The year 2002 marked the 800th anniversary of the publication of a breakthrough mathematics book, *Liber abaci* (*Book of Calculation*) by Leonardo of Pisa, in Italy. It supported a dramatic simplification of arithmetic; the way numbers were recorded and manipulated. The book begins:

The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures,

and the sign 0 ... any number may be written, as demonstrated below.

In 1202, Europe was quite different than it is today. Given the state of transportation at the time, international trade thrived over water even more than land. Air travel remained the domain of birds, bats and bugs. European merchant sailors enjoyed a flourishing trade in exotic goods along Mediterranean ports. These goods included spices from the East as well as goatskins from North Africa for tanning into leather in Italy. One result was that maritime trade created prosperous Italian city-states. Merchants' business transactions and records were cumbersome, though. Accountants calculated with an abacus and recorded their results in Roman numerals. They needed a better way and Leonardo was destined to help provide it.

Leonardo, dubbed "a solitary flame of mathematical genius during the Middle Ages," was the precocious son of a merchant sailor and diplomat, Guilielmo Bonacci. Leonardo is now more commonly called Fibonacci, meaning son of Bonacci (Latin *filius Bonacci*), although this name was not applied to him until the nineteenth century. Fibonacci was born in Pisa, Italy, just a few years before construction began on the famous Leaning Tower of Pisa. (In fact, his statue may be found on the plaza of the tower today.)

In his early years, Leonardo accompanied his father to Bugia (now Bejaia in Algeria) where he served as the director of the Pisan trading colony. Leonardo's father provided an Arab tutor to instruct him in Indian numerals and computation. Leonardo later traveled widely among Mediterranean ports in Egypt, Greece and Syria. Along the way he studied the works of more mathematically sophisticated Eastern cultures.

*Liber abaci* mainly illustrated routine calculations to solve practical problems in the new number system. Merchants needed a better way to keep accounts and exchange foreign currencies. The solutions offered in the book reveal Hindu influences of Brahmagupta and Bhaskara in methods of ratio and proportion along with Islamic influences of al-Khwarizmi and Abu Kamil with rhetorical (not yet symbolic) algebra.

After returning to Italy, Fibonacci was awarded a stipend for advising the Republic of Pisa in accounting and related mathematical matters. By decree, the Republic awarded the "'serious and learned Master Leonardo a yearly salarium of 'libre XX denariorem' in addition to the usual allowances."

The year 2002 marked the 800th anniversary of the publication of a breakthrough mathematics book, *Liber abaci* (*Book of Calculation*) by Leonardo of Pisa, in Italy. It supported a dramatic simplification of arithmetic; the way numbers were recorded and manipulated. The book begins:

The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures,

and the sign 0 ... any number may be written, as demonstrated below.

In 1202, Europe was quite different than it is today. Given the state of transportation at the time, international trade thrived over water even more than land. Air travel remained the domain of birds, bats and bugs. European merchant sailors enjoyed a flourishing trade in exotic goods along Mediterranean ports. These goods included spices from the East as well as goatskins from North Africa for tanning into leather in Italy. One result was that maritime trade created prosperous Italian city-states. Merchants' business transactions and records were cumbersome, though. Accountants calculated with an abacus and recorded their results in Roman numerals. They needed a better way and Leonardo was destined to help provide it.

Leonardo, dubbed "a solitary flame of mathematical genius during the Middle Ages," was the precocious son of a merchant sailor and diplomat, Guilielmo Bonacci. Leonardo is now more commonly called Fibonacci, meaning son of Bonacci (Latin *filius Bonacci*), although this name was not applied to him until the nineteenth century. Fibonacci was born in Pisa, Italy, just a few years before construction began on the famous Leaning Tower of Pisa. (In fact, his statue may be found on the plaza of the tower today.)

In his early years, Leonardo accompanied his father to Bugia (now Bejaia in Algeria) where he served as the director of the Pisan trading colony. Leonardo's father provided an Arab tutor to instruct him in Indian numerals and computation. Leonardo later traveled widely among Mediterranean ports in Egypt, Greece and Syria. Along the way he studied the works of more mathematically sophisticated Eastern cultures.

*Liber abaci* mainly illustrated routine calculations to solve practical problems in the new number system. Merchants needed a better way to keep accounts and exchange foreign currencies. The solutions offered in the book reveal Hindu influences of Brahmagupta and Bhaskara in methods of ratio and proportion along with Islamic influences of al-Khwarizmi and Abu Kamil with rhetorical (not yet symbolic) algebra.

After returning to Italy, Fibonacci was awarded a stipend for advising the Republic of Pisa in accounting and related mathematical matters. By decree, the Republic awarded the "'serious and learned Master Leonardo a yearly salarium of 'libre XX denariorem' in addition to the usual allowances."

Practical advances aside, the mathematical legacy of *Liber abaci* is really the fascinating set of numbers introduced by one problem posed about rabbits.

*How many rabbits can be produced from a single pair in a year if each pair begets a new pair every month, which from the second month on becomes productive, and deaths do not occur?*

To illustrate the pattern of population growth, let r represent a newborn pair of rabbits and R a mature pair. A newborn pair, r, appears in the next month as a mature pair represented by R, that is r → R. An adult pair R appears in the next month with offspring, that is R → Rr. The table below illustrates the first 4 months of population growth.

Months | Description | Symbols | Number |
---|---|---|---|

1 | Newborn | r | 1 |

2 | Mature | R | 1 |

3 | Mature & offspring | Rr | 2 |

4 | Mature, offspring & newly mature | RrR | 3 |

One way to look at the population in month 5 is to note that new offspring, r, come only from the rabbit pairs alive in month 3 and all rabbit pairs alive in month 4 continue as mature pairs, R.

Months | Symbol | Number |
---|---|---|

5 | RrRRr | 5 |

6 | RrRRrRrR | 8 |

Similarly, the population in month 6 is the sum of the populations of the two previous months, 8 = 5+3.

The pattern, that is, each month's population is the sum of the previous two months, is the key to this rabbit population growth. Thus the numbers of monthly populations is: 1, 1, 2, 3, 5, 8, 13 = 8+5, 21 = 13+8, 34 = 21+13, 55 = 34+21, 89 = 55+34, 144 = 89+55. Thus there are 144 pairs of rabbits in month 12.

The rabbit problem remained interesting over the years in mathematical circles. Henry E. Dudeney (1857–1930) thought it was too unrealistic, though. He changed Fibonacci's rabbits to cows and months to years before adding it to one of his books of puzzles, *536 Puzzles and Curious Problems:*

*If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?*

What is all the fuss about? Well, look closely at the Fibonacci numbers:

Terms | f(1) | f(2) | f(3) | f(4) | f(5) | f(6) | f(7) | f(8) | f(9) | f(10) | f(11) | f(12) | f(13) | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Values | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 |

Sums of selected Fibonacci numbers gives surprising answers. Add the 1st and 3rd elements (e.g. 1+2 = 3); you get the 4th! That is, in function notation, f(1) + f(3) = f(4). This is no aberration. By adding alternate elements we see a pattern:

f(1) + f(3) + f(5) = f(6) | 1+2+5=8 |

f(1) + f(3) + f(5) + f(7) = f(8) | 1+2+5+13=21 |

f(1) + f(3) + f(5) + f(7) + f(9) = f(10) | 1+2+5+13+34=55 |

f(1) + f(3) + f(5) + f(7) + f(9) + f(11) = f(12) | 1+2+5+13+34+89=144 |

Adding all successive Fibonacci numbers gives another interesting pattern.

f(1) + f(2) = f(4) - 1 | 1+1=3-1 |

f(1) + f(2) + f(3) = f(5) - 1 | 1+1+2=5-1 |

f(1) + f(2) + f(3) + f(4) = f(6) - 1 | 1+1+2+3=8-1 |

f(1) + f(2) + f(3) + f(4) + f(5)) = f(7) - 1 | 1+1+2+3+5=13-1 |

Actually, any number can be written as the sum of Fibonacci numbers. For example,

153 = 144 + 8 + 1 which is f(12) + f(6) + f(2).

Eduard Lucas was the first mathematician to study the Fibonacci numbers seriously for their mathematical properties. In fact, he made a general study of number sequences of this kind. They are called recursive sequences; that is, applying a fixed formula to previous terms generates new terms.

Not to be outdone, Lucas defined his own sequence, called, naturally, Lucas numbers. He began his sequence with 1 followed by 3 and then proceeded by adding terms the same way Fibonacci numbers are generated. Check them out below.

Terms | l(1) | l(2) | l(3) | l(4) | l(5) | l(6) | l(7) | l(8) | l(9) | l(10) | l(11) | l(12) | l(13) | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Values | 1 | 3 | 4 | 7 | 11 | 18 | 29 | 47 | 76 | 123 | 199 | 322 | 512 |

If you look closely, you can see that Lucas numbers are, in fact, sums of Fibonacci numbers; that is l(2) = f(1) + f(3) (i.e. 3 = 1 + 2); l(3) = f(2) + f(4) (i.e. 4 = 1 + 3); l(4) = f(3) + f(5) (i.e. 7 = 2 + 5). This pattern continues; if we jump all the way to l(12), it is equal to f(11) + f(13) (i.e. 322 = 89 + 233). As you might expect, Lucas numbers and Fibonacci numbers are also linked with interesting connections.

If you actually divide successive terms in the Fibonacci sequence, the quotients settle down to a specific value as you can see in the table below.

1/1 | = 1 | 2/1 | = 2 | |||

3/2 | = 1.5 | 5/3 | = 1.666... | |||

8/5 | = 1.6 | 13/8 | = 1.625 | |||

21/13 | = 1.61538... | 34/21 | = 1.61904 | |||

55/34 | = 1.61764 | 89/55 | = 1.61818 | |||

144/89 | = 1.61798 | 233/144 | = 1.61805 | |||

377/233 | = 1.61802 | 610/377 | = 1.61803 |

The quotients approach a number called the Golden Ratio whose value can be calculated algebraically. Using conventional subscript notation, the Fibonacci sequence is defined:

f_{1} = f_{2} = 1, f_{n+1} = f_{n} + f_{n-1}, n = 2, 3, ....

Now, the Golden Ratio is usually called Ø. We say f_{n+1}/f_{n }→ Ø as n → ∞. Noting that both f_{n+1}/f_{n} and f_{n} /f_{n-1} have the same limit Ø as n → ∞, the calculation goes like this:

Now replace the two terms: f |
||

This quadratic equation has the two solutions: |
||

The first solution is the Golden Ratio Ø, and the second is just - 1/Ø !

Why is this ratio *golden*? That's really hard to say. It is definitely related to the pleasing nature of the proportion and the uncanny way it pops up in architecture and art; for example, we find the ratio in:

- Proportions in Leonardo da Vinci's Mona Lisa
- Rectangles in Piet Mondrian's Composition 10

Many people claim that the ratio also appears in the Parthenon, in Athens. Unfortunately, it is not possible to determine the original measurements of the building due to its partial destruction over the centuries.

By the 17th century, the well-known astronomer Johann Kepler observed that the Fibonacci numbers are commonly found in plants. For example, a white lily is a one petal flower, euphorbia have 2 petals, trillium have 3, columbine have 5, bloodroot have 8, black-eyed Susan have 13 and Shasta daisies have 21. Do those numbers sound familiar? Other daisies might have 34 or 55 petals and larger sunflowers often have 89 or 144 petals. Later, in 1790, Bonnet first noted that the scales of pinecones and pineapples contain families of interlaced spirals, one winding clockwise and the other, counterclockwise. The same spirals appear in the heads of daisies and sunflowers. Depending on the species, the pairs of spirals number 21 and 34, 34 and 55, 55 and 89, or actually any two adjacent Fibonacci numbers.

This phenomenon is common throughout the plant world, but why? Understanding why was a multidisciplinary effort. Several ideas were proposed over the years, many based on the notion of natural selection. For example, the observed arrangement of spirals could provide dense packing of seeds, maximizing offspring, or perhaps the arrangement of leaves and petals allows sunlight to nourish as many as possible. The key is to determine how Fibonacci numbers promote the survival of mature plants. A daisy whose flower has 22 spirals going in one direction and 33 in the other should have about the same chance of producing viable seeds as one whose flower has 21 spirals in one direction and 34 in the other. Yet the first case is virtually unheard of while the second is quite common.

To understand we need to examine plant growth more closely. Consider the schematic diagram of the sneezewort plant . At each growth level both the number of branches and the number of leaves is a Fibonacci number.

Plants tend to grow from a single central apex. Emerging from the apex, small bumps called primordia appear. These primordia will eventually become leaves, petals, sepals or the like. They grow outward in spirals. By 1837, the pioneer crystallographer Auguste Bravais and his brother Louis observed that successive primordia make an angle of about 137.5°, as seen from the apex. This angle is supplementary to 360°/Ø (about 222.5°). Is it mere coincidence that the Golden Ratio, Ø, is involved?

In 1992, Douady and Couder from the Laboratory of Statistical Physics in Paris offered an important breakthrough. They developed a laboratory model of plant growth. According to their model, the dynamics of plant growth can account for the Fibonacci numbers. The model shows that the angle at which the primordia move away from the apex determines the spiral. If the angle is a rational fraction of 360°, then there is no spiral. Primordia would line up and, consequently, leaves and petals would eventually block the light of their predecessors. Irrational multiples of 360° create spirals that never result in overlapping leaves or petals. It makes sense that the least overlap would come from some irrational multiple of 360°, and it does! In particular, Ø is the irrational number with the simplest expression as a continued fraction.

The Douady-Couder model shows that spirals at this angle create the most efficient packing, expending the least amount of plant energy. Remember that quotients of successive Fibonacci numbers approach Ø. These quotients most closely approximate Ø. Thus, the number of spirals coming from the apex would have to be a Fibonacci number. The particular one depends only on the size of the primordia, that is, how many primordia fit around the apex before they overlap and join an existing spiral.

In the early 1930s a retired accountant, Ralph Nelson Elliott, carried out a detailed study of the changes in the Dow Jones Industrial Average. For data, Elliot had at his disposal results of the roaring bull market of the 1920s followed by the Great Depression. He concluded that stock market patterns were not random but, instead, reflected cycles of human optimism and pessimism. His treatise, *The Wave Principle,* was completed by 1935. In it he claimed to have established principles that could be used to forecast market reversals!

Basically, he recognized a simple pattern in market fluctuations. The pattern of a rising market, he said, consists of an upward trend followed by a correction. In the figure, the upward trend rises to (1) and the correction falls to (2). The upward trend is broken into three peaks, labeled 1, 3, and 5. The correction has 2 troughs, labeled A and C. This suggests that the index moves three steps forward followed by two steps back.

The entire pattern consists of 8 direction changes or waves, 5 in the impulsive phase and 3 in the corrective phase. Each of these is a Fibonacci number. Elliot claimed further that a closer examination of each wave -- for example, daily fluctuations instead of weekly -- would reveal a similar breakdown, illustrated below.

The rising side is broken into 21 wavelets (8+8+5), the falling side is broken into 13 wavelets (8+5). So far, the analysis decomposes the original wave into 34 parts, another Fibonacci number. This could go on forever but Elliot ended at 144, claiming that this is the largest number of practical value.

Later refinements of Elliot's theory relate the Golden Ratio to the depth of the wave.

Wave analysis of real life market indices is highly technical and subjective. There are many instructive websites explaining principles and applications that can be accessed for further research.

The overall outlook of Elliotticians today is very pessimistic. They believe we are entering a correction from a very large impulsive peak 5. As the theory is applied today, the correction will end at 38.2% above the level at which this great wave began. There is, however, no consensus as to exactly what that date may be. One ambitious project analyzes the last 12,000 years according to these principles!

**More information about the Fibonacci numbers and the Golden Section**