Introducing the Fibonacci Sequence

"Introducing Fibonacci Numbers" by Mike DeHaan

The phrase, “the Fibonacci sequence” usually refers to a set of numbers that starts as {0, 1, 1, 2, 3, 5, 8, 13, 21…}. At least, “The Life and Numbers of Fibonacci” starts with “zero and one”. Many others would skip the zero and simply start with “one and one”, but that does not matter.

This sequence is created by following two rules:

Actually, one must perform rule #2 over and over again, until one runs out of time, patience, paper or ink in the pen.

So, constructing the Fibonacci sequence starts from 1=1+0, then 2=1+1, then 3=2+1, then 5=3+2, 8=5+3, and so on.

Why is the Fibonacci Sequence Important?

As with many mathematical concepts, the Fibonacci sequence started as the solution to a contest. Then people began to realize it crops up in many places.

Leonardo Fibonacci Won a Contest

"Leonardo Fibonacci" by zoonabar

Leonardo Fibonacci invented his sequence to win a competition sponsored by Emperor Frederick II in 1225. The contest question was: Start with a pair of rabbits. Every month, every pair of rabbits who are over a month old gives birth to a new pair of rabbits. After “n” months, how many pairs of rabbits are there?

(It seems there must have been some more rules. These rabbits never die; they never get too old to reproduce; they form stable monogamous “married pairs” for life; and all births are synchronized on the first day of each month).

This problem can be solved by “mathematical induction”. The first step of this induction begins at the beginning. Throughout the first month there is only the one initial pair. At the start of the second month, one new pair is born, so there are two pairs. At the start of the third month, only the first pair has another pair, so the total is three pairs. But to kick off the fourth month, the oldest children produces a pair of rabbits; so do their parents. Adding these two new pairs to the existing three pairs, we have five pairs. So the pattern has begun: (1), 1, 2, 3, 5…

For the second part of this mathematical induction, we can say there are x[n-1] pairs after (n-1] months, x[n] pairs after (n) months, and x[n+1] pairs after (n+1) months. That does not say anything about how to get from x[n-1] to x[n], however.

In month (n), how many pairs of rabbits are old enough to breed? All who were alive in the previous month, numbered (n-1). So in month (n), exactly x[n-1] rabbits will be born. This number, “x[n-1]“, will be added to the “x[n]” rabbits alive in month (n) to create the number x[n+1]. To put it mathematically, x[n+1] = x[n] + x[n-1].

So the sequence was important to Fibonacci because he won a contest with it. Later, it became important for other reasons.

One Natural Example of Fibonacci Numbers

“Natural Fibonacci” presents several examples of Fibonacci numbers found in nature.

Bee genealogy is one example. The hive-dwelling honeybee has many female worker bees, one fertilized queen who is also female, and a few male drones. A male develops from an unfertilized egg, so he has only one parent. Every female bee develops from a fertilized egg, so she has two parents: the queen and one male drone.

Let’s name one female bee, “Bea”, and count “1″ for her own generation. Her parents’ generation consists of a male and a female: the number in that generation is “2″.

Bea has three grandparents. Bea’s father had only a mother; but Bea’s mother had both a mom and pop. So the grandparent’s generation has 3 members.

In Bea’s great-grandparents’ generation, her father’s mother had two parents. Her maternal grandmother also had two parents; but her maternal grandfather had only a mother. That adds up to 5 great-grandparents.

Refer to the image below to see how this works over several generations.

"Bee Genealogy" by Mike DeHaan

The image shows 6 generations. “Bea”, in generation 1, has only herself in the “Fibonacci Count”. Generation #2 are her two parents: one female (‘F’) and one male (‘M’, at the far right). Some lines join Bea’s parents to her.

Bea’s father’s only parent is ‘f‘ in the right column of the “grandparent” row. There is no line to show this relationship. But this grandmother has two parents in the Great-grand row, shown with the heavy orange background and the parenting lines.

You can see that the number of ancestors in each generation is a Fibonacci number.

Fibonacci Numbers and the Golden Ratio

Architects and artists have found that an object with a specific proportion is pleasing to the eye. If A and B are two lengths and A > B, then if the ratios A/B equals (A+B)/A, the figure is in the “golden ratio“. Surprisingly, this value is exactly “one-half of the sum of 1 plus the square root of 5″.

Even more surprisingly, there is a relationship to the Fibonacci sequence. Divide x[n+1]/x[n]. In the sixth generation, when x[n+1]=13, the result is less than 1% off from the Golden Ratio. For x[10] = 89, the error drops under 1% of 1%. This was an incredibly “easy” way to calculate the Golden Ratio before mathematicians invented general methods for determining square roots.

More Natural Fibonacci Numbers

Fibonacci numbers are found in the growth patterns of many plants and animals, particularly where something grows in a spiral. Flower petals often come in Fibonacci numbers. The example below has “only” 5 petals.

"Floral Fibonacci Number" by Mike DeHaan

Measuring the length of each coil in a garden hose may show an “extended” Fibonacci series. Each loop has to travel farther because of the width of the previous coil.

Extending the Fibonacci Family

There are two easy ways to develop a new Fibonacci series:

1. Choose different starting numbers
2. Include more generations

It is more interesting to start with numbers that are not part of the plain vanilla Fibonacci sequence. Starting with “4, 7″ continues with “11, 18, 29,47, 76, 123″ and so on.

Surprisingly, any Fibonacci sequence starting with “zero, N” gives a sequence where each number is exactly N times larger than the corresponding “plain vanilla” Fibonacci number.

The proof is as follows. We already have the plain vanilla Fibonacci series x[n+1] = x[n] + x[n-1] when x[0] = zero and x[1] = 1. Let x[1] = N for any positive integer. Then x[2] = N+0 = N rather than just 1; x[3] = N+N rather than 2. So the new sequence starts as “0, N, N, 2N”. Divide by N, and we have the “plain vanilla” start of “0, 1, 1, 2″.

By the induction principle, we assume the new Y[n-1] = Nx[n-1], and that the new Y[n] = Nx[n]. (This was true for n=1, 2 and 3). It remains to see whether Y[n+1] = Nx[n+1].

Nx[n+1] = N*(x[n] + x[n-1]) = Nx[n] + Nx[n-1] for the “plain vanilla” Fibonacci sequence, by the distribution of multiplication over addition. But we assumed that the new Y[n-1] = Nx[n-1], and that the new Y[n] = Nx[n], so “Nx[n] + Nx[n-1]” = “Y[n] + Y[n-1]“.

Therefore Nx[n+1] = Y[n] + Y[n-1], which is exactly how to calculate Y[n+1]. So Y[n+1] = Nx[n+1].

The other way to create a different Fibonacci series is to include more numbers. A triple series would use T[n+1] = T[n] + T[n-1] + T[n-2], for example. Of course, the three starting numbers could take any desired values.

Spreadsheet Exercises for the Fibonacci Series

It is easy to build a spreadsheet like the one shown below. Any cell with a non-white background colour and a number has a formula. Only the “0″ and “1″ in column A, and the “1″ in column D, were entered manually.

"Fibonacci Series and Golden Ratio" by Mike DeHaan

After building this spreadsheet, it is easy to extend the “plain vanilla” Fibonacci sequence by copying more cells down column A. Or one might change the first two values to create a new Fibonacci series.

References:
Knott and Quinney, Plus, “The life and numbers of Fibonacci”, published Sept. 1, 1997, referenced May 22, 2011.
Scott Hotton, Mission College, “Natural Fibonacci”, copyright 1999, referenced May 23, 2011.