## One Natural Example of Fibonacci Numbers

“Natural Fibonacci” presents several examples of Fibonacci numbers found in nature.

Bee genealogy is one example. The hive-dwelling honeybee has many female worker bees, one fertilized queen who is also female, and a few male drones. A male develops from an unfertilized egg, so he has only one parent. Every female bee develops from a fertilized egg, so she has two parents: the queen and one male drone.

Let’s name one female bee, “Bea”, and count “1” for her own generation. Her parents’ generation consists of a male and a female: the number in that generation is “2”.

Bea has three grandparents. Bea’s father had only a mother; but Bea’s mother had both a mom and pop. So the grandparent’s generation has 3 members.

In Bea’s great-grandparents’ generation, her father’s mother had two parents. Her maternal grandmother also had two parents; but her maternal grandfather had only a mother. That adds up to 5 great-grandparents.

Refer to the image below to see how this works over several generations.

The image shows 6 generations. “Bea”, in generation 1, has only herself in the “Fibonacci Count”. Generation #2 are her two parents: one female (‘F’) and one male (‘M’, at the far right). Some lines join Bea’s parents to her.

Bea’s father’s only parent is ‘*f*‘ in the right column of the “grandparent” row. There is no line to show this relationship. But this grandmother has two parents in the Great-grand row, shown with the heavy orange background and the parenting lines.

You can see that the number of ancestors in each generation is a Fibonacci number.

## Fibonacci Numbers and the Golden Ratio

Architects and artists have found that an object with a specific proportion is pleasing to the eye. If A and B are two lengths and A > B, then if the ratios A/B equals (A+B)/A, the figure is in the “golden ratio“. Surprisingly, this value is exactly “one-half of the sum of 1 plus the square root of 5”.

Even more surprisingly, there is a relationship to the Fibonacci sequence. Divide x/x[n]. In the sixth generation, when x=13, the result is less than 1% off from the Golden Ratio. For x[10] = 89, the error drops under 1% of 1%. This was an incredibly “easy” way to calculate the Golden Ratio before mathematicians invented general methods for determining square roots.

## More Natural Fibonacci Numbers

Fibonacci numbers are found in the growth patterns of many plants and animals, particularly where something grows in a spiral. Flower petals often come in Fibonacci numbers. The example below has “only” 5 petals.

Measuring the length of each coil in a garden hose may show an “extended” Fibonacci series. Each loop has to travel farther because of the width of the previous coil.

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