Mathematical Series and Sacred Architecture: From Fibonacci to San Galgano Abbey
$+ 1 + 2 + 3 + 5 + 8 + 13 + 21 + \cdots$ (19) where each term is obtained from the sum of the two preceding numbers. To proceed with Euler's method we must define $$f(x) = 1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + \cdots$$ (20) and note that this expression is nothing other than the Taylor series expansion at 0 of the function $$f(x) = \frac{1}{1 - x - x^2}$$ (21) Since in this new formulation $f(1) = -1$, we can deduce that $$1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + \cdots = -1$$ (22) We note the symbolic marvel by which once again the golden ratio brings us back to unity through the sum of the series that historically has been most associated with it. The appearance of the negative sign before the Unity indicates a hidden motion in opposition to an external explicitation. The phenomenal becoming of the Fibonacci sequence is an external motion of the internal and hidden Unity expressed by -1. Another sequence associated with the golden ratio, although less known than the Fibonacci sequence, is given by the Lucas numbers, i.e. $1, 3, 4, 7, 11, 18, 29, \ldots$ (23) On the symbolic importance of these numbers we will certainly write an article, but at the moment we are only interested in highlighting how, considering $$f(x) = 1 + 3x + 4x^2 + 7x^3 + 11x^4 + 18x^5 + \cdots$$ (24) and noting that this is nothing other than the series expansion of $$f(x) = \frac{1 + 2x}{1 - x - x^2}$$ (25) we can obtain the remarkable result for which $$1 + 3 + 4 + 7 + 11 + 18 + \cdots = -3$$ (26) If the sum of the Fibonacci series refers to Unity, the sum of the Lucas series refers to the Trinity. This complementarity of the two numerical series is particularly interesting and we reserve to develop it separately in a series of future articles. We note finally that with Euler's method we can find, outside of $x = 1$, the sum of divergent geometric series, i.e. $$1 + a + a^2 + a^3 + \cdots$$ (27) noting that the function $$f(x) = 1 + ax + a^2x^2 + a^3x^3 + \cdots$$ (28) is nothing other than the Taylor series development at 0 of $$f(x) = \frac{1}{1 - ax}$$ (29) and therefore $$1 + a + a^2 + a^3 + \cdots = \frac{1}{1 - a}$$ (30) for any $a \neq 1$. We do not dwell on the symbolic meaning of the relation which we leave to the reader to develop and discover. For example, setting $a$ as the golden ratio, i.e. $a = \varphi = 1.618033\ldots$, we obtain the following remarkable series $$1 + \varphi + \varphi^2 + \varphi^3 + \cdots = -\varphi$$ (31) but not written in any book of mathematical symbolism. **Where Euler's method fails.** Given these premises, we might be interested in asking what the sum of the series of natural numbers is, or that of triangular numbers, or square numbers or in general of figurate numbers. We can try to proceed with Euler's method and in this case we have that to the sequence of natural numbers $1, 2, 3, 4, 5, \ldots$ (32) is associated the function $$f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + \cdots$$ (33) which unfortunately is the series expansion of $$f(x) = \frac{x}{(1-x)^2}$$ (34) which is not defined at 1. Here then is a not insignificant limit in Euler's method. To overcome this problem and obtain the sum of all natural numbers, we will have to introduce a special function called the Riemann zeta function, which will be the object of the continuation of this article in the next issue. (to be continued) **The Module of the Abbey of San Galgano - 1st Part** (Apothegm) The great Abbey of San Galgano in the plain of Merse, near Chiusdino, had its first origin from a Chapel and a small monastery erected around the year 1185 on Mount Siepi, in the very place where, according to legend, Galgano Guidotti withdrew in the year 1180 to hermitic life by planting, in the manner of a cross, his Sword in the cracks of a boulder. The legend tells that Galgano lived in that hermitage for about a year and that he died there on December 3, 1181, at the age of 33. Around the year 1224, the construction of the great Abbey of San Galgano was begun by the Cistercian monks of Mount Siepi. The design of a real sacred building is determined -on one hand- by requirements of function and structure, on the other by spiritual realities and mystical ideas that must be translated and grafted into the purified and receptive material of the Work. Therefore, while civil architecture is only functional and illusionistic, insofar as it makes no reference to any principle and model and serves only to satisfy various aesthetic aspects, sacred architecture is instead symbolic, that is, it is characterized by being informed, or rather modeled by imitation and participation in archetypal paradigms that must be profoundly known by the Architects. Thus designed, these buildings satisfy both a physical and metaphysical need, insofar as they act as support for transmission and contemplation of supraempirical principles. From what has been mentioned, we can understand that from any point of view one approaches and analyzes a structure...