MATHS IN NATURE

AND ARTS

FIBONACCI’S SEQUENCE AND GOLDEN RATIO

FIBONACCI’S SEQUENCE

• Fibonacci’s sequence is a sequence known in the Western World thanks to Leonardo of Pisa, a XIII century Italian mathematician, also called Fibonacci. It is an infinite sequence of Natural Numbers where the first and the second terms are 1, and the rest of the terms are obtained by adding the previous two:

• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

• Examples in nature:– The number of spirals, of the 3 different

typologies, that could be found on almost any sunflower is 21, 34 and 55.

– The daisy has seeds in the form of 21 and 34 spirals.

– The pine cone has 5 and 8 spirals, or 8 and 13.

GOLDEN RATIO

• First, we divide a segment into two parts, so that the segment is to the bigger part as the bigger part is to the smaller one. Then, we call the ratio of this proportion the golden ratio or divine proportion:

...618.12

51

b

a

a

ba

• Then we can build a golden rectangle:

• Or we can build a spiral using golden rectangles:

• We can find this spiral in the Nautilus’s shell:

• We can find golden rectangles in Arts, too:– In Leonardo Da Vinci’s work:

– Or in the Parthenon:

• There are studies which say that we prefer golden rectangles, that’s why we can find it in our identity card, credit cards, packet of cigarrettes, advertisements, etcetera…

• There is a relationship between the golden ratio and the Fibonacci’s sequence:

• To finish: