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Fibonacci numbers in nature

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Fibonacci numbers

in nature

« Philosophy is written in this huge book

that I call universe which has always been opened

in front of us but we can’t understand it

if we first don’t know its language and its characters.

This book is written in the Mathematics language

and the characters are triangles, circles and geometric figures. Without them it is impossible to understand philosophy;

without them it would be like getting lost in a maze».

Galileo Galilei, Il Saggiatore

Introduction

According to the growth of flowers, the number of most of their petals is a Fibonacci number.

For example lilies have got three petals, buttercups five, chicories 21, daisies could either have 34 or 55; the head of sunflowers is formed by two spirals: one in a direction, the other in the opposite one.

The number of spirals changes between 21 and 34, 34 and 55, 55 and 89, or 89 and 144 seeds.

The same thing happens for pine cones, shells and pineapples

Who is Fibonacci?

Leonardo of Pisa, also known as Leonardo Fibonacci, was considered the most talented mathematician of the Middle Ages.

What are Fibonacci numbers?

Here’s the succession he used for a test on rabbits’ breeding:

f(1) = 1

f(2)= 1

f(n + 2) = f(n) + f(n + 1)

The numbers he found are called

Fibonacci numbers and they are:

1 1 2 3 5 8 13 21

34 55 89 144 233

377 610 ....................

They have so many applications that

there’s also a Maths magazine, called Fibonacci Quarterly.

It is the botanical name ofthe disposition ofleaves. The inflorescenceson the head of thesunflowers are formed by 55clockwise spiralsand 34 anticlockwise ones(that is f(10) e f(9));pineapples have got 21 spirals of petals in a direction and 34 in the opposite one; the same is for cauliflowers, pine cones and some kinds of cactus. Without knowing how to count, a flower can make the number of its petals a Fibonacci number.

Filotassi

Sunflowers

Sneezewort

The association of Fibonacci numbers and plants is not restricted to numbers of petals. Here we have a schematic diagram of a simple plant, the sneezewort. New shoots commonly grow out at an axil, a point where a leaf springs from the main stem of a plant.

If we draw horizontal lines through the axils, we can detect obvious stages of development in the plant. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, then produce new branch shoots at the beginning of each subsequent stage. The same law applies to all branches.

Since this pattern of development mirrors the growth of the rabbits in Fibonacci's classic problem, it is not surprising then that the number of branches at any stage of development is a Fibonacci number.

Furthermore, the number of leaves in any stage will also be a Fibonacci number.

Let's play with

these numbers!

The fifteen puzzle

It is a famous game, created by Samuel Loyd in 1878. It consists of a square table, usually in plastic, divided into four rows and four columns, with 15 square tiles, numbered from 1 in random order.

The tiles can scroll horizontally or vertically, but their shift is obviously limited by the existence of a single blank space. The goal of the game is to reorder the tiles after “mixing” them.

Now let's try to solve

the attached game with

the first 15 numbers of

Fibonacci sequence!