## Fibonacci in SunflowersBy Kaia Posey

Hi, I am Kaia Posey. In Legend EDGE Math this unit, we are working on Arithmetic Sequences, Geometric Sequences, Fibonacci Sequences, and Pascal's Triangle. I chose to focus on Nature. Well, Sunflowers. Sunflowers are such a beautiful flower; I loved researching how all the things we are learning fit into Nature and Sunflowers. I chose to dive deeper into Fibonacci Sequences and the golden ratio that we see mostly in sunflowers.

### What is Fibonacci?

Fi·bo·nac·ci se·ries noun: Fibonacci sequence: a series of numbers in which each number (Fibonacci number) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, etc.

### Examples of the Fibonacci Sequence

Each organism splits into two after an interval of maturation time characteristic of the species. This interval varies randomly but within a certain range according to external conditions, like temperature, availability of nutrients and so on. We can imagine a simplified model where, under perfect conditions, all amoebae split after the same time period of growth. So, one amoebas becomes two, two become 4, then 8, 16, 32, and so on.
Now in the Fibonacci rabbit situation, there is a lag factor; each pair requires some time to mature. So we are assuming maturation time = 1 month; gestation time = 1 month. If you were to try this in your backyard, here's what would happen.

### What is the golden ratio?

If you type into Google, "What is the golden ratio?" This will pop up in the calculator: 1.61803398875.

"The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet."

But that is confusing. In simpler terms, it is the golden ratio is a special number approximately equal to 1.618.

### How does the Golden Ratio relate to the Fibonacci Sequence?

There is a special relationship between the Golden Ratio and Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc, each number is the sum of the two numbers before it).

So, just like we naturally get seven arms when we use 0.142857 (1/7), we tend to get Fibonacci Numbers when we use the Golden Ratio.

### Why do we like to study sunflowers in Fibonacci?

Because Sunflowers show complex Fibonacci patterns and sequences, mathematicians and biologists alike want to find out more about Fibonacci and how it works in Sunflowers.

The giant flowers are one of the most obvious—as well as the prettiest—demonstrations of a hidden mathematical rule shaping the patterns and that is why all of these really smart people, like scientists and mathematicians, might want to figure it out.

### Conclusions and Generalizations

Sunflowers are a beautiful thing in nature. And so is math. We have to show people that we need sunflowers and math in the daily world or else you never know; we could never know what they were truly for.

### Cites Used

https://www.mathsisfun.com/numbers/golden-ratio.html

http://www.livescience.com/37704-phi-golden-ratio.html

http://www.sciencemag.org/news/2016/05/sunflowers-show-complex-fibonacci-sequences

Credits:

Created with images by findjeju - "sunflower let reonpam outing" • Pexels - "bloom blossom fedora" • cocoparisienne - "sunflower blossom bloom" • Pexels - "beautiful blond blonde" • Pexels - "agriculture beautiful clouds" • skeeze - "sunflowers landscape japan" • Alexas_Fotos - "sunflower bees summer" • machaq - "sunflower"