Medieval Mediterranean Cultural Exchange: Focusing on the Fibonacci Sequence

초록

In 1202, Fibonacci wrote Liber Abaci, introducing the so-called Fibonacci numbers in the context of an example involving population genetics. Fibonacci numbers, 0, 1, 1, 2, 3, 5, 8, … given by the recursive definition: fib(0)=0, fib(1)=1, fib(x+1)=fib(x)+fib(x-1). But the sequence was well known in India before Fibonacci’s times. Fibonacci numbers were used on metrical science in India by Pingala in the 5th century BCE and by Bharata in the 1st century CE. In al-Mansur’s reign (c.766), books on mathematics and astronomy were brought to the palace by an Indian scholar called Khanha (or Mankah). From these, Baghdad's scholars were introduced to Hindu astronomy and mathematics. It has been said that this was the means by which the Hindu numerals were brought into Arabic mathematics. Al-Khwarizmi was the author of two outstanding treatises devoted to arithmetic and algebra. Especially, the decimal system of numeration which came from India. His untitled arithmetic book spread the Hindu-Arab number system widely throughout the Arab and European world. Fibonacci spent much of his early life in Bugia (Algeria) learning Arabic and studying mathematics under Moslem teachers. When Fibonacci traveled to the Mediterranean with his father, he also met Islamic scholars and came into contact with the mathematical knowledge of the Islamic world. There is no record of Fibonacci's mention of having received so-called "Fibonacci numbers" from the Arab world. Also, Fibonacci only mentioned the arrangement of 12 consecutive numbers in his book (Liber Abaci) modeled on a pair of rabbits. In this paper, I would like to present various possibilities that Fibonacci learned from the Arab world about the Pingala number that originated in India. In a situation where there is no record at all, I try to make an argument based on possibility, not proof. Also, I have briefly presented how the Pingala sequence is applied in modern design.