The Sisters of the Golden Section

The golden proportion is widely believed to be extraordinarily prevalent in nature and the arts, which is often ascribed to it being the limit of the ratio between any two successive elements in the Fibonacci sequence. It is suggested here that the golden ratio may not be as exceptional as generally believed. Mathematically, some interesting properties are common to all members of a family of sequences, denoted ARS, characterised as solutions to the classic rabbit reproduction problem varying on some parameter, j, including the Fibonacci sequence as a proto-typical member—ARS₂. Furthermore, for j > 1, any limit of the ratio between successive elements in ARS _j , shares the same formal properties with all other such limits. Three actual interpretations and three further geometric applications of ARS₃, all intimately analogous to corresponding ARS₂ ones, are presented for the sake of illustration. Empirically, it is suggested here that, owing to the communality of interesting mathematical properties between ARS sequences, as well as between corresponding limits, nature might appear to have made use of some other limits, aside of its variegated use of the limit of ARS₂ — the golden ratio. Initial empirical clues are provided. Finally, the issue whether there really is special import to golden proportions in nature and the arts is revisited in view of some empirical comparisons of appearances related to Fibonacci numbers and ARS₃ numbers, particular its limit (∼1.466) and the inverse of that limit (∼0:682). It is argued that the claim that Fibonacci-related numbers are especially distinguished seems to warrant a more qualified approach than it has often met.