3 Additive properties

Let us consider the sequence of ratios of consecutive terms of the sequence (1.1). The result is a new sequence $\frac{1}{1},\frac{2}{1},\frac{3}{2},\frac{5}{3},\frac{8}{5}, \frac{{13}}{8},\frac{{21}}{{13}}, \frac{{34}}{{21}},\frac{{55}}{{34}},\frac{{89}}{{55}},\ldots$ that converges to the Golden Mean $\phi.$ If we take now a geometric progression of ratio $\phi: \ldots,\frac{1}{{\phi^2 }},\frac{1}{\phi}, 1,\phi ,\phi^2 ,\phi^3 ,\ldots$ it is easy to verify that this geometric progression is also a GSFS that satisfies relation (1.3). In fact, $\frac{1}{{\phi ^2 }} + \frac{1}{\phi } = \frac{{1 + \phi }}{{\phi^2 }} = 1.$

In the general case, the fact that $\sigma$ is a root of equation (2.1) in virtue of the equality $\sigma ^{n+1}=p\sigma ^n+q\sigma ^{n-1}$ is obviously equivalent to the assertion: The geometric progression of ratio $\sigma :\ \ldots,\frac{1}{{\sigma ^2 }},\frac{1}{\sigma }, 1,\sigma ,\sigma ^2 ,\sigma ^3 ,\ldots$ is a GSFS that satisfies relation (1.3). This fact allows us to state the following unique mathematical property:

The members of the MMF are the only positive quadratic irrational numbers that originate GSFS (with additive properties), which are, simultaneously, geometric progressions.

This curious property of satisfying both arithmetic additive and geometric properties, bestow upon all the members of the MMF interesting characteristics to become basis of different systems of geometric proportions in design (see [1] and [2]).