Generalized Golden Sections We know from preceding pages of our Museum that Fibonacci numbers are connected closely with the golden proportion. In particular, a limitation of the ratio F strives to the golden proportion._{n-1}Let us consider a limitation of the ratio of the next
Let us present now the ratio of the next
Taking into consideration definition (1) for one may replace expression (2) by the following algebraic equation:
Let us mark in t p. For p = 0 equation (3) is degenerated into the trivial equation x = 2. For p = 1 equation (3) is reduced to the algebraic equation of the golden proportion:
which has the real root Thus, equation (3) may be considered as some generalisation of the golden proportion equation (4). Here equation (3) has the following geometric interpretation (Fig.1). Let us subdivide the line segment
where
Note that proportion (5) is reduced to the dichotomy for The following property of the golden
For
Let us write down identity (7) in the following form:
It follows from (8) that the golden Note that for 2 + 2^{n-1}.^{n-1}For - We have shown that there exist a fundamental connection between Pascal Triangle, Fibonacci numbers and golden ratio!
- But investigating Pascal Triangle we have generalized the classical Fibonacci numbers and the classical golden section and introduced the notions of the generalized Fibonacci numbers (
*p*-Fibonacci numbers), the generalized golden sections (the golden*p*-sections) and the generalized golden ratios (the golden*p*-ratios). Thus, we have revealed one more secret of the Pascal Triangle, which keeps in itself a new class of irrational numbers, the golden*p*-ratios (*p*= 0, 1, 2, 3, ...).
And these results appeal us for new discoveries! We will try to give a new number definition based on the golden p-ratios and to create a new mathematics, the Mathematics of Harmony, and we welcome you to follow us! |