Symmetry pp 109-130

# Fibonacci numbers in nature

Phyllotaxis in the living world. The harmony of the built environment

## Abstract

The example of the reproduction of rabbits (or any other animal that is quick to breed) iswell-known from most people’s secondary school studies. If we begin with a pregnant rabbit, it produces another rabbit, and we have two rabbits. In the first generation, the little rabbit will not breed, so at the next stage it will only be the old rabbit that gives birth again, giving us three rabbits. In the following generation, this little rabbit will not yet breed, but we have two rabbits from the previous generation that will, which adds two rabbits to the existing three rabbits, giving us a total of five rabbits. At the next stage, the two new rabbits will not yet breed, but the older three will, which together with the existing five rabbits gives us eight rabbits in all. In the following generation, five rabbits will produce new rabbits, and there will be eight plus five, that is thirteen rabbits, of eight will breed, giving us thirteen plus eight, that is twenty-one rabbits, and so on, until this system is not upset by some external factor. The number of new-born rabbits will always equal the cumulative number of rabbits from two generations before. As we began with the second member (a2 = 1) of the sequence of numbers, there is no need for us to subtract the 1 (a1 = 1) on the right hand side of the equation $$\sum\limits_{i = 1}^n {a_i = a_{n + 2} - 1}$$.

## Keywords

Golden Section Fibonacci Number Fibonacci Sequence Lattice Order Logarithmic Spiral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.