# Day # 13: Fibonacci Numbers – From 0 and 1 to Unbounded Imagination

##### Animated GIF showing successive tilings with squares whose side lengths are successive Fibonacci numbers  +  a graph of approximations to the Golden Ratio calculated by the sequence of ratios   1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ……. , 89/55, 144/89   given by pairs of consecutive Fibonacci numbers.

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that “as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost”, and concluded that the limit approaches the golden ratio .[25][26]

X-MATHS Rabbits´ Population Growth and the

Fibonacci Sequence

# II

Recent history studies have shown that the most probable inspiration for this extraordinary sequence was not exactly rabbits but something else… Cold you tell what??…

A: 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, which is Fn−1, plus the number of male ancestors, which is Fn−2.[65] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.