What follows is a modified excerpt from Dr. Ron Knott's: The Fibonacci Numbers and Golden section in Nature - 1. I have taken the liberty of expanding this traditional, honeybee, genealogy model into the cubic, golden, number series through the agency of fantasy.

Honeybees, Fibonacci numbers and Family trees

There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee").
First, some unusual facts about honeybees such as: not all of them have two parents!
In a colony of honeybees there is one special female called the queen.
There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
There are some drone bees who are male and do no work.
Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.

So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female.

Here we follow the convention of Family Trees that parents appear above their children, so the latest generations are at the bottom and the higher up we go, the older people are. Such trees show all the ancestors (predecessors, forebears, antecedents) of the person at the bottom of the diagram. We would get quite a different tree if we listed all the descendants (progeny, offspring) of a person as we did in the rabbit problem, where we showed all the descendants of the original pair.

Let's look at the family tree of a normal queen bee.

1. She is 1 individual.
2. She has 2 parents, a male and a female.
3. She has 3 grand-parents, since her father had one parent, a female, and her mother had two parents, a male and a female.
4. She has 5 great-grand-parents: both of her grand-mothers had two parents each, while her grand-father had only one.
5. How many great-great-grand parents do you think she has?

Again we see the Fibonacci numbers:

                                       great-     great,great   gt,gt,gt
                           grand-      grand-     grand         grand
Number of       parents:   parents:    parents:   parents:      parents:
of a MALE bee:    1           2           3          5             8
of a FEMALE bee:  2           3           5          8            13
  

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Here is my extension to the above excerpt:

Continuing this Construction into
A Fantasy of a Three-Gendered Hive of Honeybees ....

First, some unusual facts about these strange, new breed of honeybees, such as: not all of them have two parents!
In a colony of three-gendered honeybees there is one special female called the queen. She arises from an egg which has been fertilized by no less than three parents: her queen mother, her type B male-father, and her type A male-drone-father.
There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
There are some drone, type A bees who are male and do no work.
Then there are some type B bees who are also male and do no work. But these are not drones.
Male drones who are type A are produced by the queen's unfertilized eggs, so male type A bees only have a mother, but no father!
Males who are type B are produced from the queen's eggs which are fertilized by a type B male, so male type B bees have a mother and a type B father!
All the females are produced when the queen has mated with two males, one of type A and one of type B, and so have three parents. Females usually end up as worker bees but some are fed with a special substance called fantasy royal jelly which makes them grow into fantasy queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.

So, female bees have 3 parents: two different males plus a female. Male type A bees have just one parent, a female. And male type B bees have two parents, a female and a male type B.

 
Let's look at the family tree of this fantasy, queen bee.

1. She is 1 individual.
2. She has 3 parents, an A male, a B male, and a female.
3. She has 6 grand-parents, since her A father had one parent: a female, her B father had two parents: a female and a B male, while her mother had three parents: an A male, a B male, and a female.
4. She has 14 great-grand-parents: well, you get the picture --- maybe.
5. How many great-great-grand parents do you think she has?

Again we see the golden numbers for the cubic polynomial:

                                       great-     great,great   gt,gt,gt
                           grand-      grand-     grand         grand
Number of       parents:   parents:    parents:   parents:      parents:
of 'A' MALE bee:  1           3           6         14            31
of 'B' MALE bee:  2           5          11         25            56
of a FEMALE bee:  3           6          14         31            70
  

Next Lesson: A modified excerpt from one of Dr. Ron Knott's web pages illustrating how to convert Fibonacci's, rabbit, breeding model into a golden, cubic, series of numbers.

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