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Drone Parent Numbers, Fibonacci Sequence (Golden Mean)

An article written by... Denis J. Horgan C.F.L Nat Cert Sc. (Apic), of Co. Dublin BKA. Originally entitled... BEES AND MATHEMATICS, which was published in the February 2004 issue of AN BEACHAIRE (THE IRISH BEEKEEPER). [The layout, graphics and the following comments are down to me.] The Fibonacci sequence or series of numbers is related to many features in biology and other branches of science, in this case it describes the number of parents of male bees (drones) that result from the haplo-diploid sex determining mechanism of fertilisation.

Leonard di Pisa, alias Fibonacci (son of Bonaccio) 1170 - 1250 was an Italian mathematician. Leonardo's father was nicknamed Bonacci - the good natured one, and so Leonardo became known as Fibonacci - son of the good natured one. He was famed for introducing the decimal number system into Europe. He has given his name to the sequence of numbers 1, 1, 2, 3, 5, 8, 13 21 34... 144, 233...

He invented the sequence when investigating a problem about the growth of a population of rabbits. While the model was not particularly realistic, it was the first of its kind. It is referred to as the "Fibonacci sequence" and will be familiar to anyone with a background in maths. If you study the sequence you will see that each number is the sum of the two previous numbers. It is a recursive sequence where the first two values are 1 and each successive term is obtained by adding together the two previous terms. Thus, the sequence begins 1, 1, 2, 3, 5, 8, .. that is 1+1=2, then 1+2=3, then 2+3=5 etc.

However at this stage I am sure you are asking yourself what this article has to do with bees and why it should be in 'An Beachaire' at all. The answer is that this sequence of numbers is found in nature. When Fibonacci was asked why he studied these numbers and their ratio he replied: "Someday these numbers will unlock the secret of nature and will explain why a drone does not have a father". Male bees (drones) are produced from a queen's unfertilised egg (parthenogenesis) so that a male bee has only one parent - a mother and no father. The female worker bees have two parents a male (drone) and a female (queen). The Fibonacci sequence is a great representation of this reproductive pattern. The ancestry of both drone and worker is shown below.

Number ofDronesWorker Bees
Parents12
Grandparents23
Great grandparents35
Great-great grandparents58
Great-great-great grandparents813


You can see from the above table that if you start by imagining one male or worker bee you can calculate how many parents, how many grandparents and great grandparents he or she would have etc. You will see that the number of bees of each generation follow a Fibonacci series exactly, both for males and females.

The ancestry can also be shown by an ancestry tree as follows:-

Ancestry Tree showing numbers per generation

If you take a calculator, using the first example mentioned at the top of the page, divide 34 by 21 and then divide 233 by 144 you will see that the ratios approach the decimal 1.61803 to five places. Over and over in nature, in both living and non-living realms, this particular number comes up again and again. Why? This number, known as the "Golden Ratio" (Golden Mean) is related to a mathematical series and a certain spiral shape that is found in nature in surprising ways. The famous astronomer, Johannes Kepler who said that scientists were "thinking God's thoughts after him" called this special number the Divine Proportion.

The Golden Ratio is sometimes denoted by the Greek letter phi. Greek letter Phi Symbols

Originated... 09 July 2004,
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