1. About Fibonacci
1. Time period (Hog 1)
2. Fibonacci's Life (Hog 1), (Vaj 9)
3. The Liber Abaci.
2. The Fibonacci Sequence
1. When discovered (Vaj 7)
2. Where it appears (Vaj 9)
3. The Golden Ratio
1. When discovered
2. Where it appears
4. Thesis: The Fibonacci Sequence and The Golden Ratio are very much a part of one another, and these two phenomena have many applications in math and nature.
3. The Simple explanations
1. How we arrive at the Fibonacci Sequence
1. Explained in English
2. The Simplest Properties of the Fibonacci Numbers
1. u1+u2+...+un=un+2-1 (Vor 6)
2. The sum of Fodd: u1+u3+u5+...+u2n-1=u2n (Vor 7)
3. Sum of Feven: u2+u4+...+u2n=u2n+1-1 (Vor 7)
4. u12+u22+...+un2=unun+1(Vor 8)
3. How we arrive at the Golden Ratio
1. Explanation of it.
1. Explanation of Golden Rectangle
2. Formula to get it. - Solve for X with a formula and Diagram
3. How to get from Fibonacci series to Golden Ratio.
4. Properties of the Golden Ratio
1. a 2=a+1 (Hog 9)
2. 1/a =a -1
3. b 2=b +1
4. Fibonacci Numbers
1. In Math
2. In Nature
5. The Golden Ratio
1. In Math
2. In Nature
One would be hard pressed to name any significant mathematicians from the middle ages, the thirteenth century. Indeed, they were few and far between. The solitary light during this dark age of mathematical and scientific void was Leonardo Fibonacci, also known as Leonardo of Pisa (Hog 1).
During his lifetime, c 1170-1250, Fibonacci produced one great work, the Liber Abaci (Vaj 9).
The title of this work, which was written in 1202, literally means "a book about the abacus" (Vor 1).
It is an excessively comprehensive work, encompassing nearly all the arithmetic and algebraic knowledge of the time. This book played an important role in the evolution of Western European mathematics, but in particular, it familiarized the Europeans with the Hindu, or Arabic numerals (1).
Although his book spoke of many different topics, Fibonacci is most known for the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., to which his name has been given. Even today, this sequence continues to be researched (Hog 1). This report should give a basic understanding of some aspects of the Fibonacci numbers.
The Fibonacci Sequence was discovered less than 800 years ago. The "deceptively simple definition implies a large variety of relationships," and appears as the solution to many intricate mathematical problems (Vaj 7).
Known of since antiquity, the Golden Ratio far predates the Fibonacci numbers. The Golden Ratio is special. A rectangle with sides of this ratio will retain this ratio when a square of its width is removed from it. I will explain more on this later. This ratio appears abundantly in nature and math, as it is a very nature-oriented number.
The Fibonacci Sequence and The Golden Ratio are very much a part of one another, and these to phenomena have many applications in math and nature, which I will discuss in this paper.
The Fibonacci Numbers are always being studied because of their complex nature. They seem simple, but are really far from it. We arrive at the Fibonacci Sequence through a method of adding the previous two terms. Starting with 1, and adding the two previous terms to get each successive term, we end up with a sequence that looks like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Therefore, if Fn is the nth term in the Fibonacci Sequence, then Fn+2=Fn+1+Fn or Fn=Fn-1+Fn-2 (Vaj 9).
Following this formula will result in the Fibonacci sequence. Fibonacci explained his sequence with a very famous problem concerning rabbits breeding. The following is a translation of pages 123-124 of the manuscript from 1228:
Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, to find out how many pairs of rabbits will be born there in the course of one year, it being assumed that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their own birth.
As the first pair produces isue in the first month, in this month there will be 2 pairs. Of these, one pair, namely the first one, gives birth in the following month so that in the second month there will be 3 pairs. Of these, 2 pairs will produce issue in the following month, so that in the third month 2 more pairs of rabbits will be born, and the number of pairs of rabbits in that month will reach 6; of which 3 pairs will produce issue in the fourth month, so that the number of pairs of rabbits will then reach 8... (Vor 2).
From this, we find the following:
- First (Month) 1 (Pairs)
- Second 2
- Third 3
- Fourth 5
- Fifth 8
- Sixth 13
- Seventh 21
- Eighth 34
- Ninth 55
- Tenth 89
- Eleventh 144
- Twelfth 377
These numbers (obviously) comprise the Fibonacci Sequence (3).
Although there are hundreds of formulas to be derived from this sequence, only few simple ones are within this paper's scope. First, we will calculate the sum of the first n Fibonacci numbers. We will show that
In English, adding all terms until Fn will equal Fn+2, or the term two ahead, minus 1. Next, we have sum formulas for Feven and Fodd. They are:
The sum of Fodd: F1+F3+F5+...+F2n-1=F2n.
Sum of Feven: F2+F4+...+F2n=F2n+1-1
Adding terms one by one will prove the equation. And finally, we square each term to come to the following equation:
Obviously, deriving these is beyond the extent of this paper, but replacing the variables with valid terms will prove it works.
The Golden Ratio, or Golden Section, is closely linked to the Fibonacci Numbers. Suppose we have a line segment AB. We must find point C on it, such that the ratio of the total length to the larger segment is the same as that of the larger segment to the smaller segment, as in the following figure:
- AB AC A C B
- AC CB 1
where none of the line segments measure 0 in length (Hog 9). Let us say that AC=1 for reasons of simplicity in finding the ratio. So, keeping in mind that AC=1 in this case,
x = AB/AC = (AC+CB)/AC = 1+CB/AC = 1+1/(AC/CB) = 1+1/(AB/AC) = 1+1/x.
So... x = 1+1/x
By bringing everything to one side and setting the equation equal to 0, we get
x2 - x - 1 = 0.
Using the quadratic formula, we find that the roots of this equation are, as you can verify,
a = (1+Ö 5)/2 and b = (1-Ö 5)/2
By computation, you can find that a » 1.618 and b » -.618. Thus we take the positive root a as the desired ratio. Now that we know that the sections AC and AB must be in the ratio 1:1.618, we may move on to the Golden Rectangle.
This shape contains sides of the same ratio, creating a figure somewhat like this:
- A B C
- E D
- G F
In this figure, AC/CF=CF/FG=FG/GE. In short, when a square is removed from the end of a Golden Rectangle, the remaining rectangle has the same proportions as the original, that is, 1:1.618 (Hog 12).
The Golden Rectangle's proportions appear frequently throughout classical Greek art and architecture (12). The German psychologists Gustav Theodor Fechner (1801-1887) and Wilhelm Max Wundt (1832-1920) have shown in a series or psychological experiments that most people do unconsciously favor "golden dimensions" when selecting pictures, cards, mirrors, wrapped parcels, and other rectangular objects. Neither artists nor psychologists fully know why, but the Golden Rectangle holds great aesthetic appeal (12).
There is also another way to find the Golden Mean: It is the ratio of any number in the Fibonacci Sequence to the previous number. The following equation will return the Golden Section number, 1.618, or a :
a » Fn+1/Fn (Hog 28).
Precisely speaking, the limit of this function is the Golden Mean; as n approaches infinity, the value of the function approaches the real value of a . The function's values do rise and fall, above and below the real a , but also get closer and closer to a .
Math and Nature exhibit the Fibonacci Numbers and the Golden Ratio very often. First I will discuss the appearances of Fibonacci Numbers. In nature, such appearances are abundant concerning the Fibonacci Numbers. On pinecones and pineapples, spirals spin out of the center when looked at from the bottom, and the number of spirals spinning one way is one Fibonacci number, and the number spinning the other way is a different Fibonacci number (Hog 81-82).
Also, many plants have leaves that extrude from the stem at various, random-seeming points, so that each leaf in the pattern has no other leaf directly above it until the pattern repeats. This is useful to the plant in that the leaves get as much sunlight and water as possible. A diagram follows:
, we will arrange Pascal's Triangle as follows:
- 1 1
- 1 2 1
- 1 3 3 1
Works Cited - Bibliography
- Hoggatt, Verner E., Jr. Fibonacci and Lucas Numbers. The Fibonacci Association, University of Santa Clara. California, 1969.
- Knott, Ron. Fibonacci Numbers and the Golden Section. Internet: http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.ht ml.
- Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. John Wiley & Sons Limited. New York, 1989.
- Vorob'ev, N. N. Fibonacci Numbers. Blaisdell Publishing Company. New York, 1961.