9 The Mathematics of Spiral Growth

2. 9 The Mathematics of Spiral Growth 9.1 Fibonacci’s Rabbits 9.2 Fibonacci Numbers 9.3 The Golden Ratio 9.4 Gnomons 9.5 Spiral Growth in Nature

3. Leonardo Fibonacci In 1202 a young Italian named Leonardo Fibonacci published a book titled LiberAbaci(which roughly translated from Latin means “The Book of Calculation”).Although not an immediate success, Liber Abaci turned out to be one of the mostimportant books in the history of Western civilization.

4. “The Book of Calculation” Liber Abaci was a remarkable book full of wonderful ideas and problems, but ourstory in this chapter focuses on just one of those problems–a purely hypotheticalquestion about the growth of a very special family of rabbits. Here is thequestion, presented in Fibonacci’s own (translated) words:

5. Fibonacci’s Rabbits A man puts one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if the nature of these rabbits issuch that every month each pair bears a new pair which fromthe second month on becomes productive?

6. Fibonacci’s Rabbits We will call P1 the number of pairs of rabbits in thefirst month, P2 the number of pairs of rabbits in the second month, P3 the number ofpairs of rabbits in the third month, and so on. With this notation the question askedby Fibonacci (...how many pairs of rabbits can be produced from [the original] pairin a year?) is answered by the value P12 (the number of pairs of rabbits in month 12).For good measure we will add one more value, P0, representing the original pairof rabbits introduced by “the man” at the start.

7. Fibonacci’s Rabbits Let’s see now how the number of pairs of rabbits grows month by month.We start with the original pair, which we will assume is a pair of young rabbits.In the first month we still have just the original pair (for convenience, let’s callthem Pair A), soP1 = 1. By the second month the original pair matures, becomes“productive,” and generates a new pair of young rabbits. Thus, by the second monthwe have the original mature Pair A plus the new young pair we will call Pair B, soP2 = 2.

8. Fibonacci’s Rabbits By the third month Pair B is still too young to breed, but Pair A generatesanother new young pair, Pair C, so P3 = 3. By the fourth month Pair C is still young,but both Pair A and Pair B are mature and generate a new pair each (Pairs Dand E). It follows that P4 = 5. We could continue this way, but our analysis can be greatly simplified by thefollowing two observations:

9. Fibonacci’s Rabbits 1. In any given month (call it month N) the number of pairs of rabbits equalsthe total number of pairs in the previous month (i.e., in month N – 1) plusthe number of mature pairs of rabbits in month N (these are the pairs thatproduce offspring–one new pair for each mature pair). 2. The number of mature rabbits in month N equals the total number of rabbitsin month N – 2(it takes two months for newborn rabbits to become mature).

10. Fibonacci’s Rabbits Observations 1 and 2 can be combined and simplified into a single mathematical formula: PN= PN – 1 + PN – 2 The above formula reads as follows: The number of pairs of rabbits in any givenmonth (PN) equals the number of pairs of rabbits the previous month (PN – 1) plusthe number of pairs of rabbits two months back (PN – 2).

11. Fibonacci’s Rabbits P5= P4 + P3 = 5 + 3 = 8 P6= P5 + P4 = 8 + 5 = 13 P7= P6 + P5 = 13 + 8 = 21 P8= P7 + P6 = 21 + 13 = 34 P9= P8 + P7 = 34 + 21 = 55 P10= P9 + P8 = 55 + 34 = 89 P11= P10 + P9 = 89 + 55 = 144 P12= P11 + P10 = 144 + 89 = 233 It follows, in order, that

12. Fibonacci’s Rabbits So there is the answer to Fibonacci’s question: In one year the man will haveraised 233 pairs of rabbits. This is the end of the story about Fibonacci’s rabbits and also the beginningof a much more interesting story about a truly remarkable sequence of numberscalled Fibonacci numbers.