The history of numbers in Western society was initiated by the impressive work of a young Italian, Leonardo Pisano, who published the “Book of Calculation” (Liber abbaci, frequently misquoted as Liber Abaci) in 1202 [6]. Pisano’s legacy is commonly known under his nickname Fibonacci, which is derived from the Latin filius Bonacci and extrapolates to “The son of the Bonacci family” [7]. Fibonacci learned the Hindu-Arabic numbering system from Arabic mathematicians and merchants during his trip to Northern Africa when he visited his father in the port city of Bugia (now Béjaïa, Algeria) as a teenager (Fig. 1). The new insights led the young Italian to question the value of the traditional Roman numbering system that had been exclusively based on plain letters and therefore did not allow for mathematical calculations. Intriguingly, the traditional Roman numeral system was devoid of a symbol for “zero,” as the number zero was not practically relevant in ancient life [8]. The major shortcoming of the lack of the numeral zero relates to the impediment of performing calculations in multiples of tens, hundreds, and etc., and the inability to calculate with negative numbers, both of which represent prerequisites for modern mathematics, technology, and science [8]. Fibonacci’s historic legacy is represented by his ability to define the “golden mean” (or “golden ratio”), a previously unresolved enigma that dated back to the times of the ancient Greek philosophers and mathematicians (Fig. 2). At Fibonacci’s times, understanding the golden mean was considered as close as unifying the principles of mathematics and science with nature and God [9].
The Fibonacci sequence is one of the groundbreaking new insights in his Liber abacci [6]. Most readers may recognize Fibonacci’s name from Dan Brown’s best-selling novel, The Da Vinci Code, where a dying man in the opening scene scrawled Fibonacci’s sequence in invisible ink on the floor of the Louvre museum in Paris [10]. The intriguing origin of the Fibonacci sequence, however, is scarcely known: As a young man, Fibonacci was challenged with the ancient task of calculating how many rabbits would be born within 1 year, originating from a single pair of rabbits. His calculation was based on the assumption that each rabbit pair will produce another pair of rabbits every month, and rabbit pairs start breeding when they are 2 months old. Fibonacci described the problem in chapter 12 of his Liber abbaci, as such [6]: “A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.”
Fibonacci’s solution to the “rabbit breeding theory” [7] set the foundation of a historic mathematical miracle (Fig. 3). In brief, there is one pair of rabbits in 1 month, one additional pair of rabbits as first offspring from the original pair, three in 3 months (an additional couple from the original pair), five in 4 months (with the first offspring now breeding as well), followed by eight, 13, 21, 34, 55, 89, 144, and a total of 233 pairs of rabbits at the end of the first year. This sequence of numbers represents the essence of the Fibonacci sequence, where each number represents the sum of the two preceding numbers. The mathematical magic about this simple series of numbers is that the “divine proportion” (golden ratio) sought by Aristotle in ancient Greece as a philosophical concept of a “desirable middle between two extremes, one of excess and the other of deficiency” is calculated by dividing any number in the Fibonacci sequence, after number 144, by its preceding number. The result will always be 1.618—the mathematical constant PHI (Fig. 2) [11]. In general terms, the golden mean implies a perfect moderate position that avoids extremes, e.g., as a ratio between the two divisions of a line such that the ratio of the smaller to the larger is identical to the ratio of the larger to the sum of both lines. We encounter the golden ratio every day in arts, culture, religion (for example in the cross of Christ), and in multiplicity of phenomena in mathematics, science, and nature, including in plants and animals [11, 12].