In mathematics, the Fibonacci numbers are the numbers in the 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.
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci.

Fibonacci numbers appear unexpectedly often in mathematics, and also in biological settings.

... the Golden Ratio ...

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence
(or any Fibonacci-like sequence), as originally shown by Kepler:
Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence,
the quotient approximates to 1.6180327868852.
Two quantities are said to be in golden ratio if (a+b)/a = a/b where a>b>0.
Its value is (1 + root 5)/2 or 1.6180339887…
Golden ratio can be found in patterns in nature like the spiral arrangement of leaves.
The Fibonacci sequence and the golden ratio are intimately interconnected.
The ratio of consecutive Fibonacci numbers converges and approaches the golden ratio and the closed-form expression for the Fibonacci sequence involves the golden ratio.

The golden ratio is an irrational number, partly because it can be defined in terms of itself.

... Bees and Rabbits ...

The bee ancestry code.
Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:
- If an egg is laid by an unmated female, it hatches a male or drone bee.
- If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee always has one parent, and a female bee has two.
If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents,
5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence.
The number of ancestors at each level, Fn, is the number of female ancestors, plus the number of male ancestors.
Leonardo also considered an unrealistic hypothetical situation where there is a pair of rabbits put in the field.
They mate at the end of one month and by the end of the second month the female produces another pair.
The rabbits never die, mate exactly after a month and the females always produce a pair (one male, one female).
The puzzle that Fibonacci posed was: how many pairs will there be in one year?
If one calculates then one will find that the number of pairs at the end of nth month would be Fn or the nth Fibonacci number.
Thus the number of rabbit pairs after 12 months would be F12 or 144.