Here’s an impressive animated movie inspired by numbers, geometry and nature.
In addition to what’s shown in the video above, the Fibonacci sequence and the golden ratio can be found in the branching of trees, the spiral arrangements of pine cones, etc. There are many unsubstantiated claims as well – like the golden ratio is found in the design of the human body – that propagates the “divinity” or universality of the golden ratio. Contrary to these claims, the golden ratio and Fibonacci sequence are probably not the magical formulas that unravel the complicated design of the universe – but they are quite fascinating phenomena that have kept scientists, mathematicians, architects and number-enthusiasts intrigued since the Italian mathematician Leonardo Pisano first introduced the sequence in 1202 . He used a simple thought experiment to explain the sequence: assuming that a female rabbit always gives birth to a pair and each pair consists of one male and one female, how many pairs of rabbits will be produced in one year? The answer leads to the Fibonacci sequence. [My previous post Nature and Numbers has a similar example of the bee ancestry.]
By the way, I came across this interesting paper (PDF) by P Singh, published in the International Journal of Mathematical Education, that claims that the so-called Fibonacci sequence was first noticed and documented by an Indian scholar Acharya Hemachandra. The discovery was originated in the metrical science of Sanskrit and Prakrit poetry. A brief description below:
The basic units in Sanskrit prosody are a letter having a single matra called laghu (light) and that having two matra called guru (heavy). The former is denoted by S and the latter by |.
Now, for one time unit you have only one short syllable available (S), so the total number of arrangements you can have is one.
For two time units, you can have two arrangement: SS and |.
For three units there are three ways to arrange the syllables: SSS, S|, and |S.
For four time units? There are five ways (not four!): SSSS, SS|, S|S, |SS and ||.
Similarly, for n time units you have F(n-1) + F(n-2) arrangements. And that’s the Fibonacci sequence!
This predates Leonardo Pisano’s discovery by at least 50 years!
While we’re on the topic of the Fibonacci sequence, here’s my personal favorite representation of the golden spiral:
P. S. Curious about the theoretical basis for the calculations in the animations? The maker of the video has detailed explanations here.