Ratio metallica

Ratio metallica (gen. Rationis metallicae) signatur ratio $1:{\frac {n+{\sqrt {n^{2}+4}}}{2}}$ (cum numero n naturale). n-ima ratio metallica definitur major radix aequationis x² - nx - 1 = 0 quadraticae. Cifra est una at prima appellatur ratio aurea, secunda ratio argentea, tertia ratio aerea.

Ratio metallica n-ima

0	(0+√4)/2	1	1.0000000000
1	(1+√5)/2	(1+√5)/2	1.6180339887...
2	(2+√8)/2	1+√2	2.4142135623...
3	(3+√13)/2	(3+√13)/2	3.3027756377...
4	(4+√20)/2	2+√5	4.2360679774...
5	(5+√29)/2	(5+√29)/2	5.1925824035...
6	(6+√40)/2	3+√10	6.1622776601...
7	(7+√53)/2	(7+√53)/2	7.1400549446...
8	(8+√68)/2	4+√17	8.1231056256...
9	(9+√85)/2	(9+√85)/2	9.1097722286...
...	...
n	$(n+{\sqrt {n^{2}+4}})/2$

Bibliographia

• Stakhov, Alekseĭ Petrovich (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, p. 228, 231. World Scientific. ISBN 9789812775832.