The Fibonacci Sequence

Leave a comment

It was in 1202 in his book Liber Abaci (Book of Calculation) that Leonardo Pisano (Leonardo of Pisa), would introduce to Western European mathematics the decimal number system that we use today.

In Liber Abaci, Leonardo Pisano, Fibonacci (a nickname attributed to Leonardo, meaning son of Bonacci, his father Guglielmo Bonacci), introduces a mathematical problem of reproducing rabbits, which in turn introduces the series of numbers which now bears his name—the Fibonacci Sequence or Series named so by French mathematician Edouard Lucas (1842–1891).

The Sequence was popularised by Fibonacci though may have been discovered earlier. Fibonacci was influenced by the “nine Indian figures” and Indian arithmetic and the Arabic numbering system, which included a zero to denote no value.

D. E. Knuth says in his book The Art of Computer Programming Vol. 1: Fundamental Algorithms errata to second edition:

“Before Fibonacci wrote his work, the sequence F(n) had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is F(n+1); therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, … explicitly. (P Singh in Historia Mathematica Vol 12 (1985) p. 229–244.)”

What is the Fibonacci Sequence?

The Fibonacci Sequence is a series of numbers starting at the number 1 and increasing by calculating the sum of the previous two numbers as follows:

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584…

The Sequence is infinite. And as the numbers increase, the ratio between succeeding numbers approaches a number that is now known as the mathematical ratio most “pleasing to the eye”: the formula for perfection or perfect beauty, the ratio of 1:1.618—the Golden Ratio or phi—noted by 1753 mathematician Robert Simson.

“The mathematician Robert Simson at the University of Glasgow in 1753 noted that, as the numbers increased in magnitude, the ratio between succeeding numbers approached the number α, the golden ratio, whose value is 1.61804…, or (1 + √5)/2.” (

Where is the Sequence found?

The Fibonacci Sequence is found in almost everything in existence. It is most commonly noted in the composition of art and design where the Golden Ratio is applied, and in music where Fibonacci numbers are the basic foundation of musical scales, notes and chords.

“According to Birken and Coon (2008, p. 59), Fibonacci numbers may appear also in the sphere of music. An octave consists of 8 notes and is represented on the piano by 8 keys. If we include sharps and flats, we add 5 black keys to the 8 white keys for a total of 13 keys, often referred to as the chromatic scale. The black keys themselves are positioned in groups of 2 and 3. All the numbers mentioned—2, 3, 5, 8, and 13—are Fibonacci Numbers.” (Management from a Natural Perspective: Discovering the meaning of Fibonacci Numbers for Management, Vlado Dimovski and Miha Uhan, undated)

By far the most obvious applications of the Fibonacci Sequence are in nature and the human body.

It was in the 19th century that scientists began to discover the Fibonacci Sequence of numbers in nature, “in the spirals of sunflower heads, in pine cones, in the regular descent (genealogy) of the male bee, in the related logarithmic (equiangular) spiral in snail shells, in the arrangement of leaf buds on a stem, and in animal horns.” (

The Sequence is seen in the branching of trees, the arrangement of leaves on a stem (phyllotaxis), the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone’s bracts. (

Fibonacci numbers in the human body’s DNA

“The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.” (

“The use of Fibonacci numbers and the Golden Mean through-out nature and by man is well established (Bachmann & Bachmann, 1979, p. 73). Fibonacci numbers appear in geometry, algebra, number theory, and many other branches of mathematics.

“However, even more spectacularly, they appear in nature; for example, the number of spirals of bracts on a pinecone is always a Fibonacci number, and, similarly, the number of spirals of bracts on a pineapple is also a Fibonacci number.

“The appearances in nature seem boundless. The Fibonacci numbers can be found in connection with the arrangement of branches on various species of trees, as well as in the number of ancestors at every generation of the male bee on its family tree.

“According to Krebs (2008, p. 185), petals on flowers, seeds on sunflowers… and the ratio of your height to the distance from your belly button to the ground provide for more of the examples.” (Source: Management from a Natural Perspective: Discovering the meaning of Fibonacci Numbers for Management, Vlado Dimovski and Miha Uhan, undated)

The identification of the Fibonacci Sequence in all of life, art, design and nature leads to some exciting analyses and interpretations.

How can we relate the Fibonacci Sequence to business?

In his book, Business Transformation Strategies: The Strategic Leader as Innovation Manager, Oswald A. J. Mascarenhas writes:

“Beauty is essential to the art of management. The more our culture becomes technology and information driven, the more do we need the emotional and metaphorical power of beauty (Neumeier 2009: 69–70).

“Buckminster Fuller once said, “When I am working on a problem, I never think about beauty. But when I have finished, if the solution is not beautiful, I know it is wrong”.”

Mascarenhas continues:

“There is ample evidence of mathematical beauty in nature, including the breathtaking complexity of fractals, the ancient sacred ratios of geometry, and the surprising concordance and harmony of theories across disciplines. Take the Fibonacci Sequence wherein each number in the sequence is the sum of the previous two. A Fibonacci Sequence looks like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. In nature, this progression is best seen in the patterns of pine cones and palm trees, in artichoke leaves and broccoli florets, in the shapes of nautilus shells (whose walls spiral outward according to the same laws). In business, the Pax Group, a home-and-office appliance design company, borrowed Fibonacci geometry to reshape its fan blades, and produced products that are 15–35 percent more energy efficient and 50–75 percent quieter.”

In their paper, Management from a Natural Perspective: Discovering the meaning of Fibonacci Numbers for Management, Vlado Dimovski and Miha Uhan give a good overview of studies towards finding tangible (and intangible) applications of the Fibonacci numbers in business and psychology.

“There have been some applications of Fibonacci to the business sphere in the past, but most of them dealt with predicting the markets in trading.

“John D. Waskom… has sensed the possibility that human development was intended to match the natural order of the material universe. Waskom loved to relate it to the spiral and its occurrence in plants, seashells, galaxies, and the DNA helix… When he called attention to the fact that young children unconsciously used phi proportions in their artwork, Waskom was affirming that unspoiled humans possessed a natural genius for living “in sync” with the universe.”

In this study, Dimovski and Uhan relate a possible application of Fibonacci numbers to the four known areas of management: planning, organising, leading and controlling, as follows:

Planning—the optimal solution for debt/equity ratio might be close to the Golden Ratio, which would mean that the optimal capital structure for a company would be 62% debt and 38% capital

Organising—determining the optimal size of your company

Leading—when to review performance of staff or staff promotions (in months 1, 3, 5, 8, etc. or in years 3, 5, 8, 13, respectively)

Controlling—when to monitor the activities of your employees (in weeks 3, 5, 8, etc.)

Fibonacci numbers in operations and marketing

While Dimovski and Uhan have hit onto something exciting, I wonder if this application can be taken even further to the day to day running and operating of business and marketing.

For example, can we use Fibonacci numbers to determine our markup or pricing? 62% markup on supplies or increasing our pricing by 10%, 20%, 30%, 50%, 80% over a period of 1, 2, 3, 5 years.

Can we use Fibonacci numbers to determine the frequency of our marketing communications in campaigns? For example, in weeks 1, 2, 3, 5, 8 or on days 1, 2, 3 then 8, 13, 21 of each month for emailing newsletters to subscribers or posting online.

If we can identify a pattern of idea formation, of innovation and of strategy and link this to Fibonacci numbers, we may be able to define something very special in the growth of the human mind and in business.

Along these same lines of thought, Waskom “sensed that if he could establish a relationship between phi and human psychological growth, he could begin to describe a natural pattern of developmental genius throughout the lifespan.” (Rose, 1991, sited by Dimovski and Uhan)