Phi Φ

This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.


(Previous Section: Fibonacci Sequence)

Master Fibonacci
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Besides the fact that rabbits produce at a “geometrical rate” (as do the numerical values of the digits in the Fibonacci sequence), there is a strange and wonderful relationship between geometry (which deals with properties, measurement, and relationships of points, lines, angles, and figures in space) and the Fibonacci sequence, which is derived from algebra (in which symbols such as letters and numbers are combined according to the rules of arithmetic) (Scotta and Marketos).

Admittedly applying his knowledge of the theories and applications that he had learned in the “niceties of Euclid’s geometric art,” the medieval Italian mathematician Leonardo Pisano “rediscovered” the arithmetic series (Gascueña). When a number in Fibonacci’s series, (1, 1, 2, 3, 5, 8, 13, …) is divided by the number preceding it, the quotients become the following series of numbers:

1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666…, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538…

The ratios approach the particular value called the “Golden Ratio” or the “Golden Number.” It has a value of approximately 1.618034 and is represented by the Greek letter Phi (Φ, φ) (Scotta and Marketos).

German mathematician, astronomer, and astrologer Johannes Kepler (1571-1630 CE), (best known today for developing laws of planetary motion) noticed this pattern, in which the ratio of consecutive Fibonacci numbers approaches the Golden or Divine Ratio (Scotta and Marketos).

What is even more fascinating about this number, 1.618 …, is the fact that it also expresses a unique relationship between two specific segments of a straight line, first defined by Euclid.

One of Euclid’s strategies in Elements was to develop results (called propositions) about geometry which were proved solely by using logic based purely on axioms and previously-proved propositions. He showed how to divide a line in mean and extreme ratio in Book 6, Proposition 30. Euclid used this phrase to mean the ratio of the smaller part of a line (CB), to the larger part (AC): CB/AC. This ratio is the SAME as the ratio of the larger part, AC, to the whole line AB (i.e. is the same as the ratio AC/AB). Therefore, CB/AC = AC/AB. The resulting quotient is 1.618 or Phi, a geometric construction (concerning the properties of figures) (Gascueña).

Thus, ratios derived from the division of successive Fibonacci’s numbers (from F(8) on) are the same number derived from dividing particular proportions of a straight line! (Livio 103).

Golden Section

The Golden Ratio Livio
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While Liber Abaci contained some geometry problems, Practica Geometriae (The Practice of Geometry) (1223) demonstrates the mathematical brilliance of Fibonacci, for it was a well-written book containing several chapters covering basic concepts of Euclidean geometry theorems with substantial, rigorous proofs that were more advanced than the geometry of others preceding him, like Boethius and Gerbert (Pope Sylvester II). More importantly, this large book contained many practice problems dealing with area and volume formulas for plane figures and bodies (Horadam, “Eight”). The solutions and explanations, however, were not written in an esoteric language. Rather, he supplied original and instructive explanations which were widely accessible, written in the vernacular language most useful to fellow citizens; he presented solutions to problems that men in common trades (such as surveyors) were able to use to make their labor more productive and profitable. Repeatedly, Fibonacci proved to be socially relevant, which is why his achievements were clearly recognized by his contemporaries and why he is considered by modern math historians to have been a genius who was able to “see the greatness in the commonplace, and to recognize the enormous potential to change the world in what seems to most people to be a mundane or obscure idea” (Devlin, Finding 21).


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Master Fibonacci: The Man Who Changed Math

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Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)
Language: English