فیبوناچی در طبیعت

In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,{\displaystyle F_{0}=0,\quad F_{1}=1,}

{\displaystyle F_{0}=0,\quad F_{1}=1,}

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

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

for n > 1.

One has F2 = 1. In some books, and particularly in old ones, F0, the “0” is omitted, and the Fibonacci sequence starts with F1 = F2 = 1. The beginning of the sequence is thus:{\displaystyle (0,)\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }

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

The Galton board

A Galton board is a vertical board with n rows of pegs onto which a ball is dropped.   Each time a ball hits a peg, it has a probability p of bouncing to the left and a probability of 1-p of bouncing to the right. The simulation’s histogram shows the distribution of x-coordinates as the balls leave the board and are collected into bins.