The Legacy & Problems with Leonardo Fibonacci: A critical look at one of the most influential mathematicians of all time

(1170?–?c. 1240–50.) In Pisa, Republic of Pisa, in the year of 1170, a famous mathematician was born. In fact later, he became known to the world as one of the most talented mathematicians of the Middle Ages and his contributions even to today's modern world are many and totally ubiquitous (Wikipedia 2023).

Leonardo Bonacci, changed, forever, how the world of numbers was seen, understood and even how the physical world around us, from the smallest structures of the universe, to the very largest, have some kind of internal "rhythm" or structural similarities which were completely unknown to scholars, to anyone, before his time.

On top of this feat, Fibonacci, the "Traveller from Pisa" also is famous for popularizing the?Indo-Arabic numeral system?in the Western world primarily through his composition in 1202 of Liber Abaci?("Book of Calculation").?He also introduced Europe to the sequence of Fibonacci Numbers, which he used as an example in?Liber Abaci (ibid. 2023). Fibonacci brought "the Indian method" to the Western hemisphere and laid the foundation for "the most common system for the symbolic representation of numbers in the world": 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. These symbols and numerals was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics?by the 9th century (ibid. 2023).

Liber Abaci ("The Book of the Abacus" - otherwise known as - "The Book of Calculation")

The lifework of Leonardo Fibonacci, Liber Abaci (1202), is a historic Latin manuscript on arithmetic. Liber Abaci?was among the first Western books to describe the Hindu-Arabic numeral system?and to use symbols resembling modern "Arabic numerals". "By addressing the applications of both commercial tradesmen and mathematicians, it promoted the superiority of the system, and the use of these glyphs" (ibid. 2023). The historian of mathematics Carl Boyer?stated in his?History of Mathematics: "The book in which Fibonacci described the new algorism is a celebrated classic, completed in 1202, but it bears a misleading title – Liber Abaci (or "book of the abacus"). It is?not?on the abacus; it is a very thorough treatise on algebraic methods and problems in which the use of the Hindu-Arabic numerals is strongly advocated" (ibid. 2023).

The Fibonacci Sequence

Liber Abaci?posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci Numbers. Although Fibonacci's?Liber Abaci?contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century (ibid. 2023) From the very start, Fibonacci intended Liber Abaci, to be about understanding the real physical world around us and providing helpful practical solutions to make calculations, expressions and solve tangible problems - not some kind of academic / numerical thought-experiment.

From the understanding and projection of (fx rabbit) population growth to the ways stars and galaxies rotate and spiral into intergalactic super-clusters or to how an accretion disc is born around a blackhole, to the structure of the bones in our ears or the spirals we see in the world of botany when flowers, trees and plants arrange the centre of their leaf- or pedal discs. All of these examples all could be expressed and explored through the Indo-Arabic numerals and it also all comes back to the characteristic spiral formation in the exact form of the Fibonacci Sequence's "Golden Ratio" - one of the most famous as well as ubiquitous patterns of nature.

Although implicit in his sequence, it was the German Mathematician Simon Jacob who noticed that "consecutive Fibonacci numbers converge to the golden ratio" (Exitus Capital LLC, 2023).

The more "popular" discovery of Fibonacci was manifested by famous Italian artist Leonardo Da Vinci (1452-1519) who illustrated Luca Pascioli's publication "Divina Proportione" (1509) and Leonardo called the Golden Ratio "Section Aurea" (The Golden Section) - This eventually became the bedrock of inspiration for many of his own world-famous works: "The Vitruvian Man" or "Mona Lisa".

To be sure, all this is NOT random - this appears to be integral to the fabric of nature itself and the physical universe around us. Leonardo Fibonacci and the Indian Mathematicians discovered this amazing truth and shared it with the rest of humanity.

In the Fibonacci Sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.?Fibonacci, himself, never mentioned the Golden Ratio?as the limit of the ratio of consecutive numbers in this sequence (ibid. 2023). He himself, likely never imagined the universal aspects of his sequence or ratio, he never mentioned it, but he did, suspect that something inexplicably, almost magical, and, certainly, something very practical was inherent within the sequence itself and the Indo-Arabic numerals. Hence, he advocated, it be suitable, as the foundation of arithmetics and of calculus.

The Problem with using "Fibonacci Retracement Theory" in Modern Financial Markets & Technical Analysis

In modern Technical Analysis of prices in free, liquid, trending markets we understand a "retracement" to be a "correction to the principal trend" (Kirkpatrik & Dahlqvist 2015:245-246). In an upward trend, for example, as prices rise, they are interrupted, periodically, by smaller counter-moves, often revealing a kind of zig-zag formation to the eyes on the chart. As time progresses, these smaller formations, in turn, create even bigger formations in prices, and ultimately form the basis of what we know today as "support" and "resistance" (ibid. 245-246). A retracement is a natural part of any healthy trend and exist on all time scales, from seconds to monthly bars and due to the fractal nature of patterns - we see the very same rhythm play out on all time scales - at all times.

"A retracement, in a healthy trend, can also represent an opportunity for the trend follower who missed the earlier stages of the trend to jump on the longer trend. A retracement, as long as it holds above the longer trendline and does not retrace more than 50%, is usually an opportunity for the trend follower to act in the direction of the larger trend" (ibid. 246).

"Unfortunately, retracements rarely reach an exact percentage". - (ibid. 246)

Free markets are both too chaotic and too volatile and also subject to too many random, exogenous factors and affected by (far) too many variables to always produce an exact absolute or relative retracement - which the Fibonacci Retracement Theory proposes fx via systematic "fanlines" plotted - or forward-projected onto the charts of the analyst - as exemplified above.

Although the logic seems sound, that a retracement is a possibility for entry or exit in trending markets, and we actually see the tendency for prices to "fan out" or "distribute" after a rise in prices when the trend exhausts, unfortunately, empirical studies have found no significant relationship between applying the Fibonacci Sequence (in any way) in price-retracements and achieving profitable risk-adjusted returns. In fact, empirical observations of trading markets found that retracements rarely follow this exact percentages of fx 33 1/3%, 50%, 66 2/3% and also the Fibonacci Sequential Percentages of 38.2% and 61.8% and so on (ibid. 246).

The point is, you have no statistically positive expectation of using the Fibonacci Sequence, but yes, you also do have a TON of cases where the Fibonacci Sequence actually plays out in price analysis - to almost perfection. But is this accurate or regular enough to make a rule-based, number-driven, or even semi-discretionary system with a positive expectation for profitable risk-adjusted returns: No, it seems.

In a 1989 paper by late Art Merril, in the Journal of Technical Analysis (Aug. 1989) it was found that the amount by which prices retrace in the Dow Jones Industrial Average (DJIA) during an advance or decline did not concentrate about any of these percentages. Later studies found similar results. As such we must conclude, anticipating exact retracements (such as those, based on Fibonacci, D. W. Gann, Andrew's Pitchfork, Speedlines, etc.) is hazardous and has no statistically significant merit, if indeed the aim of the analyst, is to make a profitable trading system.

Does that mean Fibonacci is useless, no, absolutely NOT. Leonardo Fibonacci will always be remembered for his contributions to both mathematics and also many other epistemological fields of modern research which are met with the Fibonacci Sequence or the Golden Ratio, in some way, shape or form.

From contemporary astronomers who study the movements of celestial objects, to the neuroscientist looking at our brain through a microscope, to the child on the beach who ponders the beauty and captivating symmetry of a seashell he or she found in the shallows, only for, to take home and admire it. The World of Leonardo Fibonacci is everywhere, it is all around us, everywhere and at all times. The "Traveller from Pisa", Leonardo Fibonacci, is the person we can indirectly thank for many discoveries in the future, using his sequence or discovering his patterns somewhere in nature. All of these are testament to the legacy of Leonardo Fibonacci and Indian Mathematicians who first discovered, and Fibonacci who shared with us, these magical numerals by discovering these natural patterns and, who then, brought them into our modern lives.

From art, biology, mathematics, music, history, architecture, to finance and also in many other fields (Exitus Capital LLC: 2023). Even the soundwaves from music or pulsations detected from a distant neutron star can fx all be expressed in terms of the Golden Ratio - in the form of the quintessential "spiral waves" emanating from its centre.

Many scholars today see Leonardo Fibonacci as one of the the most influential figures in the study of patterns, mathematics, ever (Benjamin 2013). Leonardo Fibonacci embodies all what is beautiful and - even magical - about mathematics, from inspiration, to calculation and finally, to application.

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DISCLAIMERS

X. The aim of this article is to inspire and educate - NOT offer investment adivce.

Y. Any factual inaccuracies found, please contact me for instant correction. The aim is to represent, nothing but the consensus truth.

Z. ANY opinions are solely my own opinions and do NOT represent those of my employer.

Bibliography / Credits / Sources

Kirkpatrik, C. & Dahlqvist, J. (2015) "Technical Analysis: The Complete Resource for Financial Market Technicians", 5th printing, Pearson Education Inc, Upper Saddle River, NJ, USA.

Wikipedia (2023) "Leonardo Fibonacci", https://en.wikipedia.org/wiki/Fibonacci.

Wikipedia (2023) "Hindi-Arabic Numerals", Link: https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_system.

Exitus Capital LLC / Author: M.B.A. Francisco Montalvo "Patterns in Nature and Finance. The Fibonacci Sequence and the Golden Ratio", Accesible via: https://exituscapital.com/patterns-in-nature-and-finance-the-fibonacci-sequence-and-the-golden-ratio/.

YouTube (2013) "Nature's Code" YouTube Link: https://www.youtube.com/watch?v=wTlw7fNcO-0&ab_channel=SciShow.

Arthur Benjamin / YouTube (2013) "The magic of Fibonacci numbers", YouTube Link: https://www.youtube.com/watch?v=SjSHVDfXHQ4&ab_channel=TED.

Popular Science / Author: Margo Milanowski (2021) "These two galaxies are locked in a cosmic battle: Galaxies millions of light years away show what could happen between the Milky Way and its neighbor, Andromeda", Link: https://www.popsci.com/space/hubble-telescope-catches-galaxy-fight/.

Google Images (2023) "Andreia Pinho & Jo?o da Silva Correia" Link: "https://sites.google.com/site/leonardofibonacci7/".