# Introducing the Fibonacci Sequence

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## 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.

“Bee Genealogy” by Mike DeHaan

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 = 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.

“Floral Fibonacci Number” by Mike DeHaan

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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