The Fibonacci sequence

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One of the most famous sequences in mathematics is the Fibonacci sequence, where  each number is the sum of the two previous numbers in the sequence. Therefore, the sequence begins with 0, and then continues on like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…

Figure 1: Leonardo Pisano

Leonardo Pisano, called Fibonacci was born in Pisa around 1170. His father (Guglielmo dei Bonacci) was a Pisan merchant and Fibonacci means “son of Bonacci”.

Around 1192 Leonardo’s father took him to Pisa. It is believed that this is where his interest in mathematics started for the young Leonardo, as he was taught by a Arabic teacher, who guided him in learning calculation techniques, especially those concerning Indo-Arabic numbers, which had not yet been introduced in Europe.

Leonardo spent close to 25 years dedicating himself to writing mathematical manuscripts. These include the Liber Abaci (1202), Practica Geometriae (1220), Flos (1225) and Liber Quadratorum (1225).

First mention of Fibonacci numbers:

Leonardo mentioned the sequence of numbers (Fibonnaci seuqence) with a problem involving rabbits to the Western world through the Liber Abaci.

The rabbit problem: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and the outcome is another male and female. After a year, how many rabbits would you have?

The answer, it turns out, is 144

The formula used to get to that answer is what’s now known as the Fibonacci sequence.

Change the angle between two consecutive points using the slider.

The number of red spirals and the number of blue spirals form two consecutive numbers in the Fibonacci sequence.

NOTICE WHAT HAPPENS:

@90°

@138° (count the spirals)

@224° (count the spirals)

@270°

@360°

Fibonacci sequence leading to the Golden Ratio: Phi (1.618)

The Fibonacci sequence is:        

  • 0 (1st term),
  • 1 (2nd term),
  • 1 (3rd term),
  • 2 (4th term),
  • 3 (5th term),
  • 5 (6th term),
  • 8 (7th term),
  • 13 (8th term),
  • 21 (9th term),
  • 34 (10th term),
  • 55 (11th term),
  • 89 (12th term)…

As the numbers increase, the consecutive numbers when divided by each other forms a value called the “golden ratio” (1.618) aka phi (. Phi value was crucial in Renaissance painting. Since painters used the ratio 1:1.618 in their work has it was believed it to be aesthetically pleasing. The same ratio can be found in nature including the anatomy of the human body. The value 1.618 shows that our universe was intelligently designed, not cosmic coincidence.

Fibonacci FACTS:
  • While Fibonacci (Leonardo Pisano) himself did not discover the Fibonacci sequence (they were named after him). The sequence of numbers originate back to ancient India, and ancient Sanskrit texts that used the Hindu-Arabic numeral system (metrical system) which predate Leonardo of Pisa by centuries.
  • It is possible to find the Fibonacci sequence in nature given by the number of petals of flowers, seashells, etc.
  • Number of flower petals according to the sequence- three (lilies and irises), five (parnassia, rose hips) or eight (cosmea), 13 (some daisies), 21 (chicory), etc.
  • Not everything is governed by the golden ratio.

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The Fibonacci sequence

One of the most famous sequences in mathematics is the Fibonacci sequence, where  each number is the sum of the two previous numbers in the