 2;
- Gomperz, Les penseurs de la Grèce, I, 116;
- Erwin Rohde, Psiche, italian version, Bari, 1914; I, 255, note 11;
- Anatolius, περί δέκαδος , 9; Delatte, Etudes …, 122, nota 1;
- Ps. Plutarch, Vita Homeri, 145;
- See Delatte, Etudes …, 120 e 122;
- Porphyrius, Life of Pythagoras, ed. Carabba, Lanciano, 1913, page 57;
- Eduard Zeller, Sibyllinische Blättern, Berlin, 1890, page 40 and follow;
- Adolf Kaegi, Die Neunzahl hei den Ostarien. Separatdruck aus den philologischen Abhandlungen;
- Dante, Par. XV, 97-98;
- See A. Meillet, Aperçu d’une histoire de la langue grecque, Paris, 1913, page 98.
- Homer, Odyssey, XIX, 175;
- Fabre d’Olivet, Les vers dorés de Pythagore expliqués, Paris, 1813, 24;
- Servius, Comm. a Vergil. – Egloga VIII, 75;
- Bungo, Numer. Mysteria, 1591, second ed., page 96;
- Bungo, Numer. Mysteria, 18.5;
- V. Pareto, Trattato di Sociologia generale, I, 499;
- See Abel Rey, La jeunesse de la science grecque, 119:
- Delatte, Etudes … 19 e cfr. Iamblichus, Vita Pithagorae, 114;
- See Proclus, ap. Taylor, I, 148;
- See Apollonius Conicals, ediz. Helberg, Lipsia, 1893, II, 170;
- Nicomachus, Introduction to Arithmetic, II, 8 page 294.
- Taylor, The Theoretic Arithmetic of the Pythagoreans, Los Angeles, 1934, page 243;
- Finally we note as an example of the archaic grouping in terns that in the Greek spoken numeration applies, as in the Italian one, the Handel law, that’s to say the words that express large numbers are formed by dividing the number in groups of three units, the class of units, the class of the thousands, the class of millions, etc;
- See Marc Haven, Le maître inconnu, 154;
- Dante, Conv. IV, 24;
- Anthol. Palatina, XIV, 1;
- Jerome Carcopino, La basilique pythagoricienne de la Porte Majeure, Paris, 1927, pag. 255;
- Eugenie Strong, The stuccoes of the underground basilica near the Porta Maggiore nel Journal of Hellenic studies, XLIV, 1924, pag. 65;
- This Pythagorean formula is an immediate consequence of the fundamental property that have the square of growing preserving the similarity of the shape. When the gnomon is a square, the two consecutive squares have to difference a square. Now the quadratic gnomons are nothing more than the odd numbers, if np odd number is a square, that’s to say if you have 2 n – 1 = m2, the sum of the first odd numbers that precede it is (n – 1) 2 and we have:
and substituting it has
Since m is odd then that is of the form m = 2 p + 1 equal the catheter can be written:
which is four times the p triangular number. This formula expresses then the Pythagorean theorem: the quadruplet of p triangular number and the (p + 1) odd number are the two catheti of a right triangle in integers in which the hypotenuse exceeds of one the even cathetus. It has namely:
for example, for p = 5 has the triangle (60, 11, 61). This formula can be deduced as a special case of the general formulas page 40 and 41 ( Arturo Reghini Sacred Pythagorean Numbers 9) by placing them in m = p + 1 and n = p; In fact, the x becomes:
.
38. (40 in the formula image) The two identities
P ( 3, 8 ) = P ( 4, 6 ) = 36 P ( 3 , 20 ) = P ( 5, 12 ) = 210
say that the 8th triangular is equal to the sixth square and the twentieth triangle is equal to the twelfth pentagonal. The problem of determining a triangle which is also a square was solved by Euler, the indeterminate equation
admits infinite integer solutions given by the double series
x 1 8 49 288 y 1 6 35 204 for which applicants apply formulas
Similarly Euler (Algebra, ed. Leipzig, page 391) has solved the problem of determining the triangular which are also pentagonal, that’s to say he solved the equation
whose infinite solutions are given by the double series x 1 20 285 3976 55385 … y 1 12 165 2296 31977. . .
for which formulas apply applicants
The triangular corresponding to odd values to the order x are also diagonal numbers, that’s to say they have the form z (z 2 – 1) with z = 1, 143, 27693. . .
The first solution, after the unity, of the two problems is that given by the two identities, that is that connected to the cube and octahedron, and the icosahedron and dodecahedron, and expresses the property that we set out on the cosmic figures.
Previous Chapter Arturo Reghini Sacred Pythagorean Numbers 11 .