## The Fibonacci Summation Series in Nature and Geometry

It is remarkable how many constant values can be calculated using Fibonacci's sequence, and how often the individual numbers of the sequence recur in myriad variations.

This is not just a numbers game, however; it is the most important mathematical representation of natural phenomena ever discovered. Generally speaking, the Fibonacci summation series is nature's law, and it is a part of the aesthetics found in any perfect shape or curve.

Fibonacci discovered how nature's law related to the summation series when he proposed that the progeny of a single pair of rabbits increased in a repeatable pattern:

Suppose there is one pair of rabbits in January, which then breed a second pair of rabbits in February, and, thereafter, these offspring produce another pair every month. The mathematical problem is to find how many pairs of rabbits there will be at the end of December.

To solve this little algebraic puzzle, we tabulate the data in four columns:

1. The total number of pairs of breeding rabbits at the beginning of each given month.

2. The total number of pairs of nonbreeding rabbits at the beginning of each month.

3. The total number of pairs of rabbits breeding during each month.

4. The total number of pairs of rabbits that have been bred at the end of 12 months.

Table 3.1 shows the progression to the total number of rabbits, based on the four criteria.

Month |
(1) |
(2) |
(3) |
(4) |

January |
0 |
1 |
0 |
1 |

February |
1 |
0 |
1 |
2 |

March |
1 |
1 |
1 |
3 |

April |
2 |
1 |
2 |
5 |

May |
3 |
2 |
3 |
8 |

June |
5 |
3 |
5 |
13 |

July |
8 |
5 |
8 |
21 |

August |
13 |
8 |
13 |
34 |

September |
21 |
13 |
21 |
55 |

October |
34 |
21 |
34 |
89 |

November |
55 |
34 |
55 |
144 |

December |
89 |
55 |
89 |
233 |

Source: The New Fibonacci Trader Workbook, by Robert Fischer (New York: Wiley, 2001), p. 20.

Source: The New Fibonacci Trader Workbook, by Robert Fischer (New York: Wiley, 2001), p. 20.

Each column contains the Fibonacci summation series, formed according to the rule that any number is the sum of the pair of immediately preceding numbers.

One needs only to look at the beauty of nature to appreciate the relevance of the Fibonacci ratio PHI as a natural constant. The number of axils on the stems of many growing plants and the number of petals on flowering plants provide many examples of the Fibonacci ratio and underlying summation series. The following illustrations depict some interesting applications of this mathematical sequence.

### Fibonacci Numbers Found in Plants

The sneezewort, a Eurasian herb, is an ideal example of the Fibonacci summation series in nature, for every new branch springs from the axil and more branches grow from a new branch.

Adding the old and the new branches together reveals a number of the Fibonacci summation series in each horizontal plane. Figure 3.1 illustrates the count.

Figure 3.1 Fibonacci numbers found in the flowers of the sneezewort. Source: The New Fibonacci Trader Workbook, by Robert Fischer (New York: Wiley, 2001), p. 4.

According to the same algebraic principle, we can easily identify Fibonacci summation series in plant life (so-called golden numbers) by counting the petals of certain common flowers. Taking the iris at 3 petals, the primrose at 5 petals, the ragwort at 13 petals, the daisy at 34 petals, and the michalmas daisy at 55 (and 89) petals, one must question whether this pattern is accidental or a particular natural law.

### Rule of Alternation in the Sunflower

The beautiful curving lines of the sunflower have existed naturally throughout thousands of centuries, and mathematicians have made them a subject of study for hundreds of years.

The sunflower has two sets of equiangular spirals superimposed and intertwined, one turning clockwise and the other turning counterclockwise. There are 21 clockwise and 34 counterclockwise spirals. Both numbers are part of the Fibonacci summation series. The order is closely related to the rule of alternation, which Elliott used in his wave principles to explain human behavior (see Figure 3.2).

Geometry of the Golden Rectangle and the Golden Section

The famous Greek mathematician Euclid of Megara (450-370 b.c.) was the first scientist to write about the golden section and to focus the analysis of a straight line.

The more complex structure of the geometry of a golden rectangle is shown in Figure 3.3. The ratio of the long side of the rectangle divided by the short side of the rectangle has the proportion of the Fibonacci ratio 1.618.

Parthenon Temple in Athens

The proportions of the Parthenon temple in Athens bear witness to the influence of the golden rectangle as well as the golden section on Greek architecture.

The proportions of the Parthenon temple fit exactly into a golden rectangle; its total width is exactly 1.618 times its height (see Figure 3.4).

Other geometric curves that are important to humankind are plentiful in nature. The most significant to civilization include the horizon of the ocean, the meteor track, the parabola of a waterfall, the arc that the sun travels in the sky, the crescent moon, and the flight of a bird.

Many of these natural curves can be geometrically modeled using ellipses. The latter finding leads into a brief description of trading tools that use the Fibonacci ratio. A basic knowledge of the construction and functions of these tools is necessary to understand the trading strategies that are introduced later in this book.

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