Fibonacci Sequence & the Golden Ratio

Fibonacci Series

Write down a sequence of numbers which starts with 0 & 1. Every subsequent number in the sequence is the sum of its two previous numbers.

Hence after 0, 1 it is 1 (0 + 1). So 0, 1, 1

The next number is 2 since 1 + 1 is 2. So, we get 0, 1, 1, 2

The next number is 3 since 1 + 2 is 3. So, 0, 1, 1, 2, 3

The next number is 5 since 2 + 3 is 5.

So the first 12 numbers in this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ....

Almost any primary school student can write down this sequence. But they would be surprised that this is arguably one of the most famous sequences in math called Fibonacci sequence.

Fibonacci in Nature

In many flowers, the seeds are arranged so that numbers from the Fibonacci Sequence appear. It is something to do with the efficient packing of seeds in a flower!

It also seems to play an important role in the formation of galaxies in the Universe!

These seem to be due to the relation between the Fibonacci Series and the Golden Ratio.

Golden Ratio

The Golden Ratio was discovered and written about by Euclid and other Greek mathematicians. The relation of the Golden Ratio to the Fibonacci Series was discovered much later.

The ratio of any number in the Fibonacci sequence, to its previous number tends towards a number which is called the Golden Ratio and has a symbol φ. It is pronounced as "Phi". The further we go in the sequence (or as the numbers become bigger & bigger) the ratio becomes closer and closer to the value of φ. For example

5/3 = 1.6666666......

233/144 = 1.61805555

377/233 = 1.618025751.....

The actual value of φ is about 1.6180339887......

Greek aestheticians said that a rectangle with sides which are in this ratio is aesthetically the most pleasing to the eye. The rectangular façade of the Parthenon in Athens seems to have this ratio. Such rectangles are also called golden rectangles.

The concept of certain rectangles being pleasing to the eye has been questioned in recent times.

A golden rectangle also has another interesting property. If we cut off the square on the smaller side, the remaining rectangle would also be a golden rectangle.

φ also has a mathematical property which can be described as φ = 1 + 1/φ.

φ is the most irrational number (degrees of irrationality).

We can use irrational numbers in computations only by approximating to some rational numbers. φ is the most difficult irrational to get a rational number approximation. Approximation to a rational number needs the consideration of a largest number of the decimal digits.

Fibonacci Numbers in Indian Mathematics

Hemachandra, a Jain sage and mathematician described the what we now call the Fibonacci sequence in around 1150, about fifty years before Fibonacci (1202) announced it.

Hemachandra was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length n − 1, or a long syllable to one of n − 2. This recursion relation F(n) = F(n − 1) + F(n − 2) is what defines the Fibonacci sequence.

He also studied the rhythms of Sanskrit poetry. Syllables in Sanskrit are either long or short. Long syllables have twice the length of short syllables. The question he asked is How many rhythm patterns with a given total length can be formed from short and long syllables?

For example, how many patterns have the length of five short syllables (i.e. five "beats")? There are eight: SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS As rhythm patterns, these are xxxxx, xxxx., xxx.x, xx.xx, x.xxx, xx.x., x.xx., x.x.x

Mathematically both the above enquires are same

An Interesting Fact from History

The relation between the Fibonacci Series & the Golden Ratio was discovered in Europe only in the 16th century. Fibonacci himself was not aware of it.

It was stated in a book written in 1560 by Simon Jacob. Kepler rediscovered this fact independently in 1608.

The Indian mathematicians who also discovered the Fibonacci Series did not mention anything about the Golden Ratio or their occurrence in nature.

Fibonacci Trivia

144 is the biggest square number which occurs in the Fibonacci Series.

Miles to Kilometers

Interestingly the adjacent numbers in the Fibonacci Series provide a rough conversion from miles to kms.

3 miles = 4.83 (≈5) kms, 5 miles = 8.04 (≈8) kms, 8m = 12.87 (≈13) km etc.