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 Fn/Fn-1 strives to the golden proportion.

Let us consider a limitation of the ratio of the next p-Fibonacci numbers for . With this aim in view let us introduce the following definition:

 (1)

Let us present now the ratio of the next p-Fibonacci numbers in the following form:

 (2)

Taking into consideration definition (1) for one may replace expression (2) by the following algebraic equation:

 xp+1 = xp + 1. (3)

Let us mark in tp the real root of algebraic equation (3). Let us investigate equation (3) for different values 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:

 x2 = x + 1. (4)

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 AB by the point C according to the following ratio:

 , (5)

where p = 0, 1, 2, 3, ... .

Figure 1. Golden p-Sections (p = 0, 1, 2, 3, ...).

Note that proportion (5) is reduced to the dichotomy for p = 0 (Fig.1-a) and to the classical Golden Section for p = 1 (Fig. 1-b). Taking into consideration the recent fact the subdivision of the line segment AB by the point C in ratio (5) was called the golden p-section but the real root of equation (3) the golden p-ratio or golden p-proportion.

The following property of the golden p-ratio emerges from algebraic equation (3):

 (6)

For n = 1 identity (6) has the following form:

 (7)

Let us write down identity (7) in the following form:

 (8)

It follows from (8) that the golden p-ratio is converted into the number, which is inverse to the p-th power of the golden p-ratio if we subtract the number 1 from it.

Note that for p = 0 we have tp = 2 and identity (6) is reduced to the following trivial identity for the binary numbers:

2n = 2n-1 + 2n-1.

For p = 1 we have and identity (6) is reduced to the following:

Thus, as result of this consideration we get a number of mathematical discoveries:
1. We have shown that there exist a fundamental connection between Pascal Triangle, Fibonacci numbers and golden ratio!
2. 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!