Nautilus Shell and Fibonacci Series

• A nautilus is a cephalopod mollusk with a spiral shell and numerous short tentacles around its mouth.
• Although more common in plants, some animals, like the nautilus, showcase Fibonacci numbers. A nautilus shell is grown in a Fibonacci spiral. The spiral occurs as the shell grows outwards and tries to maintain its proportional shape.
• Unlike humans and other animals, whose bodies change proportion as they age, the nautilus’s growth pattern allows it to maintain its shape throughout its entire life. Imagine never outgrowing your clothes or shoes. You could still be rocking those overalls your mum put you in when you were four years old.
• Not every nautilus shell makes a Fibonacci spiral, though they all adhere to some type of logarithmic spiral. Nautilus aren’t consciously aware of the way their shells grow; they are simply benefiting from an advanced evolutionary design.

Fibonacci Sequence

The Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones

${\displaystyle 1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }$

Often, especially in modern usage, the sequence is extended by one more initial term:

${\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }$

The sequence F_n of Fibonacci numbers is defined by the recurrence relation byy definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

${\displaystyle F_{n}=F_{n-1}+F_{n-2},}$

with seed values

${\displaystyle F_{1}=1,\;F_{2}=1}$

or

${\displaystyle F_{0}=0,\;F_{1}=1.}$

Source: Wikipedia, Planet Dolan