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Fibonacci Numbers

Fibonacci numbers in a sea shell
Field: Number Theory
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Fibonacci numbers in a sea shell

The spiral curve of the Nautilus sea shell follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers.

Basic Description

The Fibonacci sequence is the sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots,$ where the first two numbers are 1s and every later number is the sum of the two previous numbers. So, given two $1$'s as the first two terms, the next terms of the sequence follows as : $1+1=2, 1+2=3, 2+3=5, 3+5=8, \dots$

The Fibonacci numbers can be discovered in nature, such as the spiral of the Nautilus sea shell, the petals of the flowers, the seed head of a sunflower, and many other parts. For more information about Fibonacci patterns in nature, see Fibonacci Numbers in Nature.

Origin

Image 2

The Fibonacci sequence was studied by Leonardo of Pisa, or Fibonacci (1770-1240). In his work Liber Abacci, he introduced a problem involving the growth of the rabbit population. The assumptions were

• There is one pair of baby rabbits placed in an enclosed place on the first day of January.
• This pair will grow for two months before reproducing. Then, beginning on March 1, it will produce a new pair of rabbits on the first day of every month.
• Each new pair will similarly mature for two months and then start producing a new pair of rabbits every month, beginning on the day they turn two months old
• The rabbits never die

The problem was to find out how many pairs of rabbits there will be after one year.

The population of rabbit pairs in each month, as it turns out, matches the Fibonacci sequence. Look at Image 2. As you can see in the image, the population by month begins: 1, 1, 2, 3, 5, 8, ..., which is the same as the beginning of the Fibonacci sequence. The population continues to match the Fibonacci sequence no matter how many months out you go.

To see why this is the case, notice that to find the population of rabbit pairs in any month, we can simply add the following together:

1. the number of rabbit pairs that were already alive before the month began. This is the population of rabbit pairs from the previous month.
2. rabbit pairs that were new-born that month. This is the population of rabbit pairs from two months ago, because those are the ones that are old enough to reproduce.

This idea can also been seen in Image 2. The population in June includes all 5 pairs of rabbits that were already alive in May. The population on June 1st also includes 3 pairs of new born rabbits, because the 3 pairs that were born by April (the blue pair, the white pair, and the red pair) are old enough to reproduce.

Therefore, the population of rabbit pairs in any month is equal to the population from the previous month to the population from two months earlier. This is exactly the rule that defines the Fibonacci sequence.

Moreover, the population by month has the same starting values as the Fibonacci sequence: one pair in month 1, and one pair in month 2. Overall, then, the population has the same starting values as the Fibonacci sequence and the same rules for getting the population in any month from the population in the preceding months. The population by month, therefore, must match the Fibonacci sequence.

An interesting fact is that this problem of rabbit population was not intended to explain the Fibonacci numbers. This problem was originally intended to introduce the Hindu-Arabic numerals to Western Europe, where people were still using Roman numerals, and to help people practice addition. It was coincidence that the number of rabbits followed a certain pattern which people later named as the Fibonacci sequence.

Fibonacci Numbers in Nature

Image 3

Leaf Arrangement

Fibonacci numbers appear in the arrangement of leaves in certain plants. Take a plant, locate the lowest leaf and number that leaf as 0. Number the leaves by order of creation starting from 0, as shown in Image 3. Then, count the number of leaves you encounter until you reach the next leaf that is directly above and pointing in the same direction as the lowest leaf, which is the leaf with number 8 in this image. The number of leaves you pass, in this case, 8, will be a Fibonacci number.

Moreover, the number of rotations you make around the stem until you reach that leaf will also be a Fibonacci number. You make rotations up the stem by following ascending order of the leaf's number. In Image 3, for example, if you follow the red arrows, the number of rotations you make until you reach leaf number 8 will be 5, which is a Fibonacci number.

Spirals

Fibonacci numbers can be seen in nature through spiral forms that can be constructed by Fibonacci rectangles as shown in Image 4. Fibonacci rectangles are rectangles in which the ratio of the length to the width is the proportion of two consecutive Fibonacci numbers.

One way we can build Fibonacci rectangles is by first drawing two squares with length 1 next to each other. Then, we draw a new square with length 2 that is touching the sides of the original two squares. We draw another square with length 3 that is touching one unit square and the latest square with length 2. With each new square, a new Fibonacci rectangle is created. Its length is equal to the sum of the lengths of the latest two squares, and its width is equal to the length of the most recent square.

Image 4

After building Fibonacci rectangles, we can draw a spiral in the squares, each square containing a quarter of a circle. Such spiral is called the Fibonacci spiral, and it can be seen in sea shells, snails, the spirals of the galaxy, and other parts of nature, as shown in Image 5 and Image 6.

Image 5

Image 6

A More Mathematical Explanation

Symbolic Definition of Fibonacci Sequence

The Fibonacci sequence is the sequence $F_1, F_2, F_3, \ldots, F_n, \ldots$ where

$F_n = F_{n-1} + F_{n-2} \quad \hbox{ for } n>2$,

and

$F_1 = 1,\ F_2 = 1$.

The Fibonacci sequence is recursively definedA recursively defined sequence is one in which each term is defined by preceding terms in the sequence. For instance, $x_n=(x_{n-1})^2-3x_{n-2}$ is recursively defined. because each term is defined in terms of its two immediately preceding terms.

Sum of first $n$ Fibonacci numbers

There is a nice shortcut for computing the sum of the first $n$ Fibonacci numbers: subtracting one from the ${(n+2)}^{\rm th}$ Fibonacci number gives the sum of the first $n$ Fibonacci numbers.

For example, $F_7=13$, and subtracting one from this gives 12, which is the sum of the first 5 Fibonacci numbers.

In equation form, this property reads:

Eq. (1)        $F_1+F_2+\dots+F_n=F_{n+2}-1$

For example:

$F_1+F_2+F_3+F_4+F_5= 1 + 1 + 2 + 3 +5=12=F_7-1$

The example is demonstrated below. The total length of red bars that each correspond to $F_1, F_2, F_3, F_4, F_5$ is one unit less than the length of $F_7$.

Image 7

Greatest Common Divisor Property

There is also a nice shortcut for computing the greatest common divisor of two Fibonacci numbers. First, find the greatest common divisor of the their indicesThe index of a Fibonacci number is its position in the sequence. For example, the seventh Fibonacci has an index of 7 and the eighth Fibonacci has an index of 8.. Take that result, and find that numbered term in the Fibonacci sequence. The result is the greatest common divisor of the two original Fibonacci numbers.

For example, to find the greatest common divisor of 34 (which is $F_9$) and 144 (which is $F_{12}$), first note the greatest common divisor of 9 and 12 is 3. Then, note that $F_3$ (the third Fibonacci number) is 2. Therefore, the greatest common divisor of 34 and 144 is 2.

In equation form, this property reads:

$\gcd(F_n,F_m) = F_{\gcd(n,m)}$.

For example:

$\gcd(F_9,F_{12})=\gcd(34,144)=2=F_3=F_{\gcd(9,12)}$.

Binet's Formula for Fibonacci Numbers

Binet's Formula gives a formula for the $n^{\rm th}$ Fibonacci number as:

Here, $\varphi=\frac{1 + \sqrt{5}}{2}$ and $\bar{\varphi}=\frac{1-\sqrt{5}}{2}$.

These two numbers are the solutions to the equation

Eq. (2)        $x^2=x+1$.

A proof is given below, although it does not provide any insight into how Binet (or anyone else) would come up with this formula. It only verifies that it is correct.

Define $B_n$ by the equation:

Eq. (3)        $B_n=\frac{{\varphi}^n-{\bar{\varphi}}^n}{\sqrt5}$.

We will know Binet's formula is correct if we can show that all the values of $B_n$ are the same as the Fibonacci numbers. In order to do this, we will first show that $B_n$ follows the same recursive rule as the Fibonacci numbers, by verifying the formula:

Eq. (4)        $B_n=B_{n-1}+B_{n-2}\quad\hbox{ for } n>2$

To verify this formula, note that according to the definition of $B_n$:

Eq. (5a)        $B_{n-1}+B_{n-2} = \frac{{\varphi}^{n-1}-{\bar{\varphi}}^{n-1}}{\sqrt5}+ \frac{{\varphi}^{n-2}-{\bar{\varphi}}^{n-2}}{\sqrt5}$

Eq. (5b)        $= \frac{({\varphi}^{n-1}+{\varphi}^{n-2})-({\bar{\varphi}}^{n-1}+{\bar{\varphi}}^{n-2})}{\sqrt5}$

Eq. (5c)        $=\frac{({\varphi}+1){\varphi}^{n-2}-(\bar{\varphi}+1){\bar{\varphi}}^{n-2}}{\sqrt5}$.

Because $\varphi$ and $\bar{\varphi}$ are the two roots of Eq. (2), the above equation becomes:

Eq. (5d)        $B_{n-1}+B_{n-2}=\frac{{{\varphi}^2}{\varphi}^{n-2}-{{\bar{\varphi}}^2}{\bar{\varphi}}^{n-2}}{\sqrt5}$

Eq. (5e)        $=\frac{{\varphi}^n-{\bar{\varphi}}^n}{\sqrt5}$

Eq. (5f)        $=B_n$, as desired.

Now we know that $B_n$ follows the recursive rule of the Fibonacci numbers. If we plug $n=1$ and $n=2$ into Eq. (3), we also see that $B_1 = 1$, and $B_2 = 1$.

Thus, $B_n$ has the same recursive formula as the Fibonacci numbers and it has the same starting two values, so this formula really is a correct formula for the Fibonacci numbers.

Teaching Materials

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References

Maurer, Stephen B & Ralston, Anthony. (2004) Discrete Algorithmic Mathematics. Massachusetts : A K Peters.

Posamentier, Alfred S & Lehmann Ingmar. (2007) The Fabulous Fibonacci Numbers. New York : Prometheus Books.

Vorb'ev, N. N. (1961) Fibonacci Numbers. New York : Blaisdell Publishing Company.

Hoggatt, Verner E., Jr. (1969) Fibonacci and Lucas Numbers. Boston : Houghton Mifflin Company.

Knott, Ron. (n.d.). The Fibonacci Numbers and Golden Section in Nature. Retrieved from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

Wikipedia (Golden Ratio). (n.d.). Golden Ratio. Retrieved from http://en.wikipedia.org/wiki/Golden_ratio.

Fibonacci Numbers in Nature & the Golden Ratio. (n.d.). In World-Mysteries.com. Retrieved from http://www.world-mysteries.com/sci_17.htm

http://www.world-mysteries.com/sci_17.htm

Wikipedia (Mandelbrot Set). (n.d.). Mandelbrot Set. Retrieved from http://en.wikipedia.org/wiki/Mandelbrot_set.

Devaney, Robert L. (2006) Unveiling the Mandelbrot Set. Retrieved from http://plus.maths.org/issue40/features/devaney/.

Weisstein, Eric W. (n.d.). Mandelbrot Set. In MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/MandelbrotSet.html.

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